Go prebuilts for linux-x86, version 1.4.1
See README.ANDROID for instructions on rebuilding the prebuilts.
Change-Id: I86db7f4fb3269585078da4d38947ca9126ac813b
diff --git a/src/math/abs.go b/src/math/abs.go
new file mode 100644
index 0000000..bc41a6d
--- /dev/null
+++ b/src/math/abs.go
@@ -0,0 +1,22 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Abs returns the absolute value of x.
+//
+// Special cases are:
+// Abs(±Inf) = +Inf
+// Abs(NaN) = NaN
+func Abs(x float64) float64
+
+func abs(x float64) float64 {
+ switch {
+ case x < 0:
+ return -x
+ case x == 0:
+ return 0 // return correctly abs(-0)
+ }
+ return x
+}
diff --git a/src/math/abs_386.s b/src/math/abs_386.s
new file mode 100644
index 0000000..f30a439
--- /dev/null
+++ b/src/math/abs_386.s
@@ -0,0 +1,12 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Abs(x float64) float64
+TEXT ·Abs(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FABS // F0=|x|
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/abs_amd64.s b/src/math/abs_amd64.s
new file mode 100644
index 0000000..0424eb5
--- /dev/null
+++ b/src/math/abs_amd64.s
@@ -0,0 +1,14 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Abs(x float64) float64
+TEXT ·Abs(SB),NOSPLIT,$0
+ MOVQ $(1<<63), BX
+ MOVQ BX, X0 // movsd $(-0.0), x0
+ MOVSD x+0(FP), X1
+ ANDNPD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
diff --git a/src/math/abs_amd64p32.s b/src/math/abs_amd64p32.s
new file mode 100644
index 0000000..08c8c6b
--- /dev/null
+++ b/src/math/abs_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "abs_amd64.s"
diff --git a/src/math/abs_arm.s b/src/math/abs_arm.s
new file mode 100644
index 0000000..bfa77eb
--- /dev/null
+++ b/src/math/abs_arm.s
@@ -0,0 +1,13 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Abs(SB),NOSPLIT,$0
+ MOVW x_lo+0(FP), R0
+ MOVW x_hi+4(FP), R1
+ AND $((1<<31)-1), R1
+ MOVW R0, ret_lo+8(FP)
+ MOVW R1, ret_hi+12(FP)
+ RET
diff --git a/src/math/acosh.go b/src/math/acosh.go
new file mode 100644
index 0000000..e394008
--- /dev/null
+++ b/src/math/acosh.go
@@ -0,0 +1,60 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_acosh.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// __ieee754_acosh(x)
+// Method :
+// Based on
+// acosh(x) = log [ x + sqrt(x*x-1) ]
+// we have
+// acosh(x) := log(x)+ln2, if x is large; else
+// acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
+// acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
+//
+// Special cases:
+// acosh(x) is NaN with signal if x<1.
+// acosh(NaN) is NaN without signal.
+//
+
+// Acosh returns the inverse hyperbolic cosine of x.
+//
+// Special cases are:
+// Acosh(+Inf) = +Inf
+// Acosh(x) = NaN if x < 1
+// Acosh(NaN) = NaN
+func Acosh(x float64) float64 {
+ const (
+ Ln2 = 6.93147180559945286227e-01 // 0x3FE62E42FEFA39EF
+ Large = 1 << 28 // 2**28
+ )
+ // first case is special case
+ switch {
+ case x < 1 || IsNaN(x):
+ return NaN()
+ case x == 1:
+ return 0
+ case x >= Large:
+ return Log(x) + Ln2 // x > 2**28
+ case x > 2:
+ return Log(2*x - 1/(x+Sqrt(x*x-1))) // 2**28 > x > 2
+ }
+ t := x - 1
+ return Log1p(t + Sqrt(2*t+t*t)) // 2 >= x > 1
+}
diff --git a/src/math/all_test.go b/src/math/all_test.go
new file mode 100644
index 0000000..763efb2
--- /dev/null
+++ b/src/math/all_test.go
@@ -0,0 +1,2992 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math_test
+
+import (
+ "fmt"
+ . "math"
+ "testing"
+)
+
+var vf = []float64{
+ 4.9790119248836735e+00,
+ 7.7388724745781045e+00,
+ -2.7688005719200159e-01,
+ -5.0106036182710749e+00,
+ 9.6362937071984173e+00,
+ 2.9263772392439646e+00,
+ 5.2290834314593066e+00,
+ 2.7279399104360102e+00,
+ 1.8253080916808550e+00,
+ -8.6859247685756013e+00,
+}
+
+// The expected results below were computed by the high precision calculators
+// at http://keisan.casio.com/. More exact input values (array vf[], above)
+// were obtained by printing them with "%.26f". The answers were calculated
+// to 26 digits (by using the "Digit number" drop-down control of each
+// calculator).
+var acos = []float64{
+ 1.0496193546107222142571536e+00,
+ 6.8584012813664425171660692e-01,
+ 1.5984878714577160325521819e+00,
+ 2.0956199361475859327461799e+00,
+ 2.7053008467824138592616927e-01,
+ 1.2738121680361776018155625e+00,
+ 1.0205369421140629186287407e+00,
+ 1.2945003481781246062157835e+00,
+ 1.3872364345374451433846657e+00,
+ 2.6231510803970463967294145e+00,
+}
+var acosh = []float64{
+ 2.4743347004159012494457618e+00,
+ 2.8576385344292769649802701e+00,
+ 7.2796961502981066190593175e-01,
+ 2.4796794418831451156471977e+00,
+ 3.0552020742306061857212962e+00,
+ 2.044238592688586588942468e+00,
+ 2.5158701513104513595766636e+00,
+ 1.99050839282411638174299e+00,
+ 1.6988625798424034227205445e+00,
+ 2.9611454842470387925531875e+00,
+}
+var asin = []float64{
+ 5.2117697218417440497416805e-01,
+ 8.8495619865825236751471477e-01,
+ -02.769154466281941332086016e-02,
+ -5.2482360935268931351485822e-01,
+ 1.3002662421166552333051524e+00,
+ 2.9698415875871901741575922e-01,
+ 5.5025938468083370060258102e-01,
+ 2.7629597861677201301553823e-01,
+ 1.83559892257451475846656e-01,
+ -1.0523547536021497774980928e+00,
+}
+var asinh = []float64{
+ 2.3083139124923523427628243e+00,
+ 2.743551594301593620039021e+00,
+ -2.7345908534880091229413487e-01,
+ -2.3145157644718338650499085e+00,
+ 2.9613652154015058521951083e+00,
+ 1.7949041616585821933067568e+00,
+ 2.3564032905983506405561554e+00,
+ 1.7287118790768438878045346e+00,
+ 1.3626658083714826013073193e+00,
+ -2.8581483626513914445234004e+00,
+}
+var atan = []float64{
+ 1.372590262129621651920085e+00,
+ 1.442290609645298083020664e+00,
+ -2.7011324359471758245192595e-01,
+ -1.3738077684543379452781531e+00,
+ 1.4673921193587666049154681e+00,
+ 1.2415173565870168649117764e+00,
+ 1.3818396865615168979966498e+00,
+ 1.2194305844639670701091426e+00,
+ 1.0696031952318783760193244e+00,
+ -1.4561721938838084990898679e+00,
+}
+var atanh = []float64{
+ 5.4651163712251938116878204e-01,
+ 1.0299474112843111224914709e+00,
+ -2.7695084420740135145234906e-02,
+ -5.5072096119207195480202529e-01,
+ 1.9943940993171843235906642e+00,
+ 3.01448604578089708203017e-01,
+ 5.8033427206942188834370595e-01,
+ 2.7987997499441511013958297e-01,
+ 1.8459947964298794318714228e-01,
+ -1.3273186910532645867272502e+00,
+}
+var atan2 = []float64{
+ 1.1088291730037004444527075e+00,
+ 9.1218183188715804018797795e-01,
+ 1.5984772603216203736068915e+00,
+ 2.0352918654092086637227327e+00,
+ 8.0391819139044720267356014e-01,
+ 1.2861075249894661588866752e+00,
+ 1.0889904479131695712182587e+00,
+ 1.3044821793397925293797357e+00,
+ 1.3902530903455392306872261e+00,
+ 2.2859857424479142655411058e+00,
+}
+var cbrt = []float64{
+ 1.7075799841925094446722675e+00,
+ 1.9779982212970353936691498e+00,
+ -6.5177429017779910853339447e-01,
+ -1.7111838886544019873338113e+00,
+ 2.1279920909827937423960472e+00,
+ 1.4303536770460741452312367e+00,
+ 1.7357021059106154902341052e+00,
+ 1.3972633462554328350552916e+00,
+ 1.2221149580905388454977636e+00,
+ -2.0556003730500069110343596e+00,
+}
+var ceil = []float64{
+ 5.0000000000000000e+00,
+ 8.0000000000000000e+00,
+ 0.0000000000000000e+00,
+ -5.0000000000000000e+00,
+ 1.0000000000000000e+01,
+ 3.0000000000000000e+00,
+ 6.0000000000000000e+00,
+ 3.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ -8.0000000000000000e+00,
+}
+var copysign = []float64{
+ -4.9790119248836735e+00,
+ -7.7388724745781045e+00,
+ -2.7688005719200159e-01,
+ -5.0106036182710749e+00,
+ -9.6362937071984173e+00,
+ -2.9263772392439646e+00,
+ -5.2290834314593066e+00,
+ -2.7279399104360102e+00,
+ -1.8253080916808550e+00,
+ -8.6859247685756013e+00,
+}
+var cos = []float64{
+ 2.634752140995199110787593e-01,
+ 1.148551260848219865642039e-01,
+ 9.6191297325640768154550453e-01,
+ 2.938141150061714816890637e-01,
+ -9.777138189897924126294461e-01,
+ -9.7693041344303219127199518e-01,
+ 4.940088096948647263961162e-01,
+ -9.1565869021018925545016502e-01,
+ -2.517729313893103197176091e-01,
+ -7.39241351595676573201918e-01,
+}
+
+// Results for 100000 * Pi + vf[i]
+var cosLarge = []float64{
+ 2.634752141185559426744e-01,
+ 1.14855126055543100712e-01,
+ 9.61912973266488928113e-01,
+ 2.9381411499556122552e-01,
+ -9.777138189880161924641e-01,
+ -9.76930413445147608049e-01,
+ 4.940088097314976789841e-01,
+ -9.15658690217517835002e-01,
+ -2.51772931436786954751e-01,
+ -7.3924135157173099849e-01,
+}
+var cosh = []float64{
+ 7.2668796942212842775517446e+01,
+ 1.1479413465659254502011135e+03,
+ 1.0385767908766418550935495e+00,
+ 7.5000957789658051428857788e+01,
+ 7.655246669605357888468613e+03,
+ 9.3567491758321272072888257e+00,
+ 9.331351599270605471131735e+01,
+ 7.6833430994624643209296404e+00,
+ 3.1829371625150718153881164e+00,
+ 2.9595059261916188501640911e+03,
+}
+var erf = []float64{
+ 5.1865354817738701906913566e-01,
+ 7.2623875834137295116929844e-01,
+ -3.123458688281309990629839e-02,
+ -5.2143121110253302920437013e-01,
+ 8.2704742671312902508629582e-01,
+ 3.2101767558376376743993945e-01,
+ 5.403990312223245516066252e-01,
+ 3.0034702916738588551174831e-01,
+ 2.0369924417882241241559589e-01,
+ -7.8069386968009226729944677e-01,
+}
+var erfc = []float64{
+ 4.8134645182261298093086434e-01,
+ 2.7376124165862704883070156e-01,
+ 1.0312345868828130999062984e+00,
+ 1.5214312111025330292043701e+00,
+ 1.7295257328687097491370418e-01,
+ 6.7898232441623623256006055e-01,
+ 4.596009687776754483933748e-01,
+ 6.9965297083261411448825169e-01,
+ 7.9630075582117758758440411e-01,
+ 1.7806938696800922672994468e+00,
+}
+var exp = []float64{
+ 1.4533071302642137507696589e+02,
+ 2.2958822575694449002537581e+03,
+ 7.5814542574851666582042306e-01,
+ 6.6668778421791005061482264e-03,
+ 1.5310493273896033740861206e+04,
+ 1.8659907517999328638667732e+01,
+ 1.8662167355098714543942057e+02,
+ 1.5301332413189378961665788e+01,
+ 6.2047063430646876349125085e+00,
+ 1.6894712385826521111610438e-04,
+}
+var expm1 = []float64{
+ 5.105047796122957327384770212e-02,
+ 8.046199708567344080562675439e-02,
+ -2.764970978891639815187418703e-03,
+ -4.8871434888875355394330300273e-02,
+ 1.0115864277221467777117227494e-01,
+ 2.969616407795910726014621657e-02,
+ 5.368214487944892300914037972e-02,
+ 2.765488851131274068067445335e-02,
+ 1.842068661871398836913874273e-02,
+ -8.3193870863553801814961137573e-02,
+}
+var exp2 = []float64{
+ 3.1537839463286288034313104e+01,
+ 2.1361549283756232296144849e+02,
+ 8.2537402562185562902577219e-01,
+ 3.1021158628740294833424229e-02,
+ 7.9581744110252191462569661e+02,
+ 7.6019905892596359262696423e+00,
+ 3.7506882048388096973183084e+01,
+ 6.6250893439173561733216375e+00,
+ 3.5438267900243941544605339e+00,
+ 2.4281533133513300984289196e-03,
+}
+var fabs = []float64{
+ 4.9790119248836735e+00,
+ 7.7388724745781045e+00,
+ 2.7688005719200159e-01,
+ 5.0106036182710749e+00,
+ 9.6362937071984173e+00,
+ 2.9263772392439646e+00,
+ 5.2290834314593066e+00,
+ 2.7279399104360102e+00,
+ 1.8253080916808550e+00,
+ 8.6859247685756013e+00,
+}
+var fdim = []float64{
+ 4.9790119248836735e+00,
+ 7.7388724745781045e+00,
+ 0.0000000000000000e+00,
+ 0.0000000000000000e+00,
+ 9.6362937071984173e+00,
+ 2.9263772392439646e+00,
+ 5.2290834314593066e+00,
+ 2.7279399104360102e+00,
+ 1.8253080916808550e+00,
+ 0.0000000000000000e+00,
+}
+var floor = []float64{
+ 4.0000000000000000e+00,
+ 7.0000000000000000e+00,
+ -1.0000000000000000e+00,
+ -6.0000000000000000e+00,
+ 9.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 5.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ -9.0000000000000000e+00,
+}
+var fmod = []float64{
+ 4.197615023265299782906368e-02,
+ 2.261127525421895434476482e+00,
+ 3.231794108794261433104108e-02,
+ 4.989396381728925078391512e+00,
+ 3.637062928015826201999516e-01,
+ 1.220868282268106064236690e+00,
+ 4.770916568540693347699744e+00,
+ 1.816180268691969246219742e+00,
+ 8.734595415957246977711748e-01,
+ 1.314075231424398637614104e+00,
+}
+
+type fi struct {
+ f float64
+ i int
+}
+
+var frexp = []fi{
+ {6.2237649061045918750e-01, 3},
+ {9.6735905932226306250e-01, 3},
+ {-5.5376011438400318000e-01, -1},
+ {-6.2632545228388436250e-01, 3},
+ {6.02268356699901081250e-01, 4},
+ {7.3159430981099115000e-01, 2},
+ {6.5363542893241332500e-01, 3},
+ {6.8198497760900255000e-01, 2},
+ {9.1265404584042750000e-01, 1},
+ {-5.4287029803597508250e-01, 4},
+}
+var gamma = []float64{
+ 2.3254348370739963835386613898e+01,
+ 2.991153837155317076427529816e+03,
+ -4.561154336726758060575129109e+00,
+ 7.719403468842639065959210984e-01,
+ 1.6111876618855418534325755566e+05,
+ 1.8706575145216421164173224946e+00,
+ 3.4082787447257502836734201635e+01,
+ 1.579733951448952054898583387e+00,
+ 9.3834586598354592860187267089e-01,
+ -2.093995902923148389186189429e-05,
+}
+var j0 = []float64{
+ -1.8444682230601672018219338e-01,
+ 2.27353668906331975435892e-01,
+ 9.809259936157051116270273e-01,
+ -1.741170131426226587841181e-01,
+ -2.1389448451144143352039069e-01,
+ -2.340905848928038763337414e-01,
+ -1.0029099691890912094586326e-01,
+ -1.5466726714884328135358907e-01,
+ 3.252650187653420388714693e-01,
+ -8.72218484409407250005360235e-03,
+}
+var j1 = []float64{
+ -3.251526395295203422162967e-01,
+ 1.893581711430515718062564e-01,
+ -1.3711761352467242914491514e-01,
+ 3.287486536269617297529617e-01,
+ 1.3133899188830978473849215e-01,
+ 3.660243417832986825301766e-01,
+ -3.4436769271848174665420672e-01,
+ 4.329481396640773768835036e-01,
+ 5.8181350531954794639333955e-01,
+ -2.7030574577733036112996607e-01,
+}
+var j2 = []float64{
+ 5.3837518920137802565192769e-02,
+ -1.7841678003393207281244667e-01,
+ 9.521746934916464142495821e-03,
+ 4.28958355470987397983072e-02,
+ 2.4115371837854494725492872e-01,
+ 4.842458532394520316844449e-01,
+ -3.142145220618633390125946e-02,
+ 4.720849184745124761189957e-01,
+ 3.122312022520957042957497e-01,
+ 7.096213118930231185707277e-02,
+}
+var jM3 = []float64{
+ -3.684042080996403091021151e-01,
+ 2.8157665936340887268092661e-01,
+ 4.401005480841948348343589e-04,
+ 3.629926999056814081597135e-01,
+ 3.123672198825455192489266e-02,
+ -2.958805510589623607540455e-01,
+ -3.2033177696533233403289416e-01,
+ -2.592737332129663376736604e-01,
+ -1.0241334641061485092351251e-01,
+ -2.3762660886100206491674503e-01,
+}
+var lgamma = []fi{
+ {3.146492141244545774319734e+00, 1},
+ {8.003414490659126375852113e+00, 1},
+ {1.517575735509779707488106e+00, -1},
+ {-2.588480028182145853558748e-01, 1},
+ {1.1989897050205555002007985e+01, 1},
+ {6.262899811091257519386906e-01, 1},
+ {3.5287924899091566764846037e+00, 1},
+ {4.5725644770161182299423372e-01, 1},
+ {-6.363667087767961257654854e-02, 1},
+ {-1.077385130910300066425564e+01, -1},
+}
+var log = []float64{
+ 1.605231462693062999102599e+00,
+ 2.0462560018708770653153909e+00,
+ -1.2841708730962657801275038e+00,
+ 1.6115563905281545116286206e+00,
+ 2.2655365644872016636317461e+00,
+ 1.0737652208918379856272735e+00,
+ 1.6542360106073546632707956e+00,
+ 1.0035467127723465801264487e+00,
+ 6.0174879014578057187016475e-01,
+ 2.161703872847352815363655e+00,
+}
+var logb = []float64{
+ 2.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ -2.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 3.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ 0.0000000000000000e+00,
+ 3.0000000000000000e+00,
+}
+var log10 = []float64{
+ 6.9714316642508290997617083e-01,
+ 8.886776901739320576279124e-01,
+ -5.5770832400658929815908236e-01,
+ 6.998900476822994346229723e-01,
+ 9.8391002850684232013281033e-01,
+ 4.6633031029295153334285302e-01,
+ 7.1842557117242328821552533e-01,
+ 4.3583479968917773161304553e-01,
+ 2.6133617905227038228626834e-01,
+ 9.3881606348649405716214241e-01,
+}
+var log1p = []float64{
+ 4.8590257759797794104158205e-02,
+ 7.4540265965225865330849141e-02,
+ -2.7726407903942672823234024e-03,
+ -5.1404917651627649094953380e-02,
+ 9.1998280672258624681335010e-02,
+ 2.8843762576593352865894824e-02,
+ 5.0969534581863707268992645e-02,
+ 2.6913947602193238458458594e-02,
+ 1.8088493239630770262045333e-02,
+ -9.0865245631588989681559268e-02,
+}
+var log2 = []float64{
+ 2.3158594707062190618898251e+00,
+ 2.9521233862883917703341018e+00,
+ -1.8526669502700329984917062e+00,
+ 2.3249844127278861543568029e+00,
+ 3.268478366538305087466309e+00,
+ 1.5491157592596970278166492e+00,
+ 2.3865580889631732407886495e+00,
+ 1.447811865817085365540347e+00,
+ 8.6813999540425116282815557e-01,
+ 3.118679457227342224364709e+00,
+}
+var modf = [][2]float64{
+ {4.0000000000000000e+00, 9.7901192488367350108546816e-01},
+ {7.0000000000000000e+00, 7.3887247457810456552351752e-01},
+ {0.0000000000000000e+00, -2.7688005719200159404635997e-01},
+ {-5.0000000000000000e+00, -1.060361827107492160848778e-02},
+ {9.0000000000000000e+00, 6.3629370719841737980004837e-01},
+ {2.0000000000000000e+00, 9.2637723924396464525443662e-01},
+ {5.0000000000000000e+00, 2.2908343145930665230025625e-01},
+ {2.0000000000000000e+00, 7.2793991043601025126008608e-01},
+ {1.0000000000000000e+00, 8.2530809168085506044576505e-01},
+ {-8.0000000000000000e+00, -6.8592476857560136238589621e-01},
+}
+var nextafter32 = []float32{
+ 4.979012489318848e+00,
+ 7.738873004913330e+00,
+ -2.768800258636475e-01,
+ -5.010602951049805e+00,
+ 9.636294364929199e+00,
+ 2.926377534866333e+00,
+ 5.229084014892578e+00,
+ 2.727940082550049e+00,
+ 1.825308203697205e+00,
+ -8.685923576354980e+00,
+}
+var nextafter64 = []float64{
+ 4.97901192488367438926388786e+00,
+ 7.73887247457810545370193722e+00,
+ -2.7688005719200153853520874e-01,
+ -5.01060361827107403343006808e+00,
+ 9.63629370719841915615688777e+00,
+ 2.92637723924396508934364647e+00,
+ 5.22908343145930754047867595e+00,
+ 2.72793991043601069534929593e+00,
+ 1.82530809168085528249036997e+00,
+ -8.68592476857559958602905681e+00,
+}
+var pow = []float64{
+ 9.5282232631648411840742957e+04,
+ 5.4811599352999901232411871e+07,
+ 5.2859121715894396531132279e-01,
+ 9.7587991957286474464259698e-06,
+ 4.328064329346044846740467e+09,
+ 8.4406761805034547437659092e+02,
+ 1.6946633276191194947742146e+05,
+ 5.3449040147551939075312879e+02,
+ 6.688182138451414936380374e+01,
+ 2.0609869004248742886827439e-09,
+}
+var remainder = []float64{
+ 4.197615023265299782906368e-02,
+ 2.261127525421895434476482e+00,
+ 3.231794108794261433104108e-02,
+ -2.120723654214984321697556e-02,
+ 3.637062928015826201999516e-01,
+ 1.220868282268106064236690e+00,
+ -4.581668629186133046005125e-01,
+ -9.117596417440410050403443e-01,
+ 8.734595415957246977711748e-01,
+ 1.314075231424398637614104e+00,
+}
+var signbit = []bool{
+ false,
+ false,
+ true,
+ true,
+ false,
+ false,
+ false,
+ false,
+ false,
+ true,
+}
+var sin = []float64{
+ -9.6466616586009283766724726e-01,
+ 9.9338225271646545763467022e-01,
+ -2.7335587039794393342449301e-01,
+ 9.5586257685042792878173752e-01,
+ -2.099421066779969164496634e-01,
+ 2.135578780799860532750616e-01,
+ -8.694568971167362743327708e-01,
+ 4.019566681155577786649878e-01,
+ 9.6778633541687993721617774e-01,
+ -6.734405869050344734943028e-01,
+}
+
+// Results for 100000 * Pi + vf[i]
+var sinLarge = []float64{
+ -9.646661658548936063912e-01,
+ 9.933822527198506903752e-01,
+ -2.7335587036246899796e-01,
+ 9.55862576853689321268e-01,
+ -2.099421066862688873691e-01,
+ 2.13557878070308981163e-01,
+ -8.694568970959221300497e-01,
+ 4.01956668098863248917e-01,
+ 9.67786335404528727927e-01,
+ -6.7344058693131973066e-01,
+}
+var sinh = []float64{
+ 7.2661916084208532301448439e+01,
+ 1.1479409110035194500526446e+03,
+ -2.8043136512812518927312641e-01,
+ -7.499429091181587232835164e+01,
+ 7.6552466042906758523925934e+03,
+ 9.3031583421672014313789064e+00,
+ 9.330815755828109072810322e+01,
+ 7.6179893137269146407361477e+00,
+ 3.021769180549615819524392e+00,
+ -2.95950575724449499189888e+03,
+}
+var sqrt = []float64{
+ 2.2313699659365484748756904e+00,
+ 2.7818829009464263511285458e+00,
+ 5.2619393496314796848143251e-01,
+ 2.2384377628763938724244104e+00,
+ 3.1042380236055381099288487e+00,
+ 1.7106657298385224403917771e+00,
+ 2.286718922705479046148059e+00,
+ 1.6516476350711159636222979e+00,
+ 1.3510396336454586262419247e+00,
+ 2.9471892997524949215723329e+00,
+}
+var tan = []float64{
+ -3.661316565040227801781974e+00,
+ 8.64900232648597589369854e+00,
+ -2.8417941955033612725238097e-01,
+ 3.253290185974728640827156e+00,
+ 2.147275640380293804770778e-01,
+ -2.18600910711067004921551e-01,
+ -1.760002817872367935518928e+00,
+ -4.389808914752818126249079e-01,
+ -3.843885560201130679995041e+00,
+ 9.10988793377685105753416e-01,
+}
+
+// Results for 100000 * Pi + vf[i]
+var tanLarge = []float64{
+ -3.66131656475596512705e+00,
+ 8.6490023287202547927e+00,
+ -2.841794195104782406e-01,
+ 3.2532901861033120983e+00,
+ 2.14727564046880001365e-01,
+ -2.18600910700688062874e-01,
+ -1.760002817699722747043e+00,
+ -4.38980891453536115952e-01,
+ -3.84388555942723509071e+00,
+ 9.1098879344275101051e-01,
+}
+var tanh = []float64{
+ 9.9990531206936338549262119e-01,
+ 9.9999962057085294197613294e-01,
+ -2.7001505097318677233756845e-01,
+ -9.9991110943061718603541401e-01,
+ 9.9999999146798465745022007e-01,
+ 9.9427249436125236705001048e-01,
+ 9.9994257600983138572705076e-01,
+ 9.9149409509772875982054701e-01,
+ 9.4936501296239685514466577e-01,
+ -9.9999994291374030946055701e-01,
+}
+var trunc = []float64{
+ 4.0000000000000000e+00,
+ 7.0000000000000000e+00,
+ -0.0000000000000000e+00,
+ -5.0000000000000000e+00,
+ 9.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 5.0000000000000000e+00,
+ 2.0000000000000000e+00,
+ 1.0000000000000000e+00,
+ -8.0000000000000000e+00,
+}
+var y0 = []float64{
+ -3.053399153780788357534855e-01,
+ 1.7437227649515231515503649e-01,
+ -8.6221781263678836910392572e-01,
+ -3.100664880987498407872839e-01,
+ 1.422200649300982280645377e-01,
+ 4.000004067997901144239363e-01,
+ -3.3340749753099352392332536e-01,
+ 4.5399790746668954555205502e-01,
+ 4.8290004112497761007536522e-01,
+ 2.7036697826604756229601611e-01,
+}
+var y1 = []float64{
+ 0.15494213737457922210218611,
+ -0.2165955142081145245075746,
+ -2.4644949631241895201032829,
+ 0.1442740489541836405154505,
+ 0.2215379960518984777080163,
+ 0.3038800915160754150565448,
+ 0.0691107642452362383808547,
+ 0.2380116417809914424860165,
+ -0.20849492979459761009678934,
+ 0.0242503179793232308250804,
+}
+var y2 = []float64{
+ 0.3675780219390303613394936,
+ -0.23034826393250119879267257,
+ -16.939677983817727205631397,
+ 0.367653980523052152867791,
+ -0.0962401471767804440353136,
+ -0.1923169356184851105200523,
+ 0.35984072054267882391843766,
+ -0.2794987252299739821654982,
+ -0.7113490692587462579757954,
+ -0.2647831587821263302087457,
+}
+var yM3 = []float64{
+ -0.14035984421094849100895341,
+ -0.097535139617792072703973,
+ 242.25775994555580176377379,
+ -0.1492267014802818619511046,
+ 0.26148702629155918694500469,
+ 0.56675383593895176530394248,
+ -0.206150264009006981070575,
+ 0.64784284687568332737963658,
+ 1.3503631555901938037008443,
+ 0.1461869756579956803341844,
+}
+
+// arguments and expected results for special cases
+var vfacosSC = []float64{
+ -Pi,
+ 1,
+ Pi,
+ NaN(),
+}
+var acosSC = []float64{
+ NaN(),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+var vfacoshSC = []float64{
+ Inf(-1),
+ 0.5,
+ 1,
+ Inf(1),
+ NaN(),
+}
+var acoshSC = []float64{
+ NaN(),
+ NaN(),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfasinSC = []float64{
+ -Pi,
+ Copysign(0, -1),
+ 0,
+ Pi,
+ NaN(),
+}
+var asinSC = []float64{
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+var vfasinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var asinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfatanSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var atanSC = []float64{
+ -Pi / 2,
+ Copysign(0, -1),
+ 0,
+ Pi / 2,
+ NaN(),
+}
+
+var vfatanhSC = []float64{
+ Inf(-1),
+ -Pi,
+ -1,
+ Copysign(0, -1),
+ 0,
+ 1,
+ Pi,
+ Inf(1),
+ NaN(),
+}
+var atanhSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+ NaN(),
+ NaN(),
+}
+var vfatan2SC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), -Pi},
+ {Inf(-1), 0},
+ {Inf(-1), +Pi},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {-Pi, Inf(-1)},
+ {-Pi, 0},
+ {-Pi, Inf(1)},
+ {-Pi, NaN()},
+ {Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), -Pi},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), +Pi},
+ {Copysign(0, -1), Inf(1)},
+ {Copysign(0, -1), NaN()},
+ {0, Inf(-1)},
+ {0, -Pi},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {0, +Pi},
+ {0, Inf(1)},
+ {0, NaN()},
+ {+Pi, Inf(-1)},
+ {+Pi, 0},
+ {+Pi, Inf(1)},
+ {+Pi, NaN()},
+ {Inf(1), Inf(-1)},
+ {Inf(1), -Pi},
+ {Inf(1), 0},
+ {Inf(1), +Pi},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), NaN()},
+}
+var atan2SC = []float64{
+ -3 * Pi / 4, // atan2(-Inf, -Inf)
+ -Pi / 2, // atan2(-Inf, -Pi)
+ -Pi / 2, // atan2(-Inf, +0)
+ -Pi / 2, // atan2(-Inf, +Pi)
+ -Pi / 4, // atan2(-Inf, +Inf)
+ NaN(), // atan2(-Inf, NaN)
+ -Pi, // atan2(-Pi, -Inf)
+ -Pi / 2, // atan2(-Pi, +0)
+ Copysign(0, -1), // atan2(-Pi, Inf)
+ NaN(), // atan2(-Pi, NaN)
+ -Pi, // atan2(-0, -Inf)
+ -Pi, // atan2(-0, -Pi)
+ -Pi, // atan2(-0, -0)
+ Copysign(0, -1), // atan2(-0, +0)
+ Copysign(0, -1), // atan2(-0, +Pi)
+ Copysign(0, -1), // atan2(-0, +Inf)
+ NaN(), // atan2(-0, NaN)
+ Pi, // atan2(+0, -Inf)
+ Pi, // atan2(+0, -Pi)
+ Pi, // atan2(+0, -0)
+ 0, // atan2(+0, +0)
+ 0, // atan2(+0, +Pi)
+ 0, // atan2(+0, +Inf)
+ NaN(), // atan2(+0, NaN)
+ Pi, // atan2(+Pi, -Inf)
+ Pi / 2, // atan2(+Pi, +0)
+ 0, // atan2(+Pi, +Inf)
+ NaN(), // atan2(+Pi, NaN)
+ 3 * Pi / 4, // atan2(+Inf, -Inf)
+ Pi / 2, // atan2(+Inf, -Pi)
+ Pi / 2, // atan2(+Inf, +0)
+ Pi / 2, // atan2(+Inf, +Pi)
+ Pi / 4, // atan2(+Inf, +Inf)
+ NaN(), // atan2(+Inf, NaN)
+ NaN(), // atan2(NaN, NaN)
+}
+
+var vfcbrtSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var cbrtSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfceilSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var ceilSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfcopysignSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var copysignSC = []float64{
+ Inf(-1),
+ Inf(-1),
+ NaN(),
+}
+
+var vfcosSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var cosSC = []float64{
+ NaN(),
+ NaN(),
+ NaN(),
+}
+
+var vfcoshSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var coshSC = []float64{
+ Inf(1),
+ 1,
+ 1,
+ Inf(1),
+ NaN(),
+}
+
+var vferfSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var erfSC = []float64{
+ -1,
+ Copysign(0, -1),
+ 0,
+ 1,
+ NaN(),
+}
+
+var vferfcSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var erfcSC = []float64{
+ 2,
+ 0,
+ NaN(),
+}
+
+var vfexpSC = []float64{
+ Inf(-1),
+ -2000,
+ 2000,
+ Inf(1),
+ NaN(),
+}
+var expSC = []float64{
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ NaN(),
+}
+
+var vfexpm1SC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var expm1SC = []float64{
+ -1,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vffabsSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var fabsSC = []float64{
+ Inf(1),
+ 0,
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vffdimSC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {Inf(1), Inf(-1)},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), Inf(-1)},
+ {NaN(), Copysign(0, -1)},
+ {NaN(), 0},
+ {NaN(), Inf(1)},
+ {NaN(), NaN()},
+}
+var fdimSC = []float64{
+ NaN(),
+ 0,
+ NaN(),
+ 0,
+ 0,
+ 0,
+ 0,
+ Inf(1),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+}
+var fmaxSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ NaN(),
+ NaN(),
+ NaN(),
+ Inf(1),
+ NaN(),
+}
+var fminSC = []float64{
+ Inf(-1),
+ Inf(-1),
+ Inf(-1),
+ Copysign(0, -1),
+ Copysign(0, -1),
+ Copysign(0, -1),
+ 0,
+ Inf(-1),
+ Inf(1),
+ NaN(),
+ Inf(-1),
+ NaN(),
+ NaN(),
+ NaN(),
+ NaN(),
+}
+
+var vffmodSC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), -Pi},
+ {Inf(-1), 0},
+ {Inf(-1), Pi},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {-Pi, Inf(-1)},
+ {-Pi, 0},
+ {-Pi, Inf(1)},
+ {-Pi, NaN()},
+ {Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), Inf(1)},
+ {Copysign(0, -1), NaN()},
+ {0, Inf(-1)},
+ {0, 0},
+ {0, Inf(1)},
+ {0, NaN()},
+ {Pi, Inf(-1)},
+ {Pi, 0},
+ {Pi, Inf(1)},
+ {Pi, NaN()},
+ {Inf(1), Inf(-1)},
+ {Inf(1), -Pi},
+ {Inf(1), 0},
+ {Inf(1), Pi},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), Inf(-1)},
+ {NaN(), -Pi},
+ {NaN(), 0},
+ {NaN(), Pi},
+ {NaN(), Inf(1)},
+ {NaN(), NaN()},
+}
+var fmodSC = []float64{
+ NaN(), // fmod(-Inf, -Inf)
+ NaN(), // fmod(-Inf, -Pi)
+ NaN(), // fmod(-Inf, 0)
+ NaN(), // fmod(-Inf, Pi)
+ NaN(), // fmod(-Inf, +Inf)
+ NaN(), // fmod(-Inf, NaN)
+ -Pi, // fmod(-Pi, -Inf)
+ NaN(), // fmod(-Pi, 0)
+ -Pi, // fmod(-Pi, +Inf)
+ NaN(), // fmod(-Pi, NaN)
+ Copysign(0, -1), // fmod(-0, -Inf)
+ NaN(), // fmod(-0, 0)
+ Copysign(0, -1), // fmod(-0, Inf)
+ NaN(), // fmod(-0, NaN)
+ 0, // fmod(0, -Inf)
+ NaN(), // fmod(0, 0)
+ 0, // fmod(0, +Inf)
+ NaN(), // fmod(0, NaN)
+ Pi, // fmod(Pi, -Inf)
+ NaN(), // fmod(Pi, 0)
+ Pi, // fmod(Pi, +Inf)
+ NaN(), // fmod(Pi, NaN)
+ NaN(), // fmod(+Inf, -Inf)
+ NaN(), // fmod(+Inf, -Pi)
+ NaN(), // fmod(+Inf, 0)
+ NaN(), // fmod(+Inf, Pi)
+ NaN(), // fmod(+Inf, +Inf)
+ NaN(), // fmod(+Inf, NaN)
+ NaN(), // fmod(NaN, -Inf)
+ NaN(), // fmod(NaN, -Pi)
+ NaN(), // fmod(NaN, 0)
+ NaN(), // fmod(NaN, Pi)
+ NaN(), // fmod(NaN, +Inf)
+ NaN(), // fmod(NaN, NaN)
+}
+
+var vffrexpSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var frexpSC = []fi{
+ {Inf(-1), 0},
+ {Copysign(0, -1), 0},
+ {0, 0},
+ {Inf(1), 0},
+ {NaN(), 0},
+}
+
+var vfgammaSC = []float64{
+ Inf(-1),
+ -3,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var gammaSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Inf(1),
+ Inf(1),
+ NaN(),
+}
+
+var vfhypotSC = [][2]float64{
+ {Inf(-1), Inf(-1)},
+ {Inf(-1), 0},
+ {Inf(-1), Inf(1)},
+ {Inf(-1), NaN()},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), 0},
+ {0, Copysign(0, -1)},
+ {0, 0}, // +0, +0
+ {0, Inf(-1)},
+ {0, Inf(1)},
+ {0, NaN()},
+ {Inf(1), Inf(-1)},
+ {Inf(1), 0},
+ {Inf(1), Inf(1)},
+ {Inf(1), NaN()},
+ {NaN(), Inf(-1)},
+ {NaN(), 0},
+ {NaN(), Inf(1)},
+ {NaN(), NaN()},
+}
+var hypotSC = []float64{
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ 0,
+ 0,
+ 0,
+ 0,
+ Inf(1),
+ Inf(1),
+ NaN(),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ Inf(1),
+ NaN(),
+ Inf(1),
+ NaN(),
+}
+
+var vfilogbSC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var ilogbSC = []int{
+ MaxInt32,
+ MinInt32,
+ MaxInt32,
+ MaxInt32,
+}
+
+var vfj0SC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var j0SC = []float64{
+ 0,
+ 1,
+ 0,
+ NaN(),
+}
+var j1SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+var j2SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+var jM3SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+
+var vfldexpSC = []fi{
+ {0, 0},
+ {0, -1075},
+ {0, 1024},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), -1075},
+ {Copysign(0, -1), 1024},
+ {Inf(1), 0},
+ {Inf(1), -1024},
+ {Inf(-1), 0},
+ {Inf(-1), -1024},
+ {NaN(), -1024},
+}
+var ldexpSC = []float64{
+ 0,
+ 0,
+ 0,
+ Copysign(0, -1),
+ Copysign(0, -1),
+ Copysign(0, -1),
+ Inf(1),
+ Inf(1),
+ Inf(-1),
+ Inf(-1),
+ NaN(),
+}
+
+var vflgammaSC = []float64{
+ Inf(-1),
+ -3,
+ 0,
+ 1,
+ 2,
+ Inf(1),
+ NaN(),
+}
+var lgammaSC = []fi{
+ {Inf(-1), 1},
+ {Inf(1), 1},
+ {Inf(1), 1},
+ {0, 1},
+ {0, 1},
+ {Inf(1), 1},
+ {NaN(), 1},
+}
+
+var vflogSC = []float64{
+ Inf(-1),
+ -Pi,
+ Copysign(0, -1),
+ 0,
+ 1,
+ Inf(1),
+ NaN(),
+}
+var logSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vflogbSC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var logbSC = []float64{
+ Inf(1),
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+
+var vflog1pSC = []float64{
+ Inf(-1),
+ -Pi,
+ -1,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var log1pSC = []float64{
+ NaN(),
+ NaN(),
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfmodfSC = []float64{
+ Inf(-1),
+ Inf(1),
+ NaN(),
+}
+var modfSC = [][2]float64{
+ {Inf(-1), NaN()}, // [2]float64{Copysign(0, -1), Inf(-1)},
+ {Inf(1), NaN()}, // [2]float64{0, Inf(1)},
+ {NaN(), NaN()},
+}
+
+var vfnextafter32SC = [][2]float32{
+ {0, 0},
+ {0, float32(Copysign(0, -1))},
+ {0, -1},
+ {0, float32(NaN())},
+ {float32(Copysign(0, -1)), 1},
+ {float32(Copysign(0, -1)), 0},
+ {float32(Copysign(0, -1)), float32(Copysign(0, -1))},
+ {float32(Copysign(0, -1)), -1},
+ {float32(NaN()), 0},
+ {float32(NaN()), float32(NaN())},
+}
+var nextafter32SC = []float32{
+ 0,
+ 0,
+ -1.401298464e-45, // Float32frombits(0x80000001)
+ float32(NaN()),
+ 1.401298464e-45, // Float32frombits(0x00000001)
+ float32(Copysign(0, -1)),
+ float32(Copysign(0, -1)),
+ -1.401298464e-45, // Float32frombits(0x80000001)
+ float32(NaN()),
+ float32(NaN()),
+}
+
+var vfnextafter64SC = [][2]float64{
+ {0, 0},
+ {0, Copysign(0, -1)},
+ {0, -1},
+ {0, NaN()},
+ {Copysign(0, -1), 1},
+ {Copysign(0, -1), 0},
+ {Copysign(0, -1), Copysign(0, -1)},
+ {Copysign(0, -1), -1},
+ {NaN(), 0},
+ {NaN(), NaN()},
+}
+var nextafter64SC = []float64{
+ 0,
+ 0,
+ -4.9406564584124654418e-324, // Float64frombits(0x8000000000000001)
+ NaN(),
+ 4.9406564584124654418e-324, // Float64frombits(0x0000000000000001)
+ Copysign(0, -1),
+ Copysign(0, -1),
+ -4.9406564584124654418e-324, // Float64frombits(0x8000000000000001)
+ NaN(),
+ NaN(),
+}
+
+var vfpowSC = [][2]float64{
+ {Inf(-1), -Pi},
+ {Inf(-1), -3},
+ {Inf(-1), Copysign(0, -1)},
+ {Inf(-1), 0},
+ {Inf(-1), 1},
+ {Inf(-1), 3},
+ {Inf(-1), Pi},
+ {Inf(-1), NaN()},
+
+ {-Pi, Inf(-1)},
+ {-Pi, -Pi},
+ {-Pi, Copysign(0, -1)},
+ {-Pi, 0},
+ {-Pi, 1},
+ {-Pi, Pi},
+ {-Pi, Inf(1)},
+ {-Pi, NaN()},
+
+ {-1, Inf(-1)},
+ {-1, Inf(1)},
+ {-1, NaN()},
+ {-1 / 2, Inf(-1)},
+ {-1 / 2, Inf(1)},
+ {Copysign(0, -1), Inf(-1)},
+ {Copysign(0, -1), -Pi},
+ {Copysign(0, -1), -3},
+ {Copysign(0, -1), 3},
+ {Copysign(0, -1), Pi},
+ {Copysign(0, -1), Inf(1)},
+
+ {0, Inf(-1)},
+ {0, -Pi},
+ {0, -3},
+ {0, Copysign(0, -1)},
+ {0, 0},
+ {0, 3},
+ {0, Pi},
+ {0, Inf(1)},
+ {0, NaN()},
+
+ {1 / 2, Inf(-1)},
+ {1 / 2, Inf(1)},
+ {1, Inf(-1)},
+ {1, Inf(1)},
+ {1, NaN()},
+
+ {Pi, Inf(-1)},
+ {Pi, Copysign(0, -1)},
+ {Pi, 0},
+ {Pi, 1},
+ {Pi, Inf(1)},
+ {Pi, NaN()},
+ {Inf(1), -Pi},
+ {Inf(1), Copysign(0, -1)},
+ {Inf(1), 0},
+ {Inf(1), 1},
+ {Inf(1), Pi},
+ {Inf(1), NaN()},
+ {NaN(), -Pi},
+ {NaN(), Copysign(0, -1)},
+ {NaN(), 0},
+ {NaN(), 1},
+ {NaN(), Pi},
+ {NaN(), NaN()},
+}
+var powSC = []float64{
+ 0, // pow(-Inf, -Pi)
+ Copysign(0, -1), // pow(-Inf, -3)
+ 1, // pow(-Inf, -0)
+ 1, // pow(-Inf, +0)
+ Inf(-1), // pow(-Inf, 1)
+ Inf(-1), // pow(-Inf, 3)
+ Inf(1), // pow(-Inf, Pi)
+ NaN(), // pow(-Inf, NaN)
+ 0, // pow(-Pi, -Inf)
+ NaN(), // pow(-Pi, -Pi)
+ 1, // pow(-Pi, -0)
+ 1, // pow(-Pi, +0)
+ -Pi, // pow(-Pi, 1)
+ NaN(), // pow(-Pi, Pi)
+ Inf(1), // pow(-Pi, +Inf)
+ NaN(), // pow(-Pi, NaN)
+ 1, // pow(-1, -Inf) IEEE 754-2008
+ 1, // pow(-1, +Inf) IEEE 754-2008
+ NaN(), // pow(-1, NaN)
+ Inf(1), // pow(-1/2, -Inf)
+ 0, // pow(-1/2, +Inf)
+ Inf(1), // pow(-0, -Inf)
+ Inf(1), // pow(-0, -Pi)
+ Inf(-1), // pow(-0, -3) IEEE 754-2008
+ Copysign(0, -1), // pow(-0, 3) IEEE 754-2008
+ 0, // pow(-0, +Pi)
+ 0, // pow(-0, +Inf)
+ Inf(1), // pow(+0, -Inf)
+ Inf(1), // pow(+0, -Pi)
+ Inf(1), // pow(+0, -3)
+ 1, // pow(+0, -0)
+ 1, // pow(+0, +0)
+ 0, // pow(+0, 3)
+ 0, // pow(+0, +Pi)
+ 0, // pow(+0, +Inf)
+ NaN(), // pow(+0, NaN)
+ Inf(1), // pow(1/2, -Inf)
+ 0, // pow(1/2, +Inf)
+ 1, // pow(1, -Inf) IEEE 754-2008
+ 1, // pow(1, +Inf) IEEE 754-2008
+ 1, // pow(1, NaN) IEEE 754-2008
+ 0, // pow(+Pi, -Inf)
+ 1, // pow(+Pi, -0)
+ 1, // pow(+Pi, +0)
+ Pi, // pow(+Pi, 1)
+ Inf(1), // pow(+Pi, +Inf)
+ NaN(), // pow(+Pi, NaN)
+ 0, // pow(+Inf, -Pi)
+ 1, // pow(+Inf, -0)
+ 1, // pow(+Inf, +0)
+ Inf(1), // pow(+Inf, 1)
+ Inf(1), // pow(+Inf, Pi)
+ NaN(), // pow(+Inf, NaN)
+ NaN(), // pow(NaN, -Pi)
+ 1, // pow(NaN, -0)
+ 1, // pow(NaN, +0)
+ NaN(), // pow(NaN, 1)
+ NaN(), // pow(NaN, +Pi)
+ NaN(), // pow(NaN, NaN)
+}
+
+var vfpow10SC = []int{
+ MinInt32,
+ MaxInt32,
+ -325,
+ 309,
+}
+
+var pow10SC = []float64{
+ 0, // pow10(MinInt32)
+ Inf(1), // pow10(MaxInt32)
+ 0, // pow10(-325)
+ Inf(1), // pow10(309)
+}
+
+var vfsignbitSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var signbitSC = []bool{
+ true,
+ true,
+ false,
+ false,
+ false,
+}
+
+var vfsinSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var sinSC = []float64{
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ NaN(),
+ NaN(),
+}
+
+var vfsinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var sinhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vfsqrtSC = []float64{
+ Inf(-1),
+ -Pi,
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var sqrtSC = []float64{
+ NaN(),
+ NaN(),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+
+var vftanhSC = []float64{
+ Inf(-1),
+ Copysign(0, -1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var tanhSC = []float64{
+ -1,
+ Copysign(0, -1),
+ 0,
+ 1,
+ NaN(),
+}
+
+var vfy0SC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var y0SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+}
+var y1SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+}
+var y2SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+}
+var yM3SC = []float64{
+ NaN(),
+ Inf(1),
+ 0,
+ NaN(),
+}
+
+// arguments and expected results for boundary cases
+const (
+ SmallestNormalFloat64 = 2.2250738585072014e-308 // 2**-1022
+ LargestSubnormalFloat64 = SmallestNormalFloat64 - SmallestNonzeroFloat64
+)
+
+var vffrexpBC = []float64{
+ SmallestNormalFloat64,
+ LargestSubnormalFloat64,
+ SmallestNonzeroFloat64,
+ MaxFloat64,
+ -SmallestNormalFloat64,
+ -LargestSubnormalFloat64,
+ -SmallestNonzeroFloat64,
+ -MaxFloat64,
+}
+var frexpBC = []fi{
+ {0.5, -1021},
+ {0.99999999999999978, -1022},
+ {0.5, -1073},
+ {0.99999999999999989, 1024},
+ {-0.5, -1021},
+ {-0.99999999999999978, -1022},
+ {-0.5, -1073},
+ {-0.99999999999999989, 1024},
+}
+
+var vfldexpBC = []fi{
+ {SmallestNormalFloat64, -52},
+ {LargestSubnormalFloat64, -51},
+ {SmallestNonzeroFloat64, 1074},
+ {MaxFloat64, -(1023 + 1074)},
+ {1, -1075},
+ {-1, -1075},
+ {1, 1024},
+ {-1, 1024},
+}
+var ldexpBC = []float64{
+ SmallestNonzeroFloat64,
+ 1e-323, // 2**-1073
+ 1,
+ 1e-323, // 2**-1073
+ 0,
+ Copysign(0, -1),
+ Inf(1),
+ Inf(-1),
+}
+
+var logbBC = []float64{
+ -1022,
+ -1023,
+ -1074,
+ 1023,
+ -1022,
+ -1023,
+ -1074,
+ 1023,
+}
+
+func tolerance(a, b, e float64) bool {
+ d := a - b
+ if d < 0 {
+ d = -d
+ }
+
+ if a != 0 {
+ e = e * a
+ if e < 0 {
+ e = -e
+ }
+ }
+ return d < e
+}
+func kindaclose(a, b float64) bool { return tolerance(a, b, 1e-8) }
+func close(a, b float64) bool { return tolerance(a, b, 1e-14) }
+func veryclose(a, b float64) bool { return tolerance(a, b, 4e-16) }
+func soclose(a, b, e float64) bool { return tolerance(a, b, e) }
+func alike(a, b float64) bool {
+ switch {
+ case IsNaN(a) && IsNaN(b):
+ return true
+ case a == b:
+ return Signbit(a) == Signbit(b)
+ }
+ return false
+}
+
+func TestNaN(t *testing.T) {
+ f64 := NaN()
+ if f64 == f64 {
+ t.Fatalf("NaN() returns %g, expected NaN", f64)
+ }
+ f32 := float32(f64)
+ if f32 == f32 {
+ t.Fatalf("float32(NaN()) is %g, expected NaN", f32)
+ }
+}
+
+func TestAcos(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Acos(a); !close(acos[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", a, f, acos[i])
+ }
+ }
+ for i := 0; i < len(vfacosSC); i++ {
+ if f := Acos(vfacosSC[i]); !alike(acosSC[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", vfacosSC[i], f, acosSC[i])
+ }
+ }
+}
+
+func TestAcosh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := 1 + Abs(vf[i])
+ if f := Acosh(a); !veryclose(acosh[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", a, f, acosh[i])
+ }
+ }
+ for i := 0; i < len(vfacoshSC); i++ {
+ if f := Acosh(vfacoshSC[i]); !alike(acoshSC[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", vfacoshSC[i], f, acoshSC[i])
+ }
+ }
+}
+
+func TestAsin(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Asin(a); !veryclose(asin[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", a, f, asin[i])
+ }
+ }
+ for i := 0; i < len(vfasinSC); i++ {
+ if f := Asin(vfasinSC[i]); !alike(asinSC[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", vfasinSC[i], f, asinSC[i])
+ }
+ }
+}
+
+func TestAsinh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Asinh(vf[i]); !veryclose(asinh[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vf[i], f, asinh[i])
+ }
+ }
+ for i := 0; i < len(vfasinhSC); i++ {
+ if f := Asinh(vfasinhSC[i]); !alike(asinhSC[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vfasinhSC[i], f, asinhSC[i])
+ }
+ }
+}
+
+func TestAtan(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Atan(vf[i]); !veryclose(atan[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vf[i], f, atan[i])
+ }
+ }
+ for i := 0; i < len(vfatanSC); i++ {
+ if f := Atan(vfatanSC[i]); !alike(atanSC[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vfatanSC[i], f, atanSC[i])
+ }
+ }
+}
+
+func TestAtanh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Atanh(a); !veryclose(atanh[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", a, f, atanh[i])
+ }
+ }
+ for i := 0; i < len(vfatanhSC); i++ {
+ if f := Atanh(vfatanhSC[i]); !alike(atanhSC[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", vfatanhSC[i], f, atanhSC[i])
+ }
+ }
+}
+
+func TestAtan2(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Atan2(10, vf[i]); !veryclose(atan2[i], f) {
+ t.Errorf("Atan2(10, %g) = %g, want %g", vf[i], f, atan2[i])
+ }
+ }
+ for i := 0; i < len(vfatan2SC); i++ {
+ if f := Atan2(vfatan2SC[i][0], vfatan2SC[i][1]); !alike(atan2SC[i], f) {
+ t.Errorf("Atan2(%g, %g) = %g, want %g", vfatan2SC[i][0], vfatan2SC[i][1], f, atan2SC[i])
+ }
+ }
+}
+
+func TestCbrt(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Cbrt(vf[i]); !veryclose(cbrt[i], f) {
+ t.Errorf("Cbrt(%g) = %g, want %g", vf[i], f, cbrt[i])
+ }
+ }
+ for i := 0; i < len(vfcbrtSC); i++ {
+ if f := Cbrt(vfcbrtSC[i]); !alike(cbrtSC[i], f) {
+ t.Errorf("Cbrt(%g) = %g, want %g", vfcbrtSC[i], f, cbrtSC[i])
+ }
+ }
+}
+
+func TestCeil(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Ceil(vf[i]); ceil[i] != f {
+ t.Errorf("Ceil(%g) = %g, want %g", vf[i], f, ceil[i])
+ }
+ }
+ for i := 0; i < len(vfceilSC); i++ {
+ if f := Ceil(vfceilSC[i]); !alike(ceilSC[i], f) {
+ t.Errorf("Ceil(%g) = %g, want %g", vfceilSC[i], f, ceilSC[i])
+ }
+ }
+}
+
+func TestCopysign(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Copysign(vf[i], -1); copysign[i] != f {
+ t.Errorf("Copysign(%g, -1) = %g, want %g", vf[i], f, copysign[i])
+ }
+ }
+ for i := 0; i < len(vf); i++ {
+ if f := Copysign(vf[i], 1); -copysign[i] != f {
+ t.Errorf("Copysign(%g, 1) = %g, want %g", vf[i], f, -copysign[i])
+ }
+ }
+ for i := 0; i < len(vfcopysignSC); i++ {
+ if f := Copysign(vfcopysignSC[i], -1); !alike(copysignSC[i], f) {
+ t.Errorf("Copysign(%g, -1) = %g, want %g", vfcopysignSC[i], f, copysignSC[i])
+ }
+ }
+}
+
+func TestCos(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Cos(vf[i]); !veryclose(cos[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vf[i], f, cos[i])
+ }
+ }
+ for i := 0; i < len(vfcosSC); i++ {
+ if f := Cos(vfcosSC[i]); !alike(cosSC[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vfcosSC[i], f, cosSC[i])
+ }
+ }
+}
+
+func TestCosh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Cosh(vf[i]); !close(cosh[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vf[i], f, cosh[i])
+ }
+ }
+ for i := 0; i < len(vfcoshSC); i++ {
+ if f := Cosh(vfcoshSC[i]); !alike(coshSC[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vfcoshSC[i], f, coshSC[i])
+ }
+ }
+}
+
+func TestErf(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Erf(a); !veryclose(erf[i], f) {
+ t.Errorf("Erf(%g) = %g, want %g", a, f, erf[i])
+ }
+ }
+ for i := 0; i < len(vferfSC); i++ {
+ if f := Erf(vferfSC[i]); !alike(erfSC[i], f) {
+ t.Errorf("Erf(%g) = %g, want %g", vferfSC[i], f, erfSC[i])
+ }
+ }
+}
+
+func TestErfc(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 10
+ if f := Erfc(a); !veryclose(erfc[i], f) {
+ t.Errorf("Erfc(%g) = %g, want %g", a, f, erfc[i])
+ }
+ }
+ for i := 0; i < len(vferfcSC); i++ {
+ if f := Erfc(vferfcSC[i]); !alike(erfcSC[i], f) {
+ t.Errorf("Erfc(%g) = %g, want %g", vferfcSC[i], f, erfcSC[i])
+ }
+ }
+}
+
+func TestExp(t *testing.T) {
+ testExp(t, Exp, "Exp")
+ testExp(t, ExpGo, "ExpGo")
+}
+
+func testExp(t *testing.T, Exp func(float64) float64, name string) {
+ for i := 0; i < len(vf); i++ {
+ if f := Exp(vf[i]); !close(exp[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp[i])
+ }
+ }
+ for i := 0; i < len(vfexpSC); i++ {
+ if f := Exp(vfexpSC[i]); !alike(expSC[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vfexpSC[i], f, expSC[i])
+ }
+ }
+}
+
+func TestExpm1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 100
+ if f := Expm1(a); !veryclose(expm1[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", a, f, expm1[i])
+ }
+ }
+ for i := 0; i < len(vfexpm1SC); i++ {
+ if f := Expm1(vfexpm1SC[i]); !alike(expm1SC[i], f) {
+ t.Errorf("Expm1(%g) = %g, want %g", vfexpm1SC[i], f, expm1SC[i])
+ }
+ }
+}
+
+func TestExp2(t *testing.T) {
+ testExp2(t, Exp2, "Exp2")
+ testExp2(t, Exp2Go, "Exp2Go")
+}
+
+func testExp2(t *testing.T, Exp2 func(float64) float64, name string) {
+ for i := 0; i < len(vf); i++ {
+ if f := Exp2(vf[i]); !close(exp2[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vf[i], f, exp2[i])
+ }
+ }
+ for i := 0; i < len(vfexpSC); i++ {
+ if f := Exp2(vfexpSC[i]); !alike(expSC[i], f) {
+ t.Errorf("%s(%g) = %g, want %g", name, vfexpSC[i], f, expSC[i])
+ }
+ }
+ for n := -1074; n < 1024; n++ {
+ f := Exp2(float64(n))
+ vf := Ldexp(1, n)
+ if f != vf {
+ t.Errorf("%s(%d) = %g, want %g", name, n, f, vf)
+ }
+ }
+}
+
+func TestAbs(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Abs(vf[i]); fabs[i] != f {
+ t.Errorf("Abs(%g) = %g, want %g", vf[i], f, fabs[i])
+ }
+ }
+ for i := 0; i < len(vffabsSC); i++ {
+ if f := Abs(vffabsSC[i]); !alike(fabsSC[i], f) {
+ t.Errorf("Abs(%g) = %g, want %g", vffabsSC[i], f, fabsSC[i])
+ }
+ }
+}
+
+func TestDim(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Dim(vf[i], 0); fdim[i] != f {
+ t.Errorf("Dim(%g, %g) = %g, want %g", vf[i], 0.0, f, fdim[i])
+ }
+ }
+ for i := 0; i < len(vffdimSC); i++ {
+ if f := Dim(vffdimSC[i][0], vffdimSC[i][1]); !alike(fdimSC[i], f) {
+ t.Errorf("Dim(%g, %g) = %g, want %g", vffdimSC[i][0], vffdimSC[i][1], f, fdimSC[i])
+ }
+ }
+}
+
+func TestFloor(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Floor(vf[i]); floor[i] != f {
+ t.Errorf("Floor(%g) = %g, want %g", vf[i], f, floor[i])
+ }
+ }
+ for i := 0; i < len(vfceilSC); i++ {
+ if f := Floor(vfceilSC[i]); !alike(ceilSC[i], f) {
+ t.Errorf("Floor(%g) = %g, want %g", vfceilSC[i], f, ceilSC[i])
+ }
+ }
+}
+
+func TestMax(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Max(vf[i], ceil[i]); ceil[i] != f {
+ t.Errorf("Max(%g, %g) = %g, want %g", vf[i], ceil[i], f, ceil[i])
+ }
+ }
+ for i := 0; i < len(vffdimSC); i++ {
+ if f := Max(vffdimSC[i][0], vffdimSC[i][1]); !alike(fmaxSC[i], f) {
+ t.Errorf("Max(%g, %g) = %g, want %g", vffdimSC[i][0], vffdimSC[i][1], f, fmaxSC[i])
+ }
+ }
+}
+
+func TestMin(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Min(vf[i], floor[i]); floor[i] != f {
+ t.Errorf("Min(%g, %g) = %g, want %g", vf[i], floor[i], f, floor[i])
+ }
+ }
+ for i := 0; i < len(vffdimSC); i++ {
+ if f := Min(vffdimSC[i][0], vffdimSC[i][1]); !alike(fminSC[i], f) {
+ t.Errorf("Min(%g, %g) = %g, want %g", vffdimSC[i][0], vffdimSC[i][1], f, fminSC[i])
+ }
+ }
+}
+
+func TestMod(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Mod(10, vf[i]); fmod[i] != f {
+ t.Errorf("Mod(10, %g) = %g, want %g", vf[i], f, fmod[i])
+ }
+ }
+ for i := 0; i < len(vffmodSC); i++ {
+ if f := Mod(vffmodSC[i][0], vffmodSC[i][1]); !alike(fmodSC[i], f) {
+ t.Errorf("Mod(%g, %g) = %g, want %g", vffmodSC[i][0], vffmodSC[i][1], f, fmodSC[i])
+ }
+ }
+}
+
+func TestFrexp(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f, j := Frexp(vf[i]); !veryclose(frexp[i].f, f) || frexp[i].i != j {
+ t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vf[i], f, j, frexp[i].f, frexp[i].i)
+ }
+ }
+ for i := 0; i < len(vffrexpSC); i++ {
+ if f, j := Frexp(vffrexpSC[i]); !alike(frexpSC[i].f, f) || frexpSC[i].i != j {
+ t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vffrexpSC[i], f, j, frexpSC[i].f, frexpSC[i].i)
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if f, j := Frexp(vffrexpBC[i]); !alike(frexpBC[i].f, f) || frexpBC[i].i != j {
+ t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vffrexpBC[i], f, j, frexpBC[i].f, frexpBC[i].i)
+ }
+ }
+}
+
+func TestGamma(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Gamma(vf[i]); !close(gamma[i], f) {
+ t.Errorf("Gamma(%g) = %g, want %g", vf[i], f, gamma[i])
+ }
+ }
+ for i := 0; i < len(vfgammaSC); i++ {
+ if f := Gamma(vfgammaSC[i]); !alike(gammaSC[i], f) {
+ t.Errorf("Gamma(%g) = %g, want %g", vfgammaSC[i], f, gammaSC[i])
+ }
+ }
+}
+
+func TestHypot(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(1e200 * tanh[i] * Sqrt(2))
+ if f := Hypot(1e200*tanh[i], 1e200*tanh[i]); !veryclose(a, f) {
+ t.Errorf("Hypot(%g, %g) = %g, want %g", 1e200*tanh[i], 1e200*tanh[i], f, a)
+ }
+ }
+ for i := 0; i < len(vfhypotSC); i++ {
+ if f := Hypot(vfhypotSC[i][0], vfhypotSC[i][1]); !alike(hypotSC[i], f) {
+ t.Errorf("Hypot(%g, %g) = %g, want %g", vfhypotSC[i][0], vfhypotSC[i][1], f, hypotSC[i])
+ }
+ }
+}
+
+func TestHypotGo(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(1e200 * tanh[i] * Sqrt(2))
+ if f := HypotGo(1e200*tanh[i], 1e200*tanh[i]); !veryclose(a, f) {
+ t.Errorf("HypotGo(%g, %g) = %g, want %g", 1e200*tanh[i], 1e200*tanh[i], f, a)
+ }
+ }
+ for i := 0; i < len(vfhypotSC); i++ {
+ if f := HypotGo(vfhypotSC[i][0], vfhypotSC[i][1]); !alike(hypotSC[i], f) {
+ t.Errorf("HypotGo(%g, %g) = %g, want %g", vfhypotSC[i][0], vfhypotSC[i][1], f, hypotSC[i])
+ }
+ }
+}
+
+func TestIlogb(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := frexp[i].i - 1 // adjust because fr in the interval [½, 1)
+ if e := Ilogb(vf[i]); a != e {
+ t.Errorf("Ilogb(%g) = %d, want %d", vf[i], e, a)
+ }
+ }
+ for i := 0; i < len(vflogbSC); i++ {
+ if e := Ilogb(vflogbSC[i]); ilogbSC[i] != e {
+ t.Errorf("Ilogb(%g) = %d, want %d", vflogbSC[i], e, ilogbSC[i])
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if e := Ilogb(vffrexpBC[i]); int(logbBC[i]) != e {
+ t.Errorf("Ilogb(%g) = %d, want %d", vffrexpBC[i], e, int(logbBC[i]))
+ }
+ }
+}
+
+func TestJ0(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := J0(vf[i]); !soclose(j0[i], f, 4e-14) {
+ t.Errorf("J0(%g) = %g, want %g", vf[i], f, j0[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := J0(vfj0SC[i]); !alike(j0SC[i], f) {
+ t.Errorf("J0(%g) = %g, want %g", vfj0SC[i], f, j0SC[i])
+ }
+ }
+}
+
+func TestJ1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := J1(vf[i]); !close(j1[i], f) {
+ t.Errorf("J1(%g) = %g, want %g", vf[i], f, j1[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := J1(vfj0SC[i]); !alike(j1SC[i], f) {
+ t.Errorf("J1(%g) = %g, want %g", vfj0SC[i], f, j1SC[i])
+ }
+ }
+}
+
+func TestJn(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Jn(2, vf[i]); !close(j2[i], f) {
+ t.Errorf("Jn(2, %g) = %g, want %g", vf[i], f, j2[i])
+ }
+ if f := Jn(-3, vf[i]); !close(jM3[i], f) {
+ t.Errorf("Jn(-3, %g) = %g, want %g", vf[i], f, jM3[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := Jn(2, vfj0SC[i]); !alike(j2SC[i], f) {
+ t.Errorf("Jn(2, %g) = %g, want %g", vfj0SC[i], f, j2SC[i])
+ }
+ if f := Jn(-3, vfj0SC[i]); !alike(jM3SC[i], f) {
+ t.Errorf("Jn(-3, %g) = %g, want %g", vfj0SC[i], f, jM3SC[i])
+ }
+ }
+}
+
+func TestLdexp(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Ldexp(frexp[i].f, frexp[i].i); !veryclose(vf[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", frexp[i].f, frexp[i].i, f, vf[i])
+ }
+ }
+ for i := 0; i < len(vffrexpSC); i++ {
+ if f := Ldexp(frexpSC[i].f, frexpSC[i].i); !alike(vffrexpSC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", frexpSC[i].f, frexpSC[i].i, f, vffrexpSC[i])
+ }
+ }
+ for i := 0; i < len(vfldexpSC); i++ {
+ if f := Ldexp(vfldexpSC[i].f, vfldexpSC[i].i); !alike(ldexpSC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", vfldexpSC[i].f, vfldexpSC[i].i, f, ldexpSC[i])
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if f := Ldexp(frexpBC[i].f, frexpBC[i].i); !alike(vffrexpBC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", frexpBC[i].f, frexpBC[i].i, f, vffrexpBC[i])
+ }
+ }
+ for i := 0; i < len(vfldexpBC); i++ {
+ if f := Ldexp(vfldexpBC[i].f, vfldexpBC[i].i); !alike(ldexpBC[i], f) {
+ t.Errorf("Ldexp(%g, %d) = %g, want %g", vfldexpBC[i].f, vfldexpBC[i].i, f, ldexpBC[i])
+ }
+ }
+}
+
+func TestLgamma(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f, s := Lgamma(vf[i]); !close(lgamma[i].f, f) || lgamma[i].i != s {
+ t.Errorf("Lgamma(%g) = %g, %d, want %g, %d", vf[i], f, s, lgamma[i].f, lgamma[i].i)
+ }
+ }
+ for i := 0; i < len(vflgammaSC); i++ {
+ if f, s := Lgamma(vflgammaSC[i]); !alike(lgammaSC[i].f, f) || lgammaSC[i].i != s {
+ t.Errorf("Lgamma(%g) = %g, %d, want %g, %d", vflgammaSC[i], f, s, lgammaSC[i].f, lgammaSC[i].i)
+ }
+ }
+}
+
+func TestLog(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log(a); log[i] != f {
+ t.Errorf("Log(%g) = %g, want %g", a, f, log[i])
+ }
+ }
+ if f := Log(10); f != Ln10 {
+ t.Errorf("Log(%g) = %g, want %g", 10.0, f, Ln10)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+}
+
+func TestLogb(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Logb(vf[i]); logb[i] != f {
+ t.Errorf("Logb(%g) = %g, want %g", vf[i], f, logb[i])
+ }
+ }
+ for i := 0; i < len(vflogbSC); i++ {
+ if f := Logb(vflogbSC[i]); !alike(logbSC[i], f) {
+ t.Errorf("Logb(%g) = %g, want %g", vflogbSC[i], f, logbSC[i])
+ }
+ }
+ for i := 0; i < len(vffrexpBC); i++ {
+ if f := Logb(vffrexpBC[i]); !alike(logbBC[i], f) {
+ t.Errorf("Logb(%g) = %g, want %g", vffrexpBC[i], f, logbBC[i])
+ }
+ }
+}
+
+func TestLog10(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log10(a); !veryclose(log10[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", a, f, log10[i])
+ }
+ }
+ if f := Log10(E); f != Log10E {
+ t.Errorf("Log10(%g) = %g, want %g", E, f, Log10E)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log10(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+}
+
+func TestLog1p(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := vf[i] / 100
+ if f := Log1p(a); !veryclose(log1p[i], f) {
+ t.Errorf("Log1p(%g) = %g, want %g", a, f, log1p[i])
+ }
+ }
+ a := 9.0
+ if f := Log1p(a); f != Ln10 {
+ t.Errorf("Log1p(%g) = %g, want %g", a, f, Ln10)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log1p(vflog1pSC[i]); !alike(log1pSC[i], f) {
+ t.Errorf("Log1p(%g) = %g, want %g", vflog1pSC[i], f, log1pSC[i])
+ }
+ }
+}
+
+func TestLog2(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Log2(a); !veryclose(log2[i], f) {
+ t.Errorf("Log2(%g) = %g, want %g", a, f, log2[i])
+ }
+ }
+ if f := Log2(E); f != Log2E {
+ t.Errorf("Log2(%g) = %g, want %g", E, f, Log2E)
+ }
+ for i := 0; i < len(vflogSC); i++ {
+ if f := Log2(vflogSC[i]); !alike(logSC[i], f) {
+ t.Errorf("Log2(%g) = %g, want %g", vflogSC[i], f, logSC[i])
+ }
+ }
+ for i := -1074; i <= 1023; i++ {
+ f := Ldexp(1, i)
+ l := Log2(f)
+ if l != float64(i) {
+ t.Errorf("Log2(2**%d) = %g, want %d", i, l, i)
+ }
+ }
+}
+
+func TestModf(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f, g := Modf(vf[i]); !veryclose(modf[i][0], f) || !veryclose(modf[i][1], g) {
+ t.Errorf("Modf(%g) = %g, %g, want %g, %g", vf[i], f, g, modf[i][0], modf[i][1])
+ }
+ }
+ for i := 0; i < len(vfmodfSC); i++ {
+ if f, g := Modf(vfmodfSC[i]); !alike(modfSC[i][0], f) || !alike(modfSC[i][1], g) {
+ t.Errorf("Modf(%g) = %g, %g, want %g, %g", vfmodfSC[i], f, g, modfSC[i][0], modfSC[i][1])
+ }
+ }
+}
+
+func TestNextafter32(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ vfi := float32(vf[i])
+ if f := Nextafter32(vfi, 10); nextafter32[i] != f {
+ t.Errorf("Nextafter32(%g, %g) = %g want %g", vfi, 10.0, f, nextafter32[i])
+ }
+ }
+ for i := 0; i < len(vfnextafter32SC); i++ {
+ if f := Nextafter32(vfnextafter32SC[i][0], vfnextafter32SC[i][1]); !alike(float64(nextafter32SC[i]), float64(f)) {
+ t.Errorf("Nextafter32(%g, %g) = %g want %g", vfnextafter32SC[i][0], vfnextafter32SC[i][1], f, nextafter32SC[i])
+ }
+ }
+}
+
+func TestNextafter64(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Nextafter(vf[i], 10); nextafter64[i] != f {
+ t.Errorf("Nextafter64(%g, %g) = %g want %g", vf[i], 10.0, f, nextafter64[i])
+ }
+ }
+ for i := 0; i < len(vfnextafter64SC); i++ {
+ if f := Nextafter(vfnextafter64SC[i][0], vfnextafter64SC[i][1]); !alike(nextafter64SC[i], f) {
+ t.Errorf("Nextafter64(%g, %g) = %g want %g", vfnextafter64SC[i][0], vfnextafter64SC[i][1], f, nextafter64SC[i])
+ }
+ }
+}
+
+func TestPow(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Pow(10, vf[i]); !close(pow[i], f) {
+ t.Errorf("Pow(10, %g) = %g, want %g", vf[i], f, pow[i])
+ }
+ }
+ for i := 0; i < len(vfpowSC); i++ {
+ if f := Pow(vfpowSC[i][0], vfpowSC[i][1]); !alike(powSC[i], f) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", vfpowSC[i][0], vfpowSC[i][1], f, powSC[i])
+ }
+ }
+}
+
+func TestPow10(t *testing.T) {
+ for i := 0; i < len(vfpow10SC); i++ {
+ if f := Pow10(vfpow10SC[i]); !alike(pow10SC[i], f) {
+ t.Errorf("Pow10(%d) = %g, want %g", vfpow10SC[i], f, pow10SC[i])
+ }
+ }
+}
+
+func TestRemainder(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Remainder(10, vf[i]); remainder[i] != f {
+ t.Errorf("Remainder(10, %g) = %g, want %g", vf[i], f, remainder[i])
+ }
+ }
+ for i := 0; i < len(vffmodSC); i++ {
+ if f := Remainder(vffmodSC[i][0], vffmodSC[i][1]); !alike(fmodSC[i], f) {
+ t.Errorf("Remainder(%g, %g) = %g, want %g", vffmodSC[i][0], vffmodSC[i][1], f, fmodSC[i])
+ }
+ }
+}
+
+func TestSignbit(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Signbit(vf[i]); signbit[i] != f {
+ t.Errorf("Signbit(%g) = %t, want %t", vf[i], f, signbit[i])
+ }
+ }
+ for i := 0; i < len(vfsignbitSC); i++ {
+ if f := Signbit(vfsignbitSC[i]); signbitSC[i] != f {
+ t.Errorf("Signbit(%g) = %t, want %t", vfsignbitSC[i], f, signbitSC[i])
+ }
+ }
+}
+func TestSin(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Sin(vf[i]); !veryclose(sin[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vf[i], f, sin[i])
+ }
+ }
+ for i := 0; i < len(vfsinSC); i++ {
+ if f := Sin(vfsinSC[i]); !alike(sinSC[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
+ }
+ }
+}
+
+func TestSincos(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if s, c := Sincos(vf[i]); !veryclose(sin[i], s) || !veryclose(cos[i], c) {
+ t.Errorf("Sincos(%g) = %g, %g want %g, %g", vf[i], s, c, sin[i], cos[i])
+ }
+ }
+}
+
+func TestSinh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Sinh(vf[i]); !close(sinh[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vf[i], f, sinh[i])
+ }
+ }
+ for i := 0; i < len(vfsinhSC); i++ {
+ if f := Sinh(vfsinhSC[i]); !alike(sinhSC[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vfsinhSC[i], f, sinhSC[i])
+ }
+ }
+}
+
+func TestSqrt(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := SqrtGo(a); sqrt[i] != f {
+ t.Errorf("SqrtGo(%g) = %g, want %g", a, f, sqrt[i])
+ }
+ a = Abs(vf[i])
+ if f := Sqrt(a); sqrt[i] != f {
+ t.Errorf("Sqrt(%g) = %g, want %g", a, f, sqrt[i])
+ }
+ }
+ for i := 0; i < len(vfsqrtSC); i++ {
+ if f := SqrtGo(vfsqrtSC[i]); !alike(sqrtSC[i], f) {
+ t.Errorf("SqrtGo(%g) = %g, want %g", vfsqrtSC[i], f, sqrtSC[i])
+ }
+ if f := Sqrt(vfsqrtSC[i]); !alike(sqrtSC[i], f) {
+ t.Errorf("Sqrt(%g) = %g, want %g", vfsqrtSC[i], f, sqrtSC[i])
+ }
+ }
+}
+
+func TestTan(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Tan(vf[i]); !veryclose(tan[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vf[i], f, tan[i])
+ }
+ }
+ // same special cases as Sin
+ for i := 0; i < len(vfsinSC); i++ {
+ if f := Tan(vfsinSC[i]); !alike(sinSC[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
+ }
+ }
+}
+
+func TestTanh(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Tanh(vf[i]); !veryclose(tanh[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vf[i], f, tanh[i])
+ }
+ }
+ for i := 0; i < len(vftanhSC); i++ {
+ if f := Tanh(vftanhSC[i]); !alike(tanhSC[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vftanhSC[i], f, tanhSC[i])
+ }
+ }
+}
+
+func TestTrunc(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Trunc(vf[i]); trunc[i] != f {
+ t.Errorf("Trunc(%g) = %g, want %g", vf[i], f, trunc[i])
+ }
+ }
+ for i := 0; i < len(vfceilSC); i++ {
+ if f := Trunc(vfceilSC[i]); !alike(ceilSC[i], f) {
+ t.Errorf("Trunc(%g) = %g, want %g", vfceilSC[i], f, ceilSC[i])
+ }
+ }
+}
+
+func TestY0(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Y0(a); !close(y0[i], f) {
+ t.Errorf("Y0(%g) = %g, want %g", a, f, y0[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Y0(vfy0SC[i]); !alike(y0SC[i], f) {
+ t.Errorf("Y0(%g) = %g, want %g", vfy0SC[i], f, y0SC[i])
+ }
+ }
+}
+
+func TestY1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Y1(a); !soclose(y1[i], f, 2e-14) {
+ t.Errorf("Y1(%g) = %g, want %g", a, f, y1[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Y1(vfy0SC[i]); !alike(y1SC[i], f) {
+ t.Errorf("Y1(%g) = %g, want %g", vfy0SC[i], f, y1SC[i])
+ }
+ }
+}
+
+func TestYn(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Abs(vf[i])
+ if f := Yn(2, a); !close(y2[i], f) {
+ t.Errorf("Yn(2, %g) = %g, want %g", a, f, y2[i])
+ }
+ if f := Yn(-3, a); !close(yM3[i], f) {
+ t.Errorf("Yn(-3, %g) = %g, want %g", a, f, yM3[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Yn(2, vfy0SC[i]); !alike(y2SC[i], f) {
+ t.Errorf("Yn(2, %g) = %g, want %g", vfy0SC[i], f, y2SC[i])
+ }
+ if f := Yn(-3, vfy0SC[i]); !alike(yM3SC[i], f) {
+ t.Errorf("Yn(-3, %g) = %g, want %g", vfy0SC[i], f, yM3SC[i])
+ }
+ }
+}
+
+// Check that math functions of high angle values
+// return accurate results. [Since (vf[i] + large) - large != vf[i],
+// testing for Trig(vf[i] + large) == Trig(vf[i]), where large is
+// a multiple of 2*Pi, is misleading.]
+func TestLargeCos(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := cosLarge[i]
+ f2 := Cos(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Cos(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestLargeSin(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := sinLarge[i]
+ f2 := Sin(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Sin(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+func TestLargeSincos(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1, g1 := sinLarge[i], cosLarge[i]
+ f2, g2 := Sincos(vf[i] + large)
+ if !close(f1, f2) || !close(g1, g2) {
+ t.Errorf("Sincos(%g) = %g, %g, want %g, %g", vf[i]+large, f2, g2, f1, g1)
+ }
+ }
+}
+
+func TestLargeTan(t *testing.T) {
+ large := float64(100000 * Pi)
+ for i := 0; i < len(vf); i++ {
+ f1 := tanLarge[i]
+ f2 := Tan(vf[i] + large)
+ if !close(f1, f2) {
+ t.Errorf("Tan(%g) = %g, want %g", vf[i]+large, f2, f1)
+ }
+ }
+}
+
+// Check that math constants are accepted by compiler
+// and have right value (assumes strconv.ParseFloat works).
+// http://code.google.com/p/go/issues/detail?id=201
+
+type floatTest struct {
+ val interface{}
+ name string
+ str string
+}
+
+var floatTests = []floatTest{
+ {float64(MaxFloat64), "MaxFloat64", "1.7976931348623157e+308"},
+ {float64(SmallestNonzeroFloat64), "SmallestNonzeroFloat64", "5e-324"},
+ {float32(MaxFloat32), "MaxFloat32", "3.4028235e+38"},
+ {float32(SmallestNonzeroFloat32), "SmallestNonzeroFloat32", "1e-45"},
+}
+
+func TestFloatMinMax(t *testing.T) {
+ for _, tt := range floatTests {
+ s := fmt.Sprint(tt.val)
+ if s != tt.str {
+ t.Errorf("Sprint(%v) = %s, want %s", tt.name, s, tt.str)
+ }
+ }
+}
+
+// Benchmarks
+
+func BenchmarkAcos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Acos(.5)
+ }
+}
+
+func BenchmarkAcosh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Acosh(1.5)
+ }
+}
+
+func BenchmarkAsin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Asin(.5)
+ }
+}
+
+func BenchmarkAsinh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Asinh(.5)
+ }
+}
+
+func BenchmarkAtan(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atan(.5)
+ }
+}
+
+func BenchmarkAtanh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atanh(.5)
+ }
+}
+
+func BenchmarkAtan2(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atan2(.5, 1)
+ }
+}
+
+func BenchmarkCbrt(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cbrt(10)
+ }
+}
+
+func BenchmarkCeil(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Ceil(.5)
+ }
+}
+
+func BenchmarkCopysign(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Copysign(.5, -1)
+ }
+}
+
+func BenchmarkCos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cos(.5)
+ }
+}
+
+func BenchmarkCosh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cosh(2.5)
+ }
+}
+
+func BenchmarkErf(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Erf(.5)
+ }
+}
+
+func BenchmarkErfc(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Erfc(.5)
+ }
+}
+
+func BenchmarkExp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Exp(.5)
+ }
+}
+
+func BenchmarkExpGo(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ ExpGo(.5)
+ }
+}
+
+func BenchmarkExpm1(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Expm1(.5)
+ }
+}
+
+func BenchmarkExp2(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Exp2(.5)
+ }
+}
+
+func BenchmarkExp2Go(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Exp2Go(.5)
+ }
+}
+
+func BenchmarkAbs(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Abs(.5)
+ }
+}
+
+func BenchmarkDim(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Dim(10, 3)
+ }
+}
+
+func BenchmarkFloor(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Floor(.5)
+ }
+}
+
+func BenchmarkMax(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Max(10, 3)
+ }
+}
+
+func BenchmarkMin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Min(10, 3)
+ }
+}
+
+func BenchmarkMod(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Mod(10, 3)
+ }
+}
+
+func BenchmarkFrexp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Frexp(8)
+ }
+}
+
+func BenchmarkGamma(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Gamma(2.5)
+ }
+}
+
+func BenchmarkHypot(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Hypot(3, 4)
+ }
+}
+
+func BenchmarkHypotGo(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ HypotGo(3, 4)
+ }
+}
+
+func BenchmarkIlogb(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Ilogb(.5)
+ }
+}
+
+func BenchmarkJ0(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ J0(2.5)
+ }
+}
+
+func BenchmarkJ1(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ J1(2.5)
+ }
+}
+
+func BenchmarkJn(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Jn(2, 2.5)
+ }
+}
+
+func BenchmarkLdexp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Ldexp(.5, 2)
+ }
+}
+
+func BenchmarkLgamma(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Lgamma(2.5)
+ }
+}
+
+func BenchmarkLog(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log(.5)
+ }
+}
+
+func BenchmarkLogb(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Logb(.5)
+ }
+}
+
+func BenchmarkLog1p(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log1p(.5)
+ }
+}
+
+func BenchmarkLog10(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log10(.5)
+ }
+}
+
+func BenchmarkLog2(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log2(.5)
+ }
+}
+
+func BenchmarkModf(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Modf(1.5)
+ }
+}
+
+func BenchmarkNextafter32(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Nextafter32(.5, 1)
+ }
+}
+
+func BenchmarkNextafter64(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Nextafter(.5, 1)
+ }
+}
+
+func BenchmarkPowInt(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Pow(2, 2)
+ }
+}
+
+func BenchmarkPowFrac(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Pow(2.5, 1.5)
+ }
+}
+
+func BenchmarkPow10Pos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Pow10(300)
+ }
+}
+
+func BenchmarkPow10Neg(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Pow10(-300)
+ }
+}
+
+func BenchmarkRemainder(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Remainder(10, 3)
+ }
+}
+
+func BenchmarkSignbit(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Signbit(2.5)
+ }
+}
+
+func BenchmarkSin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sin(.5)
+ }
+}
+
+func BenchmarkSincos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sincos(.5)
+ }
+}
+
+func BenchmarkSinh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sinh(2.5)
+ }
+}
+
+func BenchmarkSqrt(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sqrt(10)
+ }
+}
+
+func BenchmarkSqrtGo(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ SqrtGo(10)
+ }
+}
+
+func BenchmarkTan(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Tan(.5)
+ }
+}
+
+func BenchmarkTanh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Tanh(2.5)
+ }
+}
+func BenchmarkTrunc(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Trunc(.5)
+ }
+}
+
+func BenchmarkY0(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Y0(2.5)
+ }
+}
+
+func BenchmarkY1(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Y1(2.5)
+ }
+}
+
+func BenchmarkYn(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Yn(2, 2.5)
+ }
+}
diff --git a/src/math/asin.go b/src/math/asin.go
new file mode 100644
index 0000000..88b851e
--- /dev/null
+++ b/src/math/asin.go
@@ -0,0 +1,55 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point arcsine and arccosine.
+
+ They are implemented by computing the arctangent
+ after appropriate range reduction.
+*/
+
+// Asin returns the arcsine, in radians, of x.
+//
+// Special cases are:
+// Asin(±0) = ±0
+// Asin(x) = NaN if x < -1 or x > 1
+func Asin(x float64) float64
+
+func asin(x float64) float64 {
+ if x == 0 {
+ return x // special case
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x > 1 {
+ return NaN() // special case
+ }
+
+ temp := Sqrt(1 - x*x)
+ if x > 0.7 {
+ temp = Pi/2 - satan(temp/x)
+ } else {
+ temp = satan(x / temp)
+ }
+
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
+
+// Acos returns the arccosine, in radians, of x.
+//
+// Special case is:
+// Acos(x) = NaN if x < -1 or x > 1
+func Acos(x float64) float64
+
+func acos(x float64) float64 {
+ return Pi/2 - Asin(x)
+}
diff --git a/src/math/asin_386.s b/src/math/asin_386.s
new file mode 100644
index 0000000..4f34e12
--- /dev/null
+++ b/src/math/asin_386.s
@@ -0,0 +1,30 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Asin(x float64) float64
+TEXT ·Asin(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=sin(x)
+ FMOVD F0, F1 // F0=sin(x), F1=sin(x)
+ FMULD F0, F0 // F0=sin(x)*sin(x), F1=sin(x)
+ FLD1 // F0=1, F1=sin(x)*sin(x), F2=sin(x)
+ FSUBRDP F0, F1 // F0=1-sin(x)*sin(x) (=cos(x)*cos(x)), F1=sin(x)
+ FSQRT // F0=cos(x), F1=sin(x)
+ FPATAN // F0=arcsin(sin(x))=x
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Acos(x float64) float64
+TEXT ·Acos(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=cos(x)
+ FMOVD F0, F1 // F0=cos(x), F1=cos(x)
+ FMULD F0, F0 // F0=cos(x)*cos(x), F1=cos(x)
+ FLD1 // F0=1, F1=cos(x)*cos(x), F2=cos(x)
+ FSUBRDP F0, F1 // F0=1-cos(x)*cos(x) (=sin(x)*sin(x)), F1=cos(x)
+ FSQRT // F0=sin(x), F1=cos(x)
+ FXCHD F0, F1 // F0=cos(x), F1=sin(x)
+ FPATAN // F0=arccos(cos(x))=x
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/asin_amd64.s b/src/math/asin_amd64.s
new file mode 100644
index 0000000..1a43d48
--- /dev/null
+++ b/src/math/asin_amd64.s
@@ -0,0 +1,11 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Asin(SB),NOSPLIT,$0
+ JMP ·asin(SB)
+
+TEXT ·Acos(SB),NOSPLIT,$0
+ JMP ·acos(SB)
diff --git a/src/math/asin_amd64p32.s b/src/math/asin_amd64p32.s
new file mode 100644
index 0000000..2751c47
--- /dev/null
+++ b/src/math/asin_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "asin_amd64.s"
diff --git a/src/math/asin_arm.s b/src/math/asin_arm.s
new file mode 100644
index 0000000..8fe03b6
--- /dev/null
+++ b/src/math/asin_arm.s
@@ -0,0 +1,11 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Asin(SB),NOSPLIT,$0
+ B ·asin(SB)
+
+TEXT ·Acos(SB),NOSPLIT,$0
+ B ·acos(SB)
diff --git a/src/math/asinh.go b/src/math/asinh.go
new file mode 100644
index 0000000..ff2de02
--- /dev/null
+++ b/src/math/asinh.go
@@ -0,0 +1,69 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/s_asinh.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// asinh(x)
+// Method :
+// Based on
+// asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
+// we have
+// asinh(x) := x if 1+x*x=1,
+// := sign(x)*(log(x)+ln2)) for large |x|, else
+// := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
+// := sign(x)*log1p(|x| + x**2/(1 + sqrt(1+x**2)))
+//
+
+// Asinh returns the inverse hyperbolic sine of x.
+//
+// Special cases are:
+// Asinh(±0) = ±0
+// Asinh(±Inf) = ±Inf
+// Asinh(NaN) = NaN
+func Asinh(x float64) float64 {
+ const (
+ Ln2 = 6.93147180559945286227e-01 // 0x3FE62E42FEFA39EF
+ NearZero = 1.0 / (1 << 28) // 2**-28
+ Large = 1 << 28 // 2**28
+ )
+ // special cases
+ if IsNaN(x) || IsInf(x, 0) {
+ return x
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ var temp float64
+ switch {
+ case x > Large:
+ temp = Log(x) + Ln2 // |x| > 2**28
+ case x > 2:
+ temp = Log(2*x + 1/(Sqrt(x*x+1)+x)) // 2**28 > |x| > 2.0
+ case x < NearZero:
+ temp = x // |x| < 2**-28
+ default:
+ temp = Log1p(x + x*x/(1+Sqrt(1+x*x))) // 2.0 > |x| > 2**-28
+ }
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
diff --git a/src/math/atan.go b/src/math/atan.go
new file mode 100644
index 0000000..7fcc90b
--- /dev/null
+++ b/src/math/atan.go
@@ -0,0 +1,105 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point arctangent.
+*/
+
+// The original C code, the long comment, and the constants below were
+// from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
+// http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a version of the original C.
+//
+// atan.c
+// Inverse circular tangent (arctangent)
+//
+// SYNOPSIS:
+// double x, y, atan();
+// y = atan( x );
+//
+// DESCRIPTION:
+// Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
+//
+// Range reduction is from three intervals into the interval from zero to 0.66.
+// The approximant uses a rational function of degree 4/5 of the form
+// x + x**3 P(x)/Q(x).
+//
+// ACCURACY:
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10, 10 50000 2.4e-17 8.3e-18
+// IEEE -10, 10 10^6 1.8e-16 5.0e-17
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// xatan evaluates a series valid in the range [0, 0.66].
+func xatan(x float64) float64 {
+ const (
+ P0 = -8.750608600031904122785e-01
+ P1 = -1.615753718733365076637e+01
+ P2 = -7.500855792314704667340e+01
+ P3 = -1.228866684490136173410e+02
+ P4 = -6.485021904942025371773e+01
+ Q0 = +2.485846490142306297962e+01
+ Q1 = +1.650270098316988542046e+02
+ Q2 = +4.328810604912902668951e+02
+ Q3 = +4.853903996359136964868e+02
+ Q4 = +1.945506571482613964425e+02
+ )
+ z := x * x
+ z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
+ z = x*z + x
+ return z
+}
+
+// satan reduces its argument (known to be positive)
+// to the range [0, 0.66] and calls xatan.
+func satan(x float64) float64 {
+ const (
+ Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
+ Tan3pio8 = 2.41421356237309504880 // tan(3*pi/8)
+ )
+ if x <= 0.66 {
+ return xatan(x)
+ }
+ if x > Tan3pio8 {
+ return Pi/2 - xatan(1/x) + Morebits
+ }
+ return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
+}
+
+// Atan returns the arctangent, in radians, of x.
+//
+// Special cases are:
+// Atan(±0) = ±0
+// Atan(±Inf) = ±Pi/2
+func Atan(x float64) float64
+
+func atan(x float64) float64 {
+ if x == 0 {
+ return x
+ }
+ if x > 0 {
+ return satan(x)
+ }
+ return -satan(-x)
+}
diff --git a/src/math/atan2.go b/src/math/atan2.go
new file mode 100644
index 0000000..d84b332
--- /dev/null
+++ b/src/math/atan2.go
@@ -0,0 +1,71 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Atan2 returns the arc tangent of y/x, using
+// the signs of the two to determine the quadrant
+// of the return value.
+//
+// Special cases are (in order):
+// Atan2(y, NaN) = NaN
+// Atan2(NaN, x) = NaN
+// Atan2(+0, x>=0) = +0
+// Atan2(-0, x>=0) = -0
+// Atan2(+0, x<=-0) = +Pi
+// Atan2(-0, x<=-0) = -Pi
+// Atan2(y>0, 0) = +Pi/2
+// Atan2(y<0, 0) = -Pi/2
+// Atan2(+Inf, +Inf) = +Pi/4
+// Atan2(-Inf, +Inf) = -Pi/4
+// Atan2(+Inf, -Inf) = 3Pi/4
+// Atan2(-Inf, -Inf) = -3Pi/4
+// Atan2(y, +Inf) = 0
+// Atan2(y>0, -Inf) = +Pi
+// Atan2(y<0, -Inf) = -Pi
+// Atan2(+Inf, x) = +Pi/2
+// Atan2(-Inf, x) = -Pi/2
+func Atan2(y, x float64) float64
+
+func atan2(y, x float64) float64 {
+ // special cases
+ switch {
+ case IsNaN(y) || IsNaN(x):
+ return NaN()
+ case y == 0:
+ if x >= 0 && !Signbit(x) {
+ return Copysign(0, y)
+ }
+ return Copysign(Pi, y)
+ case x == 0:
+ return Copysign(Pi/2, y)
+ case IsInf(x, 0):
+ if IsInf(x, 1) {
+ switch {
+ case IsInf(y, 0):
+ return Copysign(Pi/4, y)
+ default:
+ return Copysign(0, y)
+ }
+ }
+ switch {
+ case IsInf(y, 0):
+ return Copysign(3*Pi/4, y)
+ default:
+ return Copysign(Pi, y)
+ }
+ case IsInf(y, 0):
+ return Copysign(Pi/2, y)
+ }
+
+ // Call atan and determine the quadrant.
+ q := Atan(y / x)
+ if x < 0 {
+ if q <= 0 {
+ return q + Pi
+ }
+ return q - Pi
+ }
+ return q
+}
diff --git a/src/math/atan2_386.s b/src/math/atan2_386.s
new file mode 100644
index 0000000..31a74e7
--- /dev/null
+++ b/src/math/atan2_386.s
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Atan2(y, x float64) float64 // =atan(y/x)
+TEXT ·Atan2(SB),NOSPLIT,$0
+ FMOVD y+0(FP), F0 // F0=y
+ FMOVD x+8(FP), F0 // F0=x, F1=y
+ FPATAN // F0=atan(F1/F0)
+ FMOVDP F0, ret+16(FP)
+ RET
diff --git a/src/math/atan2_amd64.s b/src/math/atan2_amd64.s
new file mode 100644
index 0000000..fc471f7
--- /dev/null
+++ b/src/math/atan2_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Atan2(SB),NOSPLIT,$0
+ JMP ·atan2(SB)
diff --git a/src/math/atan2_amd64p32.s b/src/math/atan2_amd64p32.s
new file mode 100644
index 0000000..3fdc03c
--- /dev/null
+++ b/src/math/atan2_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "atan2_amd64.s"
diff --git a/src/math/atan2_arm.s b/src/math/atan2_arm.s
new file mode 100644
index 0000000..06c12ec
--- /dev/null
+++ b/src/math/atan2_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Atan2(SB),NOSPLIT,$0
+ B ·atan2(SB)
diff --git a/src/math/atan_386.s b/src/math/atan_386.s
new file mode 100644
index 0000000..f3976b1
--- /dev/null
+++ b/src/math/atan_386.s
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Atan(x float64) float64
+TEXT ·Atan(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FLD1 // F0=1, F1=x
+ FPATAN // F0=atan(F1/F0)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/atan_amd64.s b/src/math/atan_amd64.s
new file mode 100644
index 0000000..b801ae9
--- /dev/null
+++ b/src/math/atan_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Atan(SB),NOSPLIT,$0
+ JMP ·atan(SB)
diff --git a/src/math/atan_amd64p32.s b/src/math/atan_amd64p32.s
new file mode 100644
index 0000000..1c1f6ce
--- /dev/null
+++ b/src/math/atan_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "atan_amd64.s"
diff --git a/src/math/atan_arm.s b/src/math/atan_arm.s
new file mode 100644
index 0000000..d190a8b
--- /dev/null
+++ b/src/math/atan_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Atan(SB),NOSPLIT,$0
+ B ·atan(SB)
diff --git a/src/math/atanh.go b/src/math/atanh.go
new file mode 100644
index 0000000..113d5c1
--- /dev/null
+++ b/src/math/atanh.go
@@ -0,0 +1,77 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_atanh.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// __ieee754_atanh(x)
+// Method :
+// 1. Reduce x to positive by atanh(-x) = -atanh(x)
+// 2. For x>=0.5
+// 1 2x x
+// atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
+// 2 1 - x 1 - x
+//
+// For x<0.5
+// atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
+//
+// Special cases:
+// atanh(x) is NaN if |x| > 1 with signal;
+// atanh(NaN) is that NaN with no signal;
+// atanh(+-1) is +-INF with signal.
+//
+
+// Atanh returns the inverse hyperbolic tangent of x.
+//
+// Special cases are:
+// Atanh(1) = +Inf
+// Atanh(±0) = ±0
+// Atanh(-1) = -Inf
+// Atanh(x) = NaN if x < -1 or x > 1
+// Atanh(NaN) = NaN
+func Atanh(x float64) float64 {
+ const NearZero = 1.0 / (1 << 28) // 2**-28
+ // special cases
+ switch {
+ case x < -1 || x > 1 || IsNaN(x):
+ return NaN()
+ case x == 1:
+ return Inf(1)
+ case x == -1:
+ return Inf(-1)
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ var temp float64
+ switch {
+ case x < NearZero:
+ temp = x
+ case x < 0.5:
+ temp = x + x
+ temp = 0.5 * Log1p(temp+temp*x/(1-x))
+ default:
+ temp = 0.5 * Log1p((x+x)/(1-x))
+ }
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
diff --git a/src/math/big/arith.go b/src/math/big/arith.go
new file mode 100644
index 0000000..3d5a868
--- /dev/null
+++ b/src/math/big/arith.go
@@ -0,0 +1,240 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file provides Go implementations of elementary multi-precision
+// arithmetic operations on word vectors. Needed for platforms without
+// assembly implementations of these routines.
+
+package big
+
+// A Word represents a single digit of a multi-precision unsigned integer.
+type Word uintptr
+
+const (
+ // Compute the size _S of a Word in bytes.
+ _m = ^Word(0)
+ _logS = _m>>8&1 + _m>>16&1 + _m>>32&1
+ _S = 1 << _logS
+
+ _W = _S << 3 // word size in bits
+ _B = 1 << _W // digit base
+ _M = _B - 1 // digit mask
+
+ _W2 = _W / 2 // half word size in bits
+ _B2 = 1 << _W2 // half digit base
+ _M2 = _B2 - 1 // half digit mask
+)
+
+// ----------------------------------------------------------------------------
+// Elementary operations on words
+//
+// These operations are used by the vector operations below.
+
+// z1<<_W + z0 = x+y+c, with c == 0 or 1
+func addWW_g(x, y, c Word) (z1, z0 Word) {
+ yc := y + c
+ z0 = x + yc
+ if z0 < x || yc < y {
+ z1 = 1
+ }
+ return
+}
+
+// z1<<_W + z0 = x-y-c, with c == 0 or 1
+func subWW_g(x, y, c Word) (z1, z0 Word) {
+ yc := y + c
+ z0 = x - yc
+ if z0 > x || yc < y {
+ z1 = 1
+ }
+ return
+}
+
+// z1<<_W + z0 = x*y
+// Adapted from Warren, Hacker's Delight, p. 132.
+func mulWW_g(x, y Word) (z1, z0 Word) {
+ x0 := x & _M2
+ x1 := x >> _W2
+ y0 := y & _M2
+ y1 := y >> _W2
+ w0 := x0 * y0
+ t := x1*y0 + w0>>_W2
+ w1 := t & _M2
+ w2 := t >> _W2
+ w1 += x0 * y1
+ z1 = x1*y1 + w2 + w1>>_W2
+ z0 = x * y
+ return
+}
+
+// z1<<_W + z0 = x*y + c
+func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
+ z1, zz0 := mulWW(x, y)
+ if z0 = zz0 + c; z0 < zz0 {
+ z1++
+ }
+ return
+}
+
+// Length of x in bits.
+func bitLen_g(x Word) (n int) {
+ for ; x >= 0x8000; x >>= 16 {
+ n += 16
+ }
+ if x >= 0x80 {
+ x >>= 8
+ n += 8
+ }
+ if x >= 0x8 {
+ x >>= 4
+ n += 4
+ }
+ if x >= 0x2 {
+ x >>= 2
+ n += 2
+ }
+ if x >= 0x1 {
+ n++
+ }
+ return
+}
+
+// log2 computes the integer binary logarithm of x.
+// The result is the integer n for which 2^n <= x < 2^(n+1).
+// If x == 0, the result is -1.
+func log2(x Word) int {
+ return bitLen(x) - 1
+}
+
+// Number of leading zeros in x.
+func leadingZeros(x Word) uint {
+ return uint(_W - bitLen(x))
+}
+
+// q = (u1<<_W + u0 - r)/y
+// Adapted from Warren, Hacker's Delight, p. 152.
+func divWW_g(u1, u0, v Word) (q, r Word) {
+ if u1 >= v {
+ return 1<<_W - 1, 1<<_W - 1
+ }
+
+ s := leadingZeros(v)
+ v <<= s
+
+ vn1 := v >> _W2
+ vn0 := v & _M2
+ un32 := u1<<s | u0>>(_W-s)
+ un10 := u0 << s
+ un1 := un10 >> _W2
+ un0 := un10 & _M2
+ q1 := un32 / vn1
+ rhat := un32 - q1*vn1
+
+ for q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
+ q1--
+ rhat += vn1
+ if rhat >= _B2 {
+ break
+ }
+ }
+
+ un21 := un32*_B2 + un1 - q1*v
+ q0 := un21 / vn1
+ rhat = un21 - q0*vn1
+
+ for q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
+ q0--
+ rhat += vn1
+ if rhat >= _B2 {
+ break
+ }
+ }
+
+ return q1*_B2 + q0, (un21*_B2 + un0 - q0*v) >> s
+}
+
+func addVV_g(z, x, y []Word) (c Word) {
+ for i := range z {
+ c, z[i] = addWW_g(x[i], y[i], c)
+ }
+ return
+}
+
+func subVV_g(z, x, y []Word) (c Word) {
+ for i := range z {
+ c, z[i] = subWW_g(x[i], y[i], c)
+ }
+ return
+}
+
+func addVW_g(z, x []Word, y Word) (c Word) {
+ c = y
+ for i := range z {
+ c, z[i] = addWW_g(x[i], c, 0)
+ }
+ return
+}
+
+func subVW_g(z, x []Word, y Word) (c Word) {
+ c = y
+ for i := range z {
+ c, z[i] = subWW_g(x[i], c, 0)
+ }
+ return
+}
+
+func shlVU_g(z, x []Word, s uint) (c Word) {
+ if n := len(z); n > 0 {
+ ŝ := _W - s
+ w1 := x[n-1]
+ c = w1 >> ŝ
+ for i := n - 1; i > 0; i-- {
+ w := w1
+ w1 = x[i-1]
+ z[i] = w<<s | w1>>ŝ
+ }
+ z[0] = w1 << s
+ }
+ return
+}
+
+func shrVU_g(z, x []Word, s uint) (c Word) {
+ if n := len(z); n > 0 {
+ ŝ := _W - s
+ w1 := x[0]
+ c = w1 << ŝ
+ for i := 0; i < n-1; i++ {
+ w := w1
+ w1 = x[i+1]
+ z[i] = w>>s | w1<<ŝ
+ }
+ z[n-1] = w1 >> s
+ }
+ return
+}
+
+func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
+ c = r
+ for i := range z {
+ c, z[i] = mulAddWWW_g(x[i], y, c)
+ }
+ return
+}
+
+func addMulVVW_g(z, x []Word, y Word) (c Word) {
+ for i := range z {
+ z1, z0 := mulAddWWW_g(x[i], y, z[i])
+ c, z[i] = addWW_g(z0, c, 0)
+ c += z1
+ }
+ return
+}
+
+func divWVW_g(z []Word, xn Word, x []Word, y Word) (r Word) {
+ r = xn
+ for i := len(z) - 1; i >= 0; i-- {
+ z[i], r = divWW_g(r, x[i], y)
+ }
+ return
+}
diff --git a/src/math/big/arith_386.s b/src/math/big/arith_386.s
new file mode 100644
index 0000000..1b47c89
--- /dev/null
+++ b/src/math/big/arith_386.s
@@ -0,0 +1,278 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVL x+0(FP), AX
+ MULL y+4(FP)
+ MOVL DX, z1+8(FP)
+ MOVL AX, z0+12(FP)
+ RET
+
+
+// func divWW(x1, x0, y Word) (q, r Word)
+TEXT ·divWW(SB),NOSPLIT,$0
+ MOVL x1+0(FP), DX
+ MOVL x0+4(FP), AX
+ DIVL y+8(FP)
+ MOVL AX, q+12(FP)
+ MOVL DX, r+16(FP)
+ RET
+
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), CX
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ MOVL $0, DX // c = 0
+ JMP E1
+
+L1: MOVL (SI)(BX*4), AX
+ RCRL $1, DX
+ ADCL (CX)(BX*4), AX
+ RCLL $1, DX
+ MOVL AX, (DI)(BX*4)
+ ADDL $1, BX // i++
+
+E1: CMPL BX, BP // i < n
+ JL L1
+
+ MOVL DX, c+36(FP)
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBBL instead of ADCL and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), CX
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ MOVL $0, DX // c = 0
+ JMP E2
+
+L2: MOVL (SI)(BX*4), AX
+ RCRL $1, DX
+ SBBL (CX)(BX*4), AX
+ RCLL $1, DX
+ MOVL AX, (DI)(BX*4)
+ ADDL $1, BX // i++
+
+E2: CMPL BX, BP // i < n
+ JL L2
+
+ MOVL DX, c+36(FP)
+ RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), AX // c = y
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ JMP E3
+
+L3: ADDL (SI)(BX*4), AX
+ MOVL AX, (DI)(BX*4)
+ RCLL $1, AX
+ ANDL $1, AX
+ ADDL $1, BX // i++
+
+E3: CMPL BX, BP // i < n
+ JL L3
+
+ MOVL AX, c+28(FP)
+ RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), AX // c = y
+ MOVL z_len+4(FP), BP
+ MOVL $0, BX // i = 0
+ JMP E4
+
+L4: MOVL (SI)(BX*4), DX // TODO(gri) is there a reverse SUBL?
+ SUBL AX, DX
+ MOVL DX, (DI)(BX*4)
+ RCLL $1, AX
+ ANDL $1, AX
+ ADDL $1, BX // i++
+
+E4: CMPL BX, BP // i < n
+ JL L4
+
+ MOVL AX, c+28(FP)
+ RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+ MOVL z_len+4(FP), BX // i = z
+ SUBL $1, BX // i--
+ JL X8b // i < 0 (n <= 0)
+
+ // n > 0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL s+24(FP), CX
+ MOVL (SI)(BX*4), AX // w1 = x[n-1]
+ MOVL $0, DX
+ SHLL CX, DX:AX // w1>>ŝ
+ MOVL DX, c+28(FP)
+
+ CMPL BX, $0
+ JLE X8a // i <= 0
+
+ // i > 0
+L8: MOVL AX, DX // w = w1
+ MOVL -4(SI)(BX*4), AX // w1 = x[i-1]
+ SHLL CX, DX:AX // w<<s | w1>>ŝ
+ MOVL DX, (DI)(BX*4) // z[i] = w<<s | w1>>ŝ
+ SUBL $1, BX // i--
+ JG L8 // i > 0
+
+ // i <= 0
+X8a: SHLL CX, AX // w1<<s
+ MOVL AX, (DI) // z[0] = w1<<s
+ RET
+
+X8b: MOVL $0, c+28(FP)
+ RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVL z_len+4(FP), BP
+ SUBL $1, BP // n--
+ JL X9b // n < 0 (n <= 0)
+
+ // n > 0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL s+24(FP), CX
+ MOVL (SI), AX // w1 = x[0]
+ MOVL $0, DX
+ SHRL CX, DX:AX // w1<<ŝ
+ MOVL DX, c+28(FP)
+
+ MOVL $0, BX // i = 0
+ JMP E9
+
+ // i < n-1
+L9: MOVL AX, DX // w = w1
+ MOVL 4(SI)(BX*4), AX // w1 = x[i+1]
+ SHRL CX, DX:AX // w>>s | w1<<ŝ
+ MOVL DX, (DI)(BX*4) // z[i] = w>>s | w1<<ŝ
+ ADDL $1, BX // i++
+
+E9: CMPL BX, BP
+ JL L9 // i < n-1
+
+ // i >= n-1
+X9a: SHRL CX, AX // w1>>s
+ MOVL AX, (DI)(BP*4) // z[n-1] = w1>>s
+ RET
+
+X9b: MOVL $0, c+28(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), BP
+ MOVL r+28(FP), CX // c = r
+ MOVL z_len+4(FP), BX
+ LEAL (DI)(BX*4), DI
+ LEAL (SI)(BX*4), SI
+ NEGL BX // i = -n
+ JMP E5
+
+L5: MOVL (SI)(BX*4), AX
+ MULL BP
+ ADDL CX, AX
+ ADCL $0, DX
+ MOVL AX, (DI)(BX*4)
+ MOVL DX, CX
+ ADDL $1, BX // i++
+
+E5: CMPL BX, $0 // i < 0
+ JL L5
+
+ MOVL CX, c+32(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL x+12(FP), SI
+ MOVL y+24(FP), BP
+ MOVL z_len+4(FP), BX
+ LEAL (DI)(BX*4), DI
+ LEAL (SI)(BX*4), SI
+ NEGL BX // i = -n
+ MOVL $0, CX // c = 0
+ JMP E6
+
+L6: MOVL (SI)(BX*4), AX
+ MULL BP
+ ADDL CX, AX
+ ADCL $0, DX
+ ADDL AX, (DI)(BX*4)
+ ADCL $0, DX
+ MOVL DX, CX
+ ADDL $1, BX // i++
+
+E6: CMPL BX, $0 // i < 0
+ JL L6
+
+ MOVL CX, c+28(FP)
+ RET
+
+
+// func divWVW(z* Word, xn Word, x []Word, y Word) (r Word)
+TEXT ·divWVW(SB),NOSPLIT,$0
+ MOVL z+0(FP), DI
+ MOVL xn+12(FP), DX // r = xn
+ MOVL x+16(FP), SI
+ MOVL y+28(FP), CX
+ MOVL z_len+4(FP), BX // i = z
+ JMP E7
+
+L7: MOVL (SI)(BX*4), AX
+ DIVL CX
+ MOVL AX, (DI)(BX*4)
+
+E7: SUBL $1, BX // i--
+ JGE L7 // i >= 0
+
+ MOVL DX, r+32(FP)
+ RET
+
+// func bitLen(x Word) (n int)
+TEXT ·bitLen(SB),NOSPLIT,$0
+ BSRL x+0(FP), AX
+ JZ Z1
+ INCL AX
+ MOVL AX, n+4(FP)
+ RET
+
+Z1: MOVL $0, n+4(FP)
+ RET
diff --git a/src/math/big/arith_amd64.s b/src/math/big/arith_amd64.s
new file mode 100644
index 0000000..56c4cb0
--- /dev/null
+++ b/src/math/big/arith_amd64.s
@@ -0,0 +1,401 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// Literal instruction for MOVQ $0, CX.
+// (MOVQ $0, reg is translated to XORQ reg, reg and clears CF.)
+#define ZERO_CX BYTE $0x48; \
+ BYTE $0xc7; \
+ BYTE $0xc1; \
+ BYTE $0x00; \
+ BYTE $0x00; \
+ BYTE $0x00; \
+ BYTE $0x00
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MULQ y+8(FP)
+ MOVQ DX, z1+16(FP)
+ MOVQ AX, z0+24(FP)
+ RET
+
+
+// func divWW(x1, x0, y Word) (q, r Word)
+TEXT ·divWW(SB),NOSPLIT,$0
+ MOVQ x1+0(FP), DX
+ MOVQ x0+8(FP), AX
+ DIVQ y+16(FP)
+ MOVQ AX, q+24(FP)
+ MOVQ DX, r+32(FP)
+ RET
+
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, CX // c = 0
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V1 // if n < 0 goto V1
+
+U1: // n >= 0
+ // regular loop body unrolled 4x
+ RCRQ $1, CX // CF = c
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ ADCQ 0(R9)(SI*8), R11
+ ADCQ 8(R9)(SI*8), R12
+ ADCQ 16(R9)(SI*8), R13
+ ADCQ 24(R9)(SI*8), R14
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+ RCLQ $1, CX // c = CF
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U1 // if n >= 0 goto U1
+
+V1: ADDQ $4, DI // n += 4
+ JLE E1 // if n <= 0 goto E1
+
+L1: // n > 0
+ RCRQ $1, CX // CF = c
+ MOVQ 0(R8)(SI*8), R11
+ ADCQ 0(R9)(SI*8), R11
+ MOVQ R11, 0(R10)(SI*8)
+ RCLQ $1, CX // c = CF
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L1 // if n > 0 goto L1
+
+E1: MOVQ CX, c+72(FP) // return c
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBBQ instead of ADCQ and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, CX // c = 0
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V2 // if n < 0 goto V2
+
+U2: // n >= 0
+ // regular loop body unrolled 4x
+ RCRQ $1, CX // CF = c
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ SBBQ 0(R9)(SI*8), R11
+ SBBQ 8(R9)(SI*8), R12
+ SBBQ 16(R9)(SI*8), R13
+ SBBQ 24(R9)(SI*8), R14
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+ RCLQ $1, CX // c = CF
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U2 // if n >= 0 goto U2
+
+V2: ADDQ $4, DI // n += 4
+ JLE E2 // if n <= 0 goto E2
+
+L2: // n > 0
+ RCRQ $1, CX // CF = c
+ MOVQ 0(R8)(SI*8), R11
+ SBBQ 0(R9)(SI*8), R11
+ MOVQ R11, 0(R10)(SI*8)
+ RCLQ $1, CX // c = CF
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L2 // if n > 0 goto L2
+
+E2: MOVQ CX, c+72(FP) // return c
+ RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), CX // c = y
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V3 // if n < 4 goto V3
+
+U3: // n >= 0
+ // regular loop body unrolled 4x
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ ADDQ CX, R11
+ ZERO_CX
+ ADCQ $0, R12
+ ADCQ $0, R13
+ ADCQ $0, R14
+ SETCS CX // c = CF
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U3 // if n >= 0 goto U3
+
+V3: ADDQ $4, DI // n += 4
+ JLE E3 // if n <= 0 goto E3
+
+L3: // n > 0
+ ADDQ 0(R8)(SI*8), CX
+ MOVQ CX, 0(R10)(SI*8)
+ ZERO_CX
+ RCLQ $1, CX // c = CF
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L3 // if n > 0 goto L3
+
+E3: MOVQ CX, c+56(FP) // return c
+ RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+// (same as addVW except for SUBQ/SBBQ instead of ADDQ/ADCQ and label names)
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), DI
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), CX // c = y
+ MOVQ z+0(FP), R10
+
+ MOVQ $0, SI // i = 0
+
+ // s/JL/JMP/ below to disable the unrolled loop
+ SUBQ $4, DI // n -= 4
+ JL V4 // if n < 4 goto V4
+
+U4: // n >= 0
+ // regular loop body unrolled 4x
+ MOVQ 0(R8)(SI*8), R11
+ MOVQ 8(R8)(SI*8), R12
+ MOVQ 16(R8)(SI*8), R13
+ MOVQ 24(R8)(SI*8), R14
+ SUBQ CX, R11
+ ZERO_CX
+ SBBQ $0, R12
+ SBBQ $0, R13
+ SBBQ $0, R14
+ SETCS CX // c = CF
+ MOVQ R11, 0(R10)(SI*8)
+ MOVQ R12, 8(R10)(SI*8)
+ MOVQ R13, 16(R10)(SI*8)
+ MOVQ R14, 24(R10)(SI*8)
+
+ ADDQ $4, SI // i += 4
+ SUBQ $4, DI // n -= 4
+ JGE U4 // if n >= 0 goto U4
+
+V4: ADDQ $4, DI // n += 4
+ JLE E4 // if n <= 0 goto E4
+
+L4: // n > 0
+ MOVQ 0(R8)(SI*8), R11
+ SUBQ CX, R11
+ MOVQ R11, 0(R10)(SI*8)
+ ZERO_CX
+ RCLQ $1, CX // c = CF
+
+ ADDQ $1, SI // i++
+ SUBQ $1, DI // n--
+ JG L4 // if n > 0 goto L4
+
+E4: MOVQ CX, c+56(FP) // return c
+ RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), BX // i = z
+ SUBQ $1, BX // i--
+ JL X8b // i < 0 (n <= 0)
+
+ // n > 0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ s+48(FP), CX
+ MOVQ (R8)(BX*8), AX // w1 = x[n-1]
+ MOVQ $0, DX
+ SHLQ CX, DX:AX // w1>>ŝ
+ MOVQ DX, c+56(FP)
+
+ CMPQ BX, $0
+ JLE X8a // i <= 0
+
+ // i > 0
+L8: MOVQ AX, DX // w = w1
+ MOVQ -8(R8)(BX*8), AX // w1 = x[i-1]
+ SHLQ CX, DX:AX // w<<s | w1>>ŝ
+ MOVQ DX, (R10)(BX*8) // z[i] = w<<s | w1>>ŝ
+ SUBQ $1, BX // i--
+ JG L8 // i > 0
+
+ // i <= 0
+X8a: SHLQ CX, AX // w1<<s
+ MOVQ AX, (R10) // z[0] = w1<<s
+ RET
+
+X8b: MOVQ $0, c+56(FP)
+ RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVQ z_len+8(FP), R11
+ SUBQ $1, R11 // n--
+ JL X9b // n < 0 (n <= 0)
+
+ // n > 0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ s+48(FP), CX
+ MOVQ (R8), AX // w1 = x[0]
+ MOVQ $0, DX
+ SHRQ CX, DX:AX // w1<<ŝ
+ MOVQ DX, c+56(FP)
+
+ MOVQ $0, BX // i = 0
+ JMP E9
+
+ // i < n-1
+L9: MOVQ AX, DX // w = w1
+ MOVQ 8(R8)(BX*8), AX // w1 = x[i+1]
+ SHRQ CX, DX:AX // w>>s | w1<<ŝ
+ MOVQ DX, (R10)(BX*8) // z[i] = w>>s | w1<<ŝ
+ ADDQ $1, BX // i++
+
+E9: CMPQ BX, R11
+ JL L9 // i < n-1
+
+ // i >= n-1
+X9a: SHRQ CX, AX // w1>>s
+ MOVQ AX, (R10)(R11*8) // z[n-1] = w1>>s
+ RET
+
+X9b: MOVQ $0, c+56(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ r+56(FP), CX // c = r
+ MOVQ z_len+8(FP), R11
+ MOVQ $0, BX // i = 0
+ JMP E5
+
+L5: MOVQ (R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ MOVQ AX, (R10)(BX*8)
+ MOVQ DX, CX
+ ADDQ $1, BX // i++
+
+E5: CMPQ BX, R11 // i < n
+ JL L5
+
+ MOVQ CX, c+64(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ MOVQ z+0(FP), R10
+ MOVQ x+24(FP), R8
+ MOVQ y+48(FP), R9
+ MOVQ z_len+8(FP), R11
+ MOVQ $0, BX // i = 0
+ MOVQ $0, CX // c = 0
+ JMP E6
+
+L6: MOVQ (R8)(BX*8), AX
+ MULQ R9
+ ADDQ CX, AX
+ ADCQ $0, DX
+ ADDQ AX, (R10)(BX*8)
+ ADCQ $0, DX
+ MOVQ DX, CX
+ ADDQ $1, BX // i++
+
+E6: CMPQ BX, R11 // i < n
+ JL L6
+
+ MOVQ CX, c+56(FP)
+ RET
+
+
+// func divWVW(z []Word, xn Word, x []Word, y Word) (r Word)
+TEXT ·divWVW(SB),NOSPLIT,$0
+ MOVQ z+0(FP), R10
+ MOVQ xn+24(FP), DX // r = xn
+ MOVQ x+32(FP), R8
+ MOVQ y+56(FP), R9
+ MOVQ z_len+8(FP), BX // i = z
+ JMP E7
+
+L7: MOVQ (R8)(BX*8), AX
+ DIVQ R9
+ MOVQ AX, (R10)(BX*8)
+
+E7: SUBQ $1, BX // i--
+ JGE L7 // i >= 0
+
+ MOVQ DX, r+64(FP)
+ RET
+
+// func bitLen(x Word) (n int)
+TEXT ·bitLen(SB),NOSPLIT,$0
+ BSRQ x+0(FP), AX
+ JZ Z1
+ ADDQ $1, AX
+ MOVQ AX, n+8(FP)
+ RET
+
+Z1: MOVQ $0, n+8(FP)
+ RET
diff --git a/src/math/big/arith_amd64p32.s b/src/math/big/arith_amd64p32.s
new file mode 100644
index 0000000..908dbbd
--- /dev/null
+++ b/src/math/big/arith_amd64p32.s
@@ -0,0 +1,41 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·mulWW(SB),NOSPLIT,$0
+ JMP ·mulWW_g(SB)
+
+TEXT ·divWW(SB),NOSPLIT,$0
+ JMP ·divWW_g(SB)
+
+TEXT ·addVV(SB),NOSPLIT,$0
+ JMP ·addVV_g(SB)
+
+TEXT ·subVV(SB),NOSPLIT,$0
+ JMP ·subVV_g(SB)
+
+TEXT ·addVW(SB),NOSPLIT,$0
+ JMP ·addVW_g(SB)
+
+TEXT ·subVW(SB),NOSPLIT,$0
+ JMP ·subVW_g(SB)
+
+TEXT ·shlVU(SB),NOSPLIT,$0
+ JMP ·shlVU_g(SB)
+
+TEXT ·shrVU(SB),NOSPLIT,$0
+ JMP ·shrVU_g(SB)
+
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ JMP ·mulAddVWW_g(SB)
+
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ JMP ·addMulVVW_g(SB)
+
+TEXT ·divWVW(SB),NOSPLIT,$0
+ JMP ·divWVW_g(SB)
+
+TEXT ·bitLen(SB),NOSPLIT,$0
+ JMP ·bitLen_g(SB)
diff --git a/src/math/big/arith_arm.s b/src/math/big/arith_arm.s
new file mode 100644
index 0000000..a4c51c2
--- /dev/null
+++ b/src/math/big/arith_arm.s
@@ -0,0 +1,300 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// This file provides fast assembly versions for the elementary
+// arithmetic operations on vectors implemented in arith.go.
+
+// func addVV(z, x, y []Word) (c Word)
+TEXT ·addVV(SB),NOSPLIT,$0
+ ADD.S $0, R0 // clear carry flag
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ B E1
+L1:
+ MOVW.P 4(R2), R5
+ MOVW.P 4(R3), R6
+ ADC.S R6, R5
+ MOVW.P R5, 4(R1)
+E1:
+ TEQ R1, R4
+ BNE L1
+
+ MOVW $0, R0
+ MOVW.CS $1, R0
+ MOVW R0, c+36(FP)
+ RET
+
+
+// func subVV(z, x, y []Word) (c Word)
+// (same as addVV except for SBC instead of ADC and label names)
+TEXT ·subVV(SB),NOSPLIT,$0
+ SUB.S $0, R0 // clear borrow flag
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ B E2
+L2:
+ MOVW.P 4(R2), R5
+ MOVW.P 4(R3), R6
+ SBC.S R6, R5
+ MOVW.P R5, 4(R1)
+E2:
+ TEQ R1, R4
+ BNE L2
+
+ MOVW $0, R0
+ MOVW.CC $1, R0
+ MOVW R0, c+36(FP)
+ RET
+
+
+// func addVW(z, x []Word, y Word) (c Word)
+TEXT ·addVW(SB),NOSPLIT,$0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ TEQ R1, R4
+ BNE L3a
+ MOVW R3, c+28(FP)
+ RET
+L3a:
+ MOVW.P 4(R2), R5
+ ADD.S R3, R5
+ MOVW.P R5, 4(R1)
+ B E3
+L3:
+ MOVW.P 4(R2), R5
+ ADC.S $0, R5
+ MOVW.P R5, 4(R1)
+E3:
+ TEQ R1, R4
+ BNE L3
+
+ MOVW $0, R0
+ MOVW.CS $1, R0
+ MOVW R0, c+28(FP)
+ RET
+
+
+// func subVW(z, x []Word, y Word) (c Word)
+TEXT ·subVW(SB),NOSPLIT,$0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R4
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R4<<2, R1, R4
+ TEQ R1, R4
+ BNE L4a
+ MOVW R3, c+28(FP)
+ RET
+L4a:
+ MOVW.P 4(R2), R5
+ SUB.S R3, R5
+ MOVW.P R5, 4(R1)
+ B E4
+L4:
+ MOVW.P 4(R2), R5
+ SBC.S $0, R5
+ MOVW.P R5, 4(R1)
+E4:
+ TEQ R1, R4
+ BNE L4
+
+ MOVW $0, R0
+ MOVW.CC $1, R0
+ MOVW R0, c+28(FP)
+ RET
+
+
+// func shlVU(z, x []Word, s uint) (c Word)
+TEXT ·shlVU(SB),NOSPLIT,$0
+ MOVW z_len+4(FP), R5
+ TEQ $0, R5
+ BEQ X7
+
+ MOVW z+0(FP), R1
+ MOVW x+12(FP), R2
+ ADD R5<<2, R2, R2
+ ADD R5<<2, R1, R5
+ MOVW s+24(FP), R3
+ TEQ $0, R3 // shift 0 is special
+ BEQ Y7
+ ADD $4, R1 // stop one word early
+ MOVW $32, R4
+ SUB R3, R4
+ MOVW $0, R7
+
+ MOVW.W -4(R2), R6
+ MOVW R6<<R3, R7
+ MOVW R6>>R4, R6
+ MOVW R6, c+28(FP)
+ B E7
+
+L7:
+ MOVW.W -4(R2), R6
+ ORR R6>>R4, R7
+ MOVW.W R7, -4(R5)
+ MOVW R6<<R3, R7
+E7:
+ TEQ R1, R5
+ BNE L7
+
+ MOVW R7, -4(R5)
+ RET
+
+Y7: // copy loop, because shift 0 == shift 32
+ MOVW.W -4(R2), R6
+ MOVW.W R6, -4(R5)
+ TEQ R1, R5
+ BNE Y7
+
+X7:
+ MOVW $0, R1
+ MOVW R1, c+28(FP)
+ RET
+
+
+// func shrVU(z, x []Word, s uint) (c Word)
+TEXT ·shrVU(SB),NOSPLIT,$0
+ MOVW z_len+4(FP), R5
+ TEQ $0, R5
+ BEQ X6
+
+ MOVW z+0(FP), R1
+ MOVW x+12(FP), R2
+ ADD R5<<2, R1, R5
+ MOVW s+24(FP), R3
+ TEQ $0, R3 // shift 0 is special
+ BEQ Y6
+ SUB $4, R5 // stop one word early
+ MOVW $32, R4
+ SUB R3, R4
+ MOVW $0, R7
+
+ // first word
+ MOVW.P 4(R2), R6
+ MOVW R6>>R3, R7
+ MOVW R6<<R4, R6
+ MOVW R6, c+28(FP)
+ B E6
+
+ // word loop
+L6:
+ MOVW.P 4(R2), R6
+ ORR R6<<R4, R7
+ MOVW.P R7, 4(R1)
+ MOVW R6>>R3, R7
+E6:
+ TEQ R1, R5
+ BNE L6
+
+ MOVW R7, 0(R1)
+ RET
+
+Y6: // copy loop, because shift 0 == shift 32
+ MOVW.P 4(R2), R6
+ MOVW.P R6, 4(R1)
+ TEQ R1, R5
+ BNE Y6
+
+X6:
+ MOVW $0, R1
+ MOVW R1, c+28(FP)
+ RET
+
+
+// func mulAddVWW(z, x []Word, y, r Word) (c Word)
+TEXT ·mulAddVWW(SB),NOSPLIT,$0
+ MOVW $0, R0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R5
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ MOVW r+28(FP), R4
+ ADD R5<<2, R1, R5
+ B E8
+
+ // word loop
+L8:
+ MOVW.P 4(R2), R6
+ MULLU R6, R3, (R7, R6)
+ ADD.S R4, R6
+ ADC R0, R7
+ MOVW.P R6, 4(R1)
+ MOVW R7, R4
+E8:
+ TEQ R1, R5
+ BNE L8
+
+ MOVW R4, c+32(FP)
+ RET
+
+
+// func addMulVVW(z, x []Word, y Word) (c Word)
+TEXT ·addMulVVW(SB),NOSPLIT,$0
+ MOVW $0, R0
+ MOVW z+0(FP), R1
+ MOVW z_len+4(FP), R5
+ MOVW x+12(FP), R2
+ MOVW y+24(FP), R3
+ ADD R5<<2, R1, R5
+ MOVW $0, R4
+ B E9
+
+ // word loop
+L9:
+ MOVW.P 4(R2), R6
+ MULLU R6, R3, (R7, R6)
+ ADD.S R4, R6
+ ADC R0, R7
+ MOVW 0(R1), R4
+ ADD.S R4, R6
+ ADC R0, R7
+ MOVW.P R6, 4(R1)
+ MOVW R7, R4
+E9:
+ TEQ R1, R5
+ BNE L9
+
+ MOVW R4, c+28(FP)
+ RET
+
+
+// func divWVW(z* Word, xn Word, x []Word, y Word) (r Word)
+TEXT ·divWVW(SB),NOSPLIT,$0
+ // ARM has no multiword division, so use portable code.
+ B ·divWVW_g(SB)
+
+
+// func divWW(x1, x0, y Word) (q, r Word)
+TEXT ·divWW(SB),NOSPLIT,$0
+ // ARM has no multiword division, so use portable code.
+ B ·divWW_g(SB)
+
+
+// func mulWW(x, y Word) (z1, z0 Word)
+TEXT ·mulWW(SB),NOSPLIT,$0
+ MOVW x+0(FP), R1
+ MOVW y+4(FP), R2
+ MULLU R1, R2, (R4, R3)
+ MOVW R4, z1+8(FP)
+ MOVW R3, z0+12(FP)
+ RET
+
+// func bitLen(x Word) (n int)
+TEXT ·bitLen(SB),NOSPLIT,$0
+ MOVW x+0(FP), R0
+ CLZ R0, R0
+ RSB $32, R0
+ MOVW R0, n+4(FP)
+ RET
diff --git a/src/math/big/arith_decl.go b/src/math/big/arith_decl.go
new file mode 100644
index 0000000..068cc8d
--- /dev/null
+++ b/src/math/big/arith_decl.go
@@ -0,0 +1,19 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+// implemented in arith_$GOARCH.s
+func mulWW(x, y Word) (z1, z0 Word)
+func divWW(x1, x0, y Word) (q, r Word)
+func addVV(z, x, y []Word) (c Word)
+func subVV(z, x, y []Word) (c Word)
+func addVW(z, x []Word, y Word) (c Word)
+func subVW(z, x []Word, y Word) (c Word)
+func shlVU(z, x []Word, s uint) (c Word)
+func shrVU(z, x []Word, s uint) (c Word)
+func mulAddVWW(z, x []Word, y, r Word) (c Word)
+func addMulVVW(z, x []Word, y Word) (c Word)
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word)
+func bitLen(x Word) (n int)
diff --git a/src/math/big/arith_test.go b/src/math/big/arith_test.go
new file mode 100644
index 0000000..3615a65
--- /dev/null
+++ b/src/math/big/arith_test.go
@@ -0,0 +1,456 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "math/rand"
+ "testing"
+)
+
+type funWW func(x, y, c Word) (z1, z0 Word)
+type argWW struct {
+ x, y, c, z1, z0 Word
+}
+
+var sumWW = []argWW{
+ {0, 0, 0, 0, 0},
+ {0, 1, 0, 0, 1},
+ {0, 0, 1, 0, 1},
+ {0, 1, 1, 0, 2},
+ {12345, 67890, 0, 0, 80235},
+ {12345, 67890, 1, 0, 80236},
+ {_M, 1, 0, 1, 0},
+ {_M, 0, 1, 1, 0},
+ {_M, 1, 1, 1, 1},
+ {_M, _M, 0, 1, _M - 1},
+ {_M, _M, 1, 1, _M},
+}
+
+func testFunWW(t *testing.T, msg string, f funWW, a argWW) {
+ z1, z0 := f(a.x, a.y, a.c)
+ if z1 != a.z1 || z0 != a.z0 {
+ t.Errorf("%s%+v\n\tgot z1:z0 = %#x:%#x; want %#x:%#x", msg, a, z1, z0, a.z1, a.z0)
+ }
+}
+
+func TestFunWW(t *testing.T) {
+ for _, a := range sumWW {
+ arg := a
+ testFunWW(t, "addWW_g", addWW_g, arg)
+
+ arg = argWW{a.y, a.x, a.c, a.z1, a.z0}
+ testFunWW(t, "addWW_g symmetric", addWW_g, arg)
+
+ arg = argWW{a.z0, a.x, a.c, a.z1, a.y}
+ testFunWW(t, "subWW_g", subWW_g, arg)
+
+ arg = argWW{a.z0, a.y, a.c, a.z1, a.x}
+ testFunWW(t, "subWW_g symmetric", subWW_g, arg)
+ }
+}
+
+type funVV func(z, x, y []Word) (c Word)
+type argVV struct {
+ z, x, y nat
+ c Word
+}
+
+var sumVV = []argVV{
+ {},
+ {nat{0}, nat{0}, nat{0}, 0},
+ {nat{1}, nat{1}, nat{0}, 0},
+ {nat{0}, nat{_M}, nat{1}, 1},
+ {nat{80235}, nat{12345}, nat{67890}, 0},
+ {nat{_M - 1}, nat{_M}, nat{_M}, 1},
+ {nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, nat{1, 0, 0, 0}, 1},
+ {nat{0, 0, 0, _M}, nat{_M, _M, _M, _M - 1}, nat{1, 0, 0, 0}, 0},
+ {nat{0, 0, 0, 0}, nat{_M, 0, _M, 0}, nat{1, _M, 0, _M}, 1},
+}
+
+func testFunVV(t *testing.T, msg string, f funVV, a argVV) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+func TestFunVV(t *testing.T) {
+ for _, a := range sumVV {
+ arg := a
+ testFunVV(t, "addVV_g", addVV_g, arg)
+ testFunVV(t, "addVV", addVV, arg)
+
+ arg = argVV{a.z, a.y, a.x, a.c}
+ testFunVV(t, "addVV_g symmetric", addVV_g, arg)
+ testFunVV(t, "addVV symmetric", addVV, arg)
+
+ arg = argVV{a.x, a.z, a.y, a.c}
+ testFunVV(t, "subVV_g", subVV_g, arg)
+ testFunVV(t, "subVV", subVV, arg)
+
+ arg = argVV{a.y, a.z, a.x, a.c}
+ testFunVV(t, "subVV_g symmetric", subVV_g, arg)
+ testFunVV(t, "subVV symmetric", subVV, arg)
+ }
+}
+
+// Always the same seed for reproducible results.
+var rnd = rand.New(rand.NewSource(0))
+
+func rndW() Word {
+ return Word(rnd.Int63()<<1 | rnd.Int63n(2))
+}
+
+func rndV(n int) []Word {
+ v := make([]Word, n)
+ for i := range v {
+ v[i] = rndW()
+ }
+ return v
+}
+
+func benchmarkFunVV(b *testing.B, f funVV, n int) {
+ x := rndV(n)
+ y := rndV(n)
+ z := make([]Word, n)
+ b.SetBytes(int64(n * _W))
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ f(z, x, y)
+ }
+}
+
+func BenchmarkAddVV_1(b *testing.B) { benchmarkFunVV(b, addVV, 1) }
+func BenchmarkAddVV_2(b *testing.B) { benchmarkFunVV(b, addVV, 2) }
+func BenchmarkAddVV_3(b *testing.B) { benchmarkFunVV(b, addVV, 3) }
+func BenchmarkAddVV_4(b *testing.B) { benchmarkFunVV(b, addVV, 4) }
+func BenchmarkAddVV_5(b *testing.B) { benchmarkFunVV(b, addVV, 5) }
+func BenchmarkAddVV_1e1(b *testing.B) { benchmarkFunVV(b, addVV, 1e1) }
+func BenchmarkAddVV_1e2(b *testing.B) { benchmarkFunVV(b, addVV, 1e2) }
+func BenchmarkAddVV_1e3(b *testing.B) { benchmarkFunVV(b, addVV, 1e3) }
+func BenchmarkAddVV_1e4(b *testing.B) { benchmarkFunVV(b, addVV, 1e4) }
+func BenchmarkAddVV_1e5(b *testing.B) { benchmarkFunVV(b, addVV, 1e5) }
+
+type funVW func(z, x []Word, y Word) (c Word)
+type argVW struct {
+ z, x nat
+ y Word
+ c Word
+}
+
+var sumVW = []argVW{
+ {},
+ {nil, nil, 2, 2},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{1}, nat{0}, 1, 0},
+ {nat{1}, nat{1}, 0, 0},
+ {nat{0}, nat{_M}, 1, 1},
+ {nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, 1, 1},
+}
+
+var prodVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{_M}, 0, 0},
+ {nat{0}, nat{0}, _M, 0},
+ {nat{1}, nat{1}, 1, 0},
+ {nat{22793}, nat{991}, 23, 0},
+ {nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0},
+ {nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0},
+ {nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1 << 1, _M >> (_W - 1)},
+ {nat{_M << 7 & _M}, nat{_M}, 1 << 7, _M >> (_W - 7)},
+ {nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, _M >> (_W - 7)},
+}
+
+var lshVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{0}, 1, 0},
+ {nat{0}, nat{0}, 20, 0},
+
+ {nat{_M}, nat{_M}, 0, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1, 1},
+ {nat{_M << 20 & _M}, nat{_M}, 20, _M >> (_W - 20)},
+
+ {nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+ {nat{_M << 1 & _M, _M, _M}, nat{_M, _M, _M}, 1, 1},
+ {nat{_M << 20 & _M, _M, _M}, nat{_M, _M, _M}, 20, _M >> (_W - 20)},
+}
+
+var rshVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{0}, 1, 0},
+ {nat{0}, nat{0}, 20, 0},
+
+ {nat{_M}, nat{_M}, 0, 0},
+ {nat{_M >> 1}, nat{_M}, 1, _M << (_W - 1) & _M},
+ {nat{_M >> 20}, nat{_M}, 20, _M << (_W - 20) & _M},
+
+ {nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+ {nat{_M, _M, _M >> 1}, nat{_M, _M, _M}, 1, _M << (_W - 1) & _M},
+ {nat{_M, _M, _M >> 20}, nat{_M, _M, _M}, 20, _M << (_W - 20) & _M},
+}
+
+func testFunVW(t *testing.T, msg string, f funVW, a argVW) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+func makeFunVW(f func(z, x []Word, s uint) (c Word)) funVW {
+ return func(z, x []Word, s Word) (c Word) {
+ return f(z, x, uint(s))
+ }
+}
+
+func TestFunVW(t *testing.T) {
+ for _, a := range sumVW {
+ arg := a
+ testFunVW(t, "addVW_g", addVW_g, arg)
+ testFunVW(t, "addVW", addVW, arg)
+
+ arg = argVW{a.x, a.z, a.y, a.c}
+ testFunVW(t, "subVW_g", subVW_g, arg)
+ testFunVW(t, "subVW", subVW, arg)
+ }
+
+ shlVW_g := makeFunVW(shlVU_g)
+ shlVW := makeFunVW(shlVU)
+ for _, a := range lshVW {
+ arg := a
+ testFunVW(t, "shlVU_g", shlVW_g, arg)
+ testFunVW(t, "shlVU", shlVW, arg)
+ }
+
+ shrVW_g := makeFunVW(shrVU_g)
+ shrVW := makeFunVW(shrVU)
+ for _, a := range rshVW {
+ arg := a
+ testFunVW(t, "shrVU_g", shrVW_g, arg)
+ testFunVW(t, "shrVU", shrVW, arg)
+ }
+}
+
+func benchmarkFunVW(b *testing.B, f funVW, n int) {
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.SetBytes(int64(n * _W))
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ f(z, x, y)
+ }
+}
+
+func BenchmarkAddVW_1(b *testing.B) { benchmarkFunVW(b, addVW, 1) }
+func BenchmarkAddVW_2(b *testing.B) { benchmarkFunVW(b, addVW, 2) }
+func BenchmarkAddVW_3(b *testing.B) { benchmarkFunVW(b, addVW, 3) }
+func BenchmarkAddVW_4(b *testing.B) { benchmarkFunVW(b, addVW, 4) }
+func BenchmarkAddVW_5(b *testing.B) { benchmarkFunVW(b, addVW, 5) }
+func BenchmarkAddVW_1e1(b *testing.B) { benchmarkFunVW(b, addVW, 1e1) }
+func BenchmarkAddVW_1e2(b *testing.B) { benchmarkFunVW(b, addVW, 1e2) }
+func BenchmarkAddVW_1e3(b *testing.B) { benchmarkFunVW(b, addVW, 1e3) }
+func BenchmarkAddVW_1e4(b *testing.B) { benchmarkFunVW(b, addVW, 1e4) }
+func BenchmarkAddVW_1e5(b *testing.B) { benchmarkFunVW(b, addVW, 1e5) }
+
+type funVWW func(z, x []Word, y, r Word) (c Word)
+type argVWW struct {
+ z, x nat
+ y, r Word
+ c Word
+}
+
+var prodVWW = []argVWW{
+ {},
+ {nat{0}, nat{0}, 0, 0, 0},
+ {nat{991}, nat{0}, 0, 991, 0},
+ {nat{0}, nat{_M}, 0, 0, 0},
+ {nat{991}, nat{_M}, 0, 991, 0},
+ {nat{0}, nat{0}, _M, 0, 0},
+ {nat{991}, nat{0}, _M, 991, 0},
+ {nat{1}, nat{1}, 1, 0, 0},
+ {nat{992}, nat{1}, 1, 991, 0},
+ {nat{22793}, nat{991}, 23, 0, 0},
+ {nat{22800}, nat{991}, 23, 7, 0},
+ {nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0, 0},
+ {nat{7, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 7, 0},
+ {nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0, 0},
+ {nat{991, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 991, 0},
+ {nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0, 0},
+ {nat{991, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 991, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1 << 1, 0, _M >> (_W - 1)},
+ {nat{_M<<1&_M + 1}, nat{_M}, 1 << 1, 1, _M >> (_W - 1)},
+ {nat{_M << 7 & _M}, nat{_M}, 1 << 7, 0, _M >> (_W - 7)},
+ {nat{_M<<7&_M + 1<<6}, nat{_M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+ {nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 0, _M >> (_W - 7)},
+ {nat{_M<<7&_M + 1<<6, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+}
+
+func testFunVWW(t *testing.T, msg string, f funVWW, a argVWW) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y, a.r)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+// TODO(gri) mulAddVWW and divWVW are symmetric operations but
+// their signature is not symmetric. Try to unify.
+
+type funWVW func(z []Word, xn Word, x []Word, y Word) (r Word)
+type argWVW struct {
+ z nat
+ xn Word
+ x nat
+ y Word
+ r Word
+}
+
+func testFunWVW(t *testing.T, msg string, f funWVW, a argWVW) {
+ z := make(nat, len(a.z))
+ r := f(z, a.xn, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if r != a.r {
+ t.Errorf("%s%+v\n\tgot r = %#x; want %#x", msg, a, r, a.r)
+ }
+}
+
+func TestFunVWW(t *testing.T) {
+ for _, a := range prodVWW {
+ arg := a
+ testFunVWW(t, "mulAddVWW_g", mulAddVWW_g, arg)
+ testFunVWW(t, "mulAddVWW", mulAddVWW, arg)
+
+ if a.y != 0 && a.r < a.y {
+ arg := argWVW{a.x, a.c, a.z, a.y, a.r}
+ testFunWVW(t, "divWVW_g", divWVW_g, arg)
+ testFunWVW(t, "divWVW", divWVW, arg)
+ }
+ }
+}
+
+var mulWWTests = []struct {
+ x, y Word
+ q, r Word
+}{
+ {_M, _M, _M - 1, 1},
+ // 32 bit only: {0xc47dfa8c, 50911, 0x98a4, 0x998587f4},
+}
+
+func TestMulWW(t *testing.T) {
+ for i, test := range mulWWTests {
+ q, r := mulWW_g(test.x, test.y)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+}
+
+var mulAddWWWTests = []struct {
+ x, y, c Word
+ q, r Word
+}{
+ // TODO(agl): These will only work on 64-bit platforms.
+ // {15064310297182388543, 0xe7df04d2d35d5d80, 13537600649892366549, 13644450054494335067, 10832252001440893781},
+ // {15064310297182388543, 0xdab2f18048baa68d, 13644450054494335067, 12869334219691522700, 14233854684711418382},
+ {_M, _M, 0, _M - 1, 1},
+ {_M, _M, _M, _M, 0},
+}
+
+func TestMulAddWWW(t *testing.T) {
+ for i, test := range mulAddWWWTests {
+ q, r := mulAddWWW_g(test.x, test.y, test.c)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+}
+
+func benchmarkAddMulVVW(b *testing.B, n int) {
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.SetBytes(int64(n * _W))
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ addMulVVW(z, x, y)
+ }
+}
+
+func BenchmarkAddMulVVW_1(b *testing.B) { benchmarkAddMulVVW(b, 1) }
+func BenchmarkAddMulVVW_2(b *testing.B) { benchmarkAddMulVVW(b, 2) }
+func BenchmarkAddMulVVW_3(b *testing.B) { benchmarkAddMulVVW(b, 3) }
+func BenchmarkAddMulVVW_4(b *testing.B) { benchmarkAddMulVVW(b, 4) }
+func BenchmarkAddMulVVW_5(b *testing.B) { benchmarkAddMulVVW(b, 5) }
+func BenchmarkAddMulVVW_1e1(b *testing.B) { benchmarkAddMulVVW(b, 1e1) }
+func BenchmarkAddMulVVW_1e2(b *testing.B) { benchmarkAddMulVVW(b, 1e2) }
+func BenchmarkAddMulVVW_1e3(b *testing.B) { benchmarkAddMulVVW(b, 1e3) }
+func BenchmarkAddMulVVW_1e4(b *testing.B) { benchmarkAddMulVVW(b, 1e4) }
+func BenchmarkAddMulVVW_1e5(b *testing.B) { benchmarkAddMulVVW(b, 1e5) }
+
+func testWordBitLen(t *testing.T, fname string, f func(Word) int) {
+ for i := 0; i <= _W; i++ {
+ x := Word(1) << uint(i-1) // i == 0 => x == 0
+ n := f(x)
+ if n != i {
+ t.Errorf("got %d; want %d for %s(%#x)", n, i, fname, x)
+ }
+ }
+}
+
+func TestWordBitLen(t *testing.T) {
+ testWordBitLen(t, "bitLen", bitLen)
+ testWordBitLen(t, "bitLen_g", bitLen_g)
+}
+
+// runs b.N iterations of bitLen called on a Word containing (1 << nbits)-1.
+func benchmarkBitLenN(b *testing.B, nbits uint) {
+ testword := Word((uint64(1) << nbits) - 1)
+ for i := 0; i < b.N; i++ {
+ bitLen(testword)
+ }
+}
+
+// Individual bitLen tests. Numbers chosen to examine both sides
+// of powers-of-two boundaries.
+func BenchmarkBitLen0(b *testing.B) { benchmarkBitLenN(b, 0) }
+func BenchmarkBitLen1(b *testing.B) { benchmarkBitLenN(b, 1) }
+func BenchmarkBitLen2(b *testing.B) { benchmarkBitLenN(b, 2) }
+func BenchmarkBitLen3(b *testing.B) { benchmarkBitLenN(b, 3) }
+func BenchmarkBitLen4(b *testing.B) { benchmarkBitLenN(b, 4) }
+func BenchmarkBitLen5(b *testing.B) { benchmarkBitLenN(b, 5) }
+func BenchmarkBitLen8(b *testing.B) { benchmarkBitLenN(b, 8) }
+func BenchmarkBitLen9(b *testing.B) { benchmarkBitLenN(b, 9) }
+func BenchmarkBitLen16(b *testing.B) { benchmarkBitLenN(b, 16) }
+func BenchmarkBitLen17(b *testing.B) { benchmarkBitLenN(b, 17) }
+func BenchmarkBitLen31(b *testing.B) { benchmarkBitLenN(b, 31) }
diff --git a/src/math/big/calibrate_test.go b/src/math/big/calibrate_test.go
new file mode 100644
index 0000000..f69ffbf
--- /dev/null
+++ b/src/math/big/calibrate_test.go
@@ -0,0 +1,88 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file prints execution times for the Mul benchmark
+// given different Karatsuba thresholds. The result may be
+// used to manually fine-tune the threshold constant. The
+// results are somewhat fragile; use repeated runs to get
+// a clear picture.
+
+// Usage: go test -run=TestCalibrate -calibrate
+
+package big
+
+import (
+ "flag"
+ "fmt"
+ "testing"
+ "time"
+)
+
+var calibrate = flag.Bool("calibrate", false, "run calibration test")
+
+func karatsubaLoad(b *testing.B) {
+ BenchmarkMul(b)
+}
+
+// measureKaratsuba returns the time to run a Karatsuba-relevant benchmark
+// given Karatsuba threshold th.
+func measureKaratsuba(th int) time.Duration {
+ th, karatsubaThreshold = karatsubaThreshold, th
+ res := testing.Benchmark(karatsubaLoad)
+ karatsubaThreshold = th
+ return time.Duration(res.NsPerOp())
+}
+
+func computeThresholds() {
+ fmt.Printf("Multiplication times for varying Karatsuba thresholds\n")
+ fmt.Printf("(run repeatedly for good results)\n")
+
+ // determine Tk, the work load execution time using basic multiplication
+ Tb := measureKaratsuba(1e9) // th == 1e9 => Karatsuba multiplication disabled
+ fmt.Printf("Tb = %10s\n", Tb)
+
+ // thresholds
+ th := 4
+ th1 := -1
+ th2 := -1
+
+ var deltaOld time.Duration
+ for count := -1; count != 0 && th < 128; count-- {
+ // determine Tk, the work load execution time using Karatsuba multiplication
+ Tk := measureKaratsuba(th)
+
+ // improvement over Tb
+ delta := (Tb - Tk) * 100 / Tb
+
+ fmt.Printf("th = %3d Tk = %10s %4d%%", th, Tk, delta)
+
+ // determine break-even point
+ if Tk < Tb && th1 < 0 {
+ th1 = th
+ fmt.Print(" break-even point")
+ }
+
+ // determine diminishing return
+ if 0 < delta && delta < deltaOld && th2 < 0 {
+ th2 = th
+ fmt.Print(" diminishing return")
+ }
+ deltaOld = delta
+
+ fmt.Println()
+
+ // trigger counter
+ if th1 >= 0 && th2 >= 0 && count < 0 {
+ count = 10 // this many extra measurements after we got both thresholds
+ }
+
+ th++
+ }
+}
+
+func TestCalibrate(t *testing.T) {
+ if *calibrate {
+ computeThresholds()
+ }
+}
diff --git a/src/math/big/example_test.go b/src/math/big/example_test.go
new file mode 100644
index 0000000..078be47
--- /dev/null
+++ b/src/math/big/example_test.go
@@ -0,0 +1,51 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big_test
+
+import (
+ "fmt"
+ "log"
+ "math/big"
+)
+
+func ExampleRat_SetString() {
+ r := new(big.Rat)
+ r.SetString("355/113")
+ fmt.Println(r.FloatString(3))
+ // Output: 3.142
+}
+
+func ExampleInt_SetString() {
+ i := new(big.Int)
+ i.SetString("644", 8) // octal
+ fmt.Println(i)
+ // Output: 420
+}
+
+func ExampleRat_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ r := new(big.Rat)
+ _, err := fmt.Sscan("1.5000", r)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(r)
+ }
+ // Output: 3/2
+}
+
+func ExampleInt_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ i := new(big.Int)
+ _, err := fmt.Sscan("18446744073709551617", i)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(i)
+ }
+ // Output: 18446744073709551617
+}
diff --git a/src/math/big/gcd_test.go b/src/math/big/gcd_test.go
new file mode 100644
index 0000000..c0b9f58
--- /dev/null
+++ b/src/math/big/gcd_test.go
@@ -0,0 +1,47 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements a GCD benchmark.
+// Usage: go test math/big -test.bench GCD
+
+package big
+
+import (
+ "math/rand"
+ "testing"
+)
+
+// randInt returns a pseudo-random Int in the range [1<<(size-1), (1<<size) - 1]
+func randInt(r *rand.Rand, size uint) *Int {
+ n := new(Int).Lsh(intOne, size-1)
+ x := new(Int).Rand(r, n)
+ return x.Add(x, n) // make sure result > 1<<(size-1)
+}
+
+func runGCD(b *testing.B, aSize, bSize uint) {
+ b.StopTimer()
+ var r = rand.New(rand.NewSource(1234))
+ aa := randInt(r, aSize)
+ bb := randInt(r, bSize)
+ b.StartTimer()
+ for i := 0; i < b.N; i++ {
+ new(Int).GCD(nil, nil, aa, bb)
+ }
+}
+
+func BenchmarkGCD10x10(b *testing.B) { runGCD(b, 10, 10) }
+func BenchmarkGCD10x100(b *testing.B) { runGCD(b, 10, 100) }
+func BenchmarkGCD10x1000(b *testing.B) { runGCD(b, 10, 1000) }
+func BenchmarkGCD10x10000(b *testing.B) { runGCD(b, 10, 10000) }
+func BenchmarkGCD10x100000(b *testing.B) { runGCD(b, 10, 100000) }
+func BenchmarkGCD100x100(b *testing.B) { runGCD(b, 100, 100) }
+func BenchmarkGCD100x1000(b *testing.B) { runGCD(b, 100, 1000) }
+func BenchmarkGCD100x10000(b *testing.B) { runGCD(b, 100, 10000) }
+func BenchmarkGCD100x100000(b *testing.B) { runGCD(b, 100, 100000) }
+func BenchmarkGCD1000x1000(b *testing.B) { runGCD(b, 1000, 1000) }
+func BenchmarkGCD1000x10000(b *testing.B) { runGCD(b, 1000, 10000) }
+func BenchmarkGCD1000x100000(b *testing.B) { runGCD(b, 1000, 100000) }
+func BenchmarkGCD10000x10000(b *testing.B) { runGCD(b, 10000, 10000) }
+func BenchmarkGCD10000x100000(b *testing.B) { runGCD(b, 10000, 100000) }
+func BenchmarkGCD100000x100000(b *testing.B) { runGCD(b, 100000, 100000) }
diff --git a/src/math/big/hilbert_test.go b/src/math/big/hilbert_test.go
new file mode 100644
index 0000000..1a84341
--- /dev/null
+++ b/src/math/big/hilbert_test.go
@@ -0,0 +1,160 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// A little test program and benchmark for rational arithmetics.
+// Computes a Hilbert matrix, its inverse, multiplies them
+// and verifies that the product is the identity matrix.
+
+package big
+
+import (
+ "fmt"
+ "testing"
+)
+
+type matrix struct {
+ n, m int
+ a []*Rat
+}
+
+func (a *matrix) at(i, j int) *Rat {
+ if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+ panic("index out of range")
+ }
+ return a.a[i*a.m+j]
+}
+
+func (a *matrix) set(i, j int, x *Rat) {
+ if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+ panic("index out of range")
+ }
+ a.a[i*a.m+j] = x
+}
+
+func newMatrix(n, m int) *matrix {
+ if !(0 <= n && 0 <= m) {
+ panic("illegal matrix")
+ }
+ a := new(matrix)
+ a.n = n
+ a.m = m
+ a.a = make([]*Rat, n*m)
+ return a
+}
+
+func newUnit(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ x := NewRat(0, 1)
+ if i == j {
+ x.SetInt64(1)
+ }
+ a.set(i, j, x)
+ }
+ }
+ return a
+}
+
+func newHilbert(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ a.set(i, j, NewRat(1, int64(i+j+1)))
+ }
+ }
+ return a
+}
+
+func newInverseHilbert(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ x1 := new(Rat).SetInt64(int64(i + j + 1))
+ x2 := new(Rat).SetInt(new(Int).Binomial(int64(n+i), int64(n-j-1)))
+ x3 := new(Rat).SetInt(new(Int).Binomial(int64(n+j), int64(n-i-1)))
+ x4 := new(Rat).SetInt(new(Int).Binomial(int64(i+j), int64(i)))
+
+ x1.Mul(x1, x2)
+ x1.Mul(x1, x3)
+ x1.Mul(x1, x4)
+ x1.Mul(x1, x4)
+
+ if (i+j)&1 != 0 {
+ x1.Neg(x1)
+ }
+
+ a.set(i, j, x1)
+ }
+ }
+ return a
+}
+
+func (a *matrix) mul(b *matrix) *matrix {
+ if a.m != b.n {
+ panic("illegal matrix multiply")
+ }
+ c := newMatrix(a.n, b.m)
+ for i := 0; i < c.n; i++ {
+ for j := 0; j < c.m; j++ {
+ x := NewRat(0, 1)
+ for k := 0; k < a.m; k++ {
+ x.Add(x, new(Rat).Mul(a.at(i, k), b.at(k, j)))
+ }
+ c.set(i, j, x)
+ }
+ }
+ return c
+}
+
+func (a *matrix) eql(b *matrix) bool {
+ if a.n != b.n || a.m != b.m {
+ return false
+ }
+ for i := 0; i < a.n; i++ {
+ for j := 0; j < a.m; j++ {
+ if a.at(i, j).Cmp(b.at(i, j)) != 0 {
+ return false
+ }
+ }
+ }
+ return true
+}
+
+func (a *matrix) String() string {
+ s := ""
+ for i := 0; i < a.n; i++ {
+ for j := 0; j < a.m; j++ {
+ s += fmt.Sprintf("\t%s", a.at(i, j))
+ }
+ s += "\n"
+ }
+ return s
+}
+
+func doHilbert(t *testing.T, n int) {
+ a := newHilbert(n)
+ b := newInverseHilbert(n)
+ I := newUnit(n)
+ ab := a.mul(b)
+ if !ab.eql(I) {
+ if t == nil {
+ panic("Hilbert failed")
+ }
+ t.Errorf("a = %s\n", a)
+ t.Errorf("b = %s\n", b)
+ t.Errorf("a*b = %s\n", ab)
+ t.Errorf("I = %s\n", I)
+ }
+}
+
+func TestHilbert(t *testing.T) {
+ doHilbert(t, 10)
+}
+
+func BenchmarkHilbert(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ doHilbert(nil, 10)
+ }
+}
diff --git a/src/math/big/int.go b/src/math/big/int.go
new file mode 100644
index 0000000..d22e39e
--- /dev/null
+++ b/src/math/big/int.go
@@ -0,0 +1,1031 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements signed multi-precision integers.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "math/rand"
+ "strings"
+)
+
+// An Int represents a signed multi-precision integer.
+// The zero value for an Int represents the value 0.
+type Int struct {
+ neg bool // sign
+ abs nat // absolute value of the integer
+}
+
+var intOne = &Int{false, natOne}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x == 0
+// +1 if x > 0
+//
+func (x *Int) Sign() int {
+ if len(x.abs) == 0 {
+ return 0
+ }
+ if x.neg {
+ return -1
+ }
+ return 1
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Int) SetInt64(x int64) *Int {
+ neg := false
+ if x < 0 {
+ neg = true
+ x = -x
+ }
+ z.abs = z.abs.setUint64(uint64(x))
+ z.neg = neg
+ return z
+}
+
+// SetUint64 sets z to x and returns z.
+func (z *Int) SetUint64(x uint64) *Int {
+ z.abs = z.abs.setUint64(x)
+ z.neg = false
+ return z
+}
+
+// NewInt allocates and returns a new Int set to x.
+func NewInt(x int64) *Int {
+ return new(Int).SetInt64(x)
+}
+
+// Set sets z to x and returns z.
+func (z *Int) Set(x *Int) *Int {
+ if z != x {
+ z.abs = z.abs.set(x.abs)
+ z.neg = x.neg
+ }
+ return z
+}
+
+// Bits provides raw (unchecked but fast) access to x by returning its
+// absolute value as a little-endian Word slice. The result and x share
+// the same underlying array.
+// Bits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (x *Int) Bits() []Word {
+ return x.abs
+}
+
+// SetBits provides raw (unchecked but fast) access to z by setting its
+// value to abs, interpreted as a little-endian Word slice, and returning
+// z. The result and abs share the same underlying array.
+// SetBits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (z *Int) SetBits(abs []Word) *Int {
+ z.abs = nat(abs).norm()
+ z.neg = false
+ return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Int) Abs(x *Int) *Int {
+ z.Set(x)
+ z.neg = false
+ return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Int) Neg(x *Int) *Int {
+ z.Set(x)
+ z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
+ return z
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Int) Add(x, y *Int) *Int {
+ neg := x.neg
+ if x.neg == y.neg {
+ // x + y == x + y
+ // (-x) + (-y) == -(x + y)
+ z.abs = z.abs.add(x.abs, y.abs)
+ } else {
+ // x + (-y) == x - y == -(y - x)
+ // (-x) + y == y - x == -(x - y)
+ if x.abs.cmp(y.abs) >= 0 {
+ z.abs = z.abs.sub(x.abs, y.abs)
+ } else {
+ neg = !neg
+ z.abs = z.abs.sub(y.abs, x.abs)
+ }
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+ return z
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Int) Sub(x, y *Int) *Int {
+ neg := x.neg
+ if x.neg != y.neg {
+ // x - (-y) == x + y
+ // (-x) - y == -(x + y)
+ z.abs = z.abs.add(x.abs, y.abs)
+ } else {
+ // x - y == x - y == -(y - x)
+ // (-x) - (-y) == y - x == -(x - y)
+ if x.abs.cmp(y.abs) >= 0 {
+ z.abs = z.abs.sub(x.abs, y.abs)
+ } else {
+ neg = !neg
+ z.abs = z.abs.sub(y.abs, x.abs)
+ }
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+ return z
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Int) Mul(x, y *Int) *Int {
+ // x * y == x * y
+ // x * (-y) == -(x * y)
+ // (-x) * y == -(x * y)
+ // (-x) * (-y) == x * y
+ z.abs = z.abs.mul(x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+ return z
+}
+
+// MulRange sets z to the product of all integers
+// in the range [a, b] inclusively and returns z.
+// If a > b (empty range), the result is 1.
+func (z *Int) MulRange(a, b int64) *Int {
+ switch {
+ case a > b:
+ return z.SetInt64(1) // empty range
+ case a <= 0 && b >= 0:
+ return z.SetInt64(0) // range includes 0
+ }
+ // a <= b && (b < 0 || a > 0)
+
+ neg := false
+ if a < 0 {
+ neg = (b-a)&1 == 0
+ a, b = -b, -a
+ }
+
+ z.abs = z.abs.mulRange(uint64(a), uint64(b))
+ z.neg = neg
+ return z
+}
+
+// Binomial sets z to the binomial coefficient of (n, k) and returns z.
+func (z *Int) Binomial(n, k int64) *Int {
+ var a, b Int
+ a.MulRange(n-k+1, n)
+ b.MulRange(1, k)
+ return z.Quo(&a, &b)
+}
+
+// Quo sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Quo implements truncated division (like Go); see QuoRem for more details.
+func (z *Int) Quo(x, y *Int) *Int {
+ z.abs, _ = z.abs.div(nil, x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+ return z
+}
+
+// Rem sets z to the remainder x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Rem implements truncated modulus (like Go); see QuoRem for more details.
+func (z *Int) Rem(x, y *Int) *Int {
+ _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
+ return z
+}
+
+// QuoRem sets z to the quotient x/y and r to the remainder x%y
+// and returns the pair (z, r) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// QuoRem implements T-division and modulus (like Go):
+//
+// q = x/y with the result truncated to zero
+// r = x - y*q
+//
+// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
+// See DivMod for Euclidean division and modulus (unlike Go).
+//
+func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
+ z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
+ z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
+ return z, r
+}
+
+// Div sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Div implements Euclidean division (unlike Go); see DivMod for more details.
+func (z *Int) Div(x, y *Int) *Int {
+ y_neg := y.neg // z may be an alias for y
+ var r Int
+ z.QuoRem(x, y, &r)
+ if r.neg {
+ if y_neg {
+ z.Add(z, intOne)
+ } else {
+ z.Sub(z, intOne)
+ }
+ }
+ return z
+}
+
+// Mod sets z to the modulus x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
+func (z *Int) Mod(x, y *Int) *Int {
+ y0 := y // save y
+ if z == y || alias(z.abs, y.abs) {
+ y0 = new(Int).Set(y)
+ }
+ var q Int
+ q.QuoRem(x, y, z)
+ if z.neg {
+ if y0.neg {
+ z.Sub(z, y0)
+ } else {
+ z.Add(z, y0)
+ }
+ }
+ return z
+}
+
+// DivMod sets z to the quotient x div y and m to the modulus x mod y
+// and returns the pair (z, m) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// DivMod implements Euclidean division and modulus (unlike Go):
+//
+// q = x div y such that
+// m = x - y*q with 0 <= m < |q|
+//
+// (See Raymond T. Boute, ``The Euclidean definition of the functions
+// div and mod''. ACM Transactions on Programming Languages and
+// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
+// ACM press.)
+// See QuoRem for T-division and modulus (like Go).
+//
+func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
+ y0 := y // save y
+ if z == y || alias(z.abs, y.abs) {
+ y0 = new(Int).Set(y)
+ }
+ z.QuoRem(x, y, m)
+ if m.neg {
+ if y0.neg {
+ z.Add(z, intOne)
+ m.Sub(m, y0)
+ } else {
+ z.Sub(z, intOne)
+ m.Add(m, y0)
+ }
+ }
+ return z, m
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y
+// +1 if x > y
+//
+func (x *Int) Cmp(y *Int) (r int) {
+ // x cmp y == x cmp y
+ // x cmp (-y) == x
+ // (-x) cmp y == y
+ // (-x) cmp (-y) == -(x cmp y)
+ switch {
+ case x.neg == y.neg:
+ r = x.abs.cmp(y.abs)
+ if x.neg {
+ r = -r
+ }
+ case x.neg:
+ r = -1
+ default:
+ r = 1
+ }
+ return
+}
+
+func (x *Int) String() string {
+ switch {
+ case x == nil:
+ return "<nil>"
+ case x.neg:
+ return "-" + x.abs.decimalString()
+ }
+ return x.abs.decimalString()
+}
+
+func charset(ch rune) string {
+ switch ch {
+ case 'b':
+ return lowercaseDigits[0:2]
+ case 'o':
+ return lowercaseDigits[0:8]
+ case 'd', 's', 'v':
+ return lowercaseDigits[0:10]
+ case 'x':
+ return lowercaseDigits[0:16]
+ case 'X':
+ return uppercaseDigits[0:16]
+ }
+ return "" // unknown format
+}
+
+// write count copies of text to s
+func writeMultiple(s fmt.State, text string, count int) {
+ if len(text) > 0 {
+ b := []byte(text)
+ for ; count > 0; count-- {
+ s.Write(b)
+ }
+ }
+}
+
+// Format is a support routine for fmt.Formatter. It accepts
+// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
+// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+// Also supported are the full suite of package fmt's format
+// verbs for integral types, including '+', '-', and ' '
+// for sign control, '#' for leading zero in octal and for
+// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
+// respectively, specification of minimum digits precision,
+// output field width, space or zero padding, and left or
+// right justification.
+//
+func (x *Int) Format(s fmt.State, ch rune) {
+ cs := charset(ch)
+
+ // special cases
+ switch {
+ case cs == "":
+ // unknown format
+ fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
+ return
+ case x == nil:
+ fmt.Fprint(s, "<nil>")
+ return
+ }
+
+ // determine sign character
+ sign := ""
+ switch {
+ case x.neg:
+ sign = "-"
+ case s.Flag('+'): // supersedes ' ' when both specified
+ sign = "+"
+ case s.Flag(' '):
+ sign = " "
+ }
+
+ // determine prefix characters for indicating output base
+ prefix := ""
+ if s.Flag('#') {
+ switch ch {
+ case 'o': // octal
+ prefix = "0"
+ case 'x': // hexadecimal
+ prefix = "0x"
+ case 'X':
+ prefix = "0X"
+ }
+ }
+
+ // determine digits with base set by len(cs) and digit characters from cs
+ digits := x.abs.string(cs)
+
+ // number of characters for the three classes of number padding
+ var left int // space characters to left of digits for right justification ("%8d")
+ var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
+ var right int // space characters to right of digits for left justification ("%-8d")
+
+ // determine number padding from precision: the least number of digits to output
+ precision, precisionSet := s.Precision()
+ if precisionSet {
+ switch {
+ case len(digits) < precision:
+ zeroes = precision - len(digits) // count of zero padding
+ case digits == "0" && precision == 0:
+ return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
+ }
+ }
+
+ // determine field pad from width: the least number of characters to output
+ length := len(sign) + len(prefix) + zeroes + len(digits)
+ if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
+ switch d := width - length; {
+ case s.Flag('-'):
+ // pad on the right with spaces; supersedes '0' when both specified
+ right = d
+ case s.Flag('0') && !precisionSet:
+ // pad with zeroes unless precision also specified
+ zeroes = d
+ default:
+ // pad on the left with spaces
+ left = d
+ }
+ }
+
+ // print number as [left pad][sign][prefix][zero pad][digits][right pad]
+ writeMultiple(s, " ", left)
+ writeMultiple(s, sign, 1)
+ writeMultiple(s, prefix, 1)
+ writeMultiple(s, "0", zeroes)
+ writeMultiple(s, digits, 1)
+ writeMultiple(s, " ", right)
+}
+
+// scan sets z to the integer value corresponding to the longest possible prefix
+// read from r representing a signed integer number in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined but the returned value is nil. The
+// syntax follows the syntax of integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
+ // determine sign
+ ch, _, err := r.ReadRune()
+ if err != nil {
+ return nil, 0, err
+ }
+ neg := false
+ switch ch {
+ case '-':
+ neg = true
+ case '+': // nothing to do
+ default:
+ r.UnreadRune()
+ }
+
+ // determine mantissa
+ z.abs, base, err = z.abs.scan(r, base)
+ if err != nil {
+ return nil, base, err
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+
+ return z, base, nil
+}
+
+// Scan is a support routine for fmt.Scanner; it sets z to the value of
+// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
+// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+func (z *Int) Scan(s fmt.ScanState, ch rune) error {
+ s.SkipSpace() // skip leading space characters
+ base := 0
+ switch ch {
+ case 'b':
+ base = 2
+ case 'o':
+ base = 8
+ case 'd':
+ base = 10
+ case 'x', 'X':
+ base = 16
+ case 's', 'v':
+ // let scan determine the base
+ default:
+ return errors.New("Int.Scan: invalid verb")
+ }
+ _, _, err := z.scan(s, base)
+ return err
+}
+
+// low32 returns the least significant 32 bits of z.
+func low32(z nat) uint32 {
+ if len(z) == 0 {
+ return 0
+ }
+ return uint32(z[0])
+}
+
+// low64 returns the least significant 64 bits of z.
+func low64(z nat) uint64 {
+ if len(z) == 0 {
+ return 0
+ }
+ v := uint64(z[0])
+ if _W == 32 && len(z) > 1 {
+ v |= uint64(z[1]) << 32
+ }
+ return v
+}
+
+// Int64 returns the int64 representation of x.
+// If x cannot be represented in an int64, the result is undefined.
+func (x *Int) Int64() int64 {
+ v := int64(low64(x.abs))
+ if x.neg {
+ v = -v
+ }
+ return v
+}
+
+// Uint64 returns the uint64 representation of x.
+// If x cannot be represented in a uint64, the result is undefined.
+func (x *Int) Uint64() uint64 {
+ return low64(x.abs)
+}
+
+// SetString sets z to the value of s, interpreted in the given base,
+// and returns z and a boolean indicating success. If SetString fails,
+// the value of z is undefined but the returned value is nil.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z *Int) SetString(s string, base int) (*Int, bool) {
+ r := strings.NewReader(s)
+ _, _, err := z.scan(r, base)
+ if err != nil {
+ return nil, false
+ }
+ _, _, err = r.ReadRune()
+ if err != io.EOF {
+ return nil, false
+ }
+ return z, true // err == io.EOF => scan consumed all of s
+}
+
+// SetBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z *Int) SetBytes(buf []byte) *Int {
+ z.abs = z.abs.setBytes(buf)
+ z.neg = false
+ return z
+}
+
+// Bytes returns the absolute value of x as a big-endian byte slice.
+func (x *Int) Bytes() []byte {
+ buf := make([]byte, len(x.abs)*_S)
+ return buf[x.abs.bytes(buf):]
+}
+
+// BitLen returns the length of the absolute value of x in bits.
+// The bit length of 0 is 0.
+func (x *Int) BitLen() int {
+ return x.abs.bitLen()
+}
+
+// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
+// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
+// See Knuth, volume 2, section 4.6.3.
+func (z *Int) Exp(x, y, m *Int) *Int {
+ var yWords nat
+ if !y.neg {
+ yWords = y.abs
+ }
+ // y >= 0
+
+ var mWords nat
+ if m != nil {
+ mWords = m.abs // m.abs may be nil for m == 0
+ }
+
+ z.abs = z.abs.expNN(x.abs, yWords, mWords)
+ z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
+ if z.neg && len(mWords) > 0 {
+ // make modulus result positive
+ z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
+ z.neg = false
+ }
+
+ return z
+}
+
+// GCD sets z to the greatest common divisor of a and b, which both must
+// be > 0, and returns z.
+// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
+// If either a or b is <= 0, GCD sets z = x = y = 0.
+func (z *Int) GCD(x, y, a, b *Int) *Int {
+ if a.Sign() <= 0 || b.Sign() <= 0 {
+ z.SetInt64(0)
+ if x != nil {
+ x.SetInt64(0)
+ }
+ if y != nil {
+ y.SetInt64(0)
+ }
+ return z
+ }
+ if x == nil && y == nil {
+ return z.binaryGCD(a, b)
+ }
+
+ A := new(Int).Set(a)
+ B := new(Int).Set(b)
+
+ X := new(Int)
+ Y := new(Int).SetInt64(1)
+
+ lastX := new(Int).SetInt64(1)
+ lastY := new(Int)
+
+ q := new(Int)
+ temp := new(Int)
+
+ for len(B.abs) > 0 {
+ r := new(Int)
+ q, r = q.QuoRem(A, B, r)
+
+ A, B = B, r
+
+ temp.Set(X)
+ X.Mul(X, q)
+ X.neg = !X.neg
+ X.Add(X, lastX)
+ lastX.Set(temp)
+
+ temp.Set(Y)
+ Y.Mul(Y, q)
+ Y.neg = !Y.neg
+ Y.Add(Y, lastY)
+ lastY.Set(temp)
+ }
+
+ if x != nil {
+ *x = *lastX
+ }
+
+ if y != nil {
+ *y = *lastY
+ }
+
+ *z = *A
+ return z
+}
+
+// binaryGCD sets z to the greatest common divisor of a and b, which both must
+// be > 0, and returns z.
+// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
+func (z *Int) binaryGCD(a, b *Int) *Int {
+ u := z
+ v := new(Int)
+
+ // use one Euclidean iteration to ensure that u and v are approx. the same size
+ switch {
+ case len(a.abs) > len(b.abs):
+ u.Set(b)
+ v.Rem(a, b)
+ case len(a.abs) < len(b.abs):
+ u.Set(a)
+ v.Rem(b, a)
+ default:
+ u.Set(a)
+ v.Set(b)
+ }
+
+ // v might be 0 now
+ if len(v.abs) == 0 {
+ return u
+ }
+ // u > 0 && v > 0
+
+ // determine largest k such that u = u' << k, v = v' << k
+ k := u.abs.trailingZeroBits()
+ if vk := v.abs.trailingZeroBits(); vk < k {
+ k = vk
+ }
+ u.Rsh(u, k)
+ v.Rsh(v, k)
+
+ // determine t (we know that u > 0)
+ t := new(Int)
+ if u.abs[0]&1 != 0 {
+ // u is odd
+ t.Neg(v)
+ } else {
+ t.Set(u)
+ }
+
+ for len(t.abs) > 0 {
+ // reduce t
+ t.Rsh(t, t.abs.trailingZeroBits())
+ if t.neg {
+ v, t = t, v
+ v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
+ } else {
+ u, t = t, u
+ }
+ t.Sub(u, v)
+ }
+
+ return z.Lsh(u, k)
+}
+
+// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
+// If it returns true, x is prime with probability 1 - 1/4^n.
+// If it returns false, x is not prime.
+func (x *Int) ProbablyPrime(n int) bool {
+ return !x.neg && x.abs.probablyPrime(n)
+}
+
+// Rand sets z to a pseudo-random number in [0, n) and returns z.
+func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
+ z.neg = false
+ if n.neg == true || len(n.abs) == 0 {
+ z.abs = nil
+ return z
+ }
+ z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
+ return z
+}
+
+// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
+// and returns z. If g and n are not relatively prime, the result is undefined.
+func (z *Int) ModInverse(g, n *Int) *Int {
+ var d Int
+ d.GCD(z, nil, g, n)
+ // x and y are such that g*x + n*y = d. Since g and n are
+ // relatively prime, d = 1. Taking that modulo n results in
+ // g*x = 1, therefore x is the inverse element.
+ if z.neg {
+ z.Add(z, n)
+ }
+ return z
+}
+
+// Lsh sets z = x << n and returns z.
+func (z *Int) Lsh(x *Int, n uint) *Int {
+ z.abs = z.abs.shl(x.abs, n)
+ z.neg = x.neg
+ return z
+}
+
+// Rsh sets z = x >> n and returns z.
+func (z *Int) Rsh(x *Int, n uint) *Int {
+ if x.neg {
+ // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
+ t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
+ t = t.shr(t, n)
+ z.abs = t.add(t, natOne)
+ z.neg = true // z cannot be zero if x is negative
+ return z
+ }
+
+ z.abs = z.abs.shr(x.abs, n)
+ z.neg = false
+ return z
+}
+
+// Bit returns the value of the i'th bit of x. That is, it
+// returns (x>>i)&1. The bit index i must be >= 0.
+func (x *Int) Bit(i int) uint {
+ if i == 0 {
+ // optimization for common case: odd/even test of x
+ if len(x.abs) > 0 {
+ return uint(x.abs[0] & 1) // bit 0 is same for -x
+ }
+ return 0
+ }
+ if i < 0 {
+ panic("negative bit index")
+ }
+ if x.neg {
+ t := nat(nil).sub(x.abs, natOne)
+ return t.bit(uint(i)) ^ 1
+ }
+
+ return x.abs.bit(uint(i))
+}
+
+// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
+// That is, if b is 1 SetBit sets z = x | (1 << i);
+// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
+// SetBit will panic.
+func (z *Int) SetBit(x *Int, i int, b uint) *Int {
+ if i < 0 {
+ panic("negative bit index")
+ }
+ if x.neg {
+ t := z.abs.sub(x.abs, natOne)
+ t = t.setBit(t, uint(i), b^1)
+ z.abs = t.add(t, natOne)
+ z.neg = len(z.abs) > 0
+ return z
+ }
+ z.abs = z.abs.setBit(x.abs, uint(i), b)
+ z.neg = false
+ return z
+}
+
+// And sets z = x & y and returns z.
+func (z *Int) And(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
+ z.neg = true // z cannot be zero if x and y are negative
+ return z
+ }
+
+ // x & y == x & y
+ z.abs = z.abs.and(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // & is symmetric
+ }
+
+ // x & (-y) == x & ^(y-1) == x &^ (y-1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.andNot(x.abs, y1)
+ z.neg = false
+ return z
+}
+
+// AndNot sets z = x &^ y and returns z.
+func (z *Int) AndNot(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.andNot(y1, x1)
+ z.neg = false
+ return z
+ }
+
+ // x &^ y == x &^ y
+ z.abs = z.abs.andNot(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ if x.neg {
+ // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
+ z.neg = true // z cannot be zero if x is negative and y is positive
+ return z
+ }
+
+ // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
+ y1 := nat(nil).add(y.abs, natOne)
+ z.abs = z.abs.and(x.abs, y1)
+ z.neg = false
+ return z
+}
+
+// Or sets z = x | y and returns z.
+func (z *Int) Or(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
+ z.neg = true // z cannot be zero if x and y are negative
+ return z
+ }
+
+ // x | y == x | y
+ z.abs = z.abs.or(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // | is symmetric
+ }
+
+ // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
+ z.neg = true // z cannot be zero if one of x or y is negative
+ return z
+}
+
+// Xor sets z = x ^ y and returns z.
+func (z *Int) Xor(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.xor(x1, y1)
+ z.neg = false
+ return z
+ }
+
+ // x ^ y == x ^ y
+ z.abs = z.abs.xor(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // ^ is symmetric
+ }
+
+ // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
+ z.neg = true // z cannot be zero if only one of x or y is negative
+ return z
+}
+
+// Not sets z = ^x and returns z.
+func (z *Int) Not(x *Int) *Int {
+ if x.neg {
+ // ^(-x) == ^(^(x-1)) == x-1
+ z.abs = z.abs.sub(x.abs, natOne)
+ z.neg = false
+ return z
+ }
+
+ // ^x == -x-1 == -(x+1)
+ z.abs = z.abs.add(x.abs, natOne)
+ z.neg = true // z cannot be zero if x is positive
+ return z
+}
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const intGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Int) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+ buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
+ i := x.abs.bytes(buf) - 1 // i >= 0
+ b := intGobVersion << 1 // make space for sign bit
+ if x.neg {
+ b |= 1
+ }
+ buf[i] = b
+ return buf[i:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Int) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Int{}
+ return nil
+ }
+ b := buf[0]
+ if b>>1 != intGobVersion {
+ return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
+ }
+ z.neg = b&1 != 0
+ z.abs = z.abs.setBytes(buf[1:])
+ return nil
+}
+
+// MarshalJSON implements the json.Marshaler interface.
+func (z *Int) MarshalJSON() ([]byte, error) {
+ // TODO(gri): get rid of the []byte/string conversions
+ return []byte(z.String()), nil
+}
+
+// UnmarshalJSON implements the json.Unmarshaler interface.
+func (z *Int) UnmarshalJSON(text []byte) error {
+ // TODO(gri): get rid of the []byte/string conversions
+ if _, ok := z.SetString(string(text), 0); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+ }
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+func (z *Int) MarshalText() (text []byte, err error) {
+ return []byte(z.String()), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+func (z *Int) UnmarshalText(text []byte) error {
+ if _, ok := z.SetString(string(text), 0); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+ }
+ return nil
+}
diff --git a/src/math/big/int_test.go b/src/math/big/int_test.go
new file mode 100644
index 0000000..6070cf3
--- /dev/null
+++ b/src/math/big/int_test.go
@@ -0,0 +1,1625 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/hex"
+ "encoding/json"
+ "encoding/xml"
+ "fmt"
+ "math/rand"
+ "testing"
+ "testing/quick"
+)
+
+func isNormalized(x *Int) bool {
+ if len(x.abs) == 0 {
+ return !x.neg
+ }
+ // len(x.abs) > 0
+ return x.abs[len(x.abs)-1] != 0
+}
+
+type funZZ func(z, x, y *Int) *Int
+type argZZ struct {
+ z, x, y *Int
+}
+
+var sumZZ = []argZZ{
+ {NewInt(0), NewInt(0), NewInt(0)},
+ {NewInt(1), NewInt(1), NewInt(0)},
+ {NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
+ {NewInt(-1), NewInt(-1), NewInt(0)},
+ {NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
+ {NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
+}
+
+var prodZZ = []argZZ{
+ {NewInt(0), NewInt(0), NewInt(0)},
+ {NewInt(0), NewInt(1), NewInt(0)},
+ {NewInt(1), NewInt(1), NewInt(1)},
+ {NewInt(-991 * 991), NewInt(991), NewInt(-991)},
+ // TODO(gri) add larger products
+}
+
+func TestSignZ(t *testing.T) {
+ var zero Int
+ for _, a := range sumZZ {
+ s := a.z.Sign()
+ e := a.z.Cmp(&zero)
+ if s != e {
+ t.Errorf("got %d; want %d for z = %v", s, e, a.z)
+ }
+ }
+}
+
+func TestSetZ(t *testing.T) {
+ for _, a := range sumZZ {
+ var z Int
+ z.Set(a.z)
+ if !isNormalized(&z) {
+ t.Errorf("%v is not normalized", z)
+ }
+ if (&z).Cmp(a.z) != 0 {
+ t.Errorf("got z = %v; want %v", z, a.z)
+ }
+ }
+}
+
+func TestAbsZ(t *testing.T) {
+ var zero Int
+ for _, a := range sumZZ {
+ var z Int
+ z.Abs(a.z)
+ var e Int
+ e.Set(a.z)
+ if e.Cmp(&zero) < 0 {
+ e.Sub(&zero, &e)
+ }
+ if z.Cmp(&e) != 0 {
+ t.Errorf("got z = %v; want %v", z, e)
+ }
+ }
+}
+
+func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
+ var z Int
+ f(&z, a.x, a.y)
+ if !isNormalized(&z) {
+ t.Errorf("%s%v is not normalized", msg, z)
+ }
+ if (&z).Cmp(a.z) != 0 {
+ t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
+ }
+}
+
+func TestSumZZ(t *testing.T) {
+ AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
+ SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
+ for _, a := range sumZZ {
+ arg := a
+ testFunZZ(t, "AddZZ", AddZZ, arg)
+
+ arg = argZZ{a.z, a.y, a.x}
+ testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
+
+ arg = argZZ{a.x, a.z, a.y}
+ testFunZZ(t, "SubZZ", SubZZ, arg)
+
+ arg = argZZ{a.y, a.z, a.x}
+ testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
+ }
+}
+
+func TestProdZZ(t *testing.T) {
+ MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
+ for _, a := range prodZZ {
+ arg := a
+ testFunZZ(t, "MulZZ", MulZZ, arg)
+
+ arg = argZZ{a.z, a.y, a.x}
+ testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
+ }
+}
+
+// mulBytes returns x*y via grade school multiplication. Both inputs
+// and the result are assumed to be in big-endian representation (to
+// match the semantics of Int.Bytes and Int.SetBytes).
+func mulBytes(x, y []byte) []byte {
+ z := make([]byte, len(x)+len(y))
+
+ // multiply
+ k0 := len(z) - 1
+ for j := len(y) - 1; j >= 0; j-- {
+ d := int(y[j])
+ if d != 0 {
+ k := k0
+ carry := 0
+ for i := len(x) - 1; i >= 0; i-- {
+ t := int(z[k]) + int(x[i])*d + carry
+ z[k], carry = byte(t), t>>8
+ k--
+ }
+ z[k] = byte(carry)
+ }
+ k0--
+ }
+
+ // normalize (remove leading 0's)
+ i := 0
+ for i < len(z) && z[i] == 0 {
+ i++
+ }
+
+ return z[i:]
+}
+
+func checkMul(a, b []byte) bool {
+ var x, y, z1 Int
+ x.SetBytes(a)
+ y.SetBytes(b)
+ z1.Mul(&x, &y)
+
+ var z2 Int
+ z2.SetBytes(mulBytes(a, b))
+
+ return z1.Cmp(&z2) == 0
+}
+
+func TestMul(t *testing.T) {
+ if err := quick.Check(checkMul, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+var mulRangesZ = []struct {
+ a, b int64
+ prod string
+}{
+ // entirely positive ranges are covered by mulRangesN
+ {-1, 1, "0"},
+ {-2, -1, "2"},
+ {-3, -2, "6"},
+ {-3, -1, "-6"},
+ {1, 3, "6"},
+ {-10, -10, "-10"},
+ {0, -1, "1"}, // empty range
+ {-1, -100, "1"}, // empty range
+ {-1, 1, "0"}, // range includes 0
+ {-1e9, 0, "0"}, // range includes 0
+ {-1e9, 1e9, "0"}, // range includes 0
+ {-10, -1, "3628800"}, // 10!
+ {-20, -2, "-2432902008176640000"}, // -20!
+ {-99, -1,
+ "-933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "511852109168640000000000000000000000", // -99!
+ },
+}
+
+func TestMulRangeZ(t *testing.T) {
+ var tmp Int
+ // test entirely positive ranges
+ for i, r := range mulRangesN {
+ prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
+ if prod != r.prod {
+ t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
+ }
+ }
+ // test other ranges
+ for i, r := range mulRangesZ {
+ prod := tmp.MulRange(r.a, r.b).String()
+ if prod != r.prod {
+ t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+var stringTests = []struct {
+ in string
+ out string
+ base int
+ val int64
+ ok bool
+}{
+ {in: "", ok: false},
+ {in: "a", ok: false},
+ {in: "z", ok: false},
+ {in: "+", ok: false},
+ {in: "-", ok: false},
+ {in: "0b", ok: false},
+ {in: "0x", ok: false},
+ {in: "2", base: 2, ok: false},
+ {in: "0b2", base: 0, ok: false},
+ {in: "08", ok: false},
+ {in: "8", base: 8, ok: false},
+ {in: "0xg", base: 0, ok: false},
+ {in: "g", base: 16, ok: false},
+ {"0", "0", 0, 0, true},
+ {"0", "0", 10, 0, true},
+ {"0", "0", 16, 0, true},
+ {"+0", "0", 0, 0, true},
+ {"-0", "0", 0, 0, true},
+ {"10", "10", 0, 10, true},
+ {"10", "10", 10, 10, true},
+ {"10", "10", 16, 16, true},
+ {"-10", "-10", 16, -16, true},
+ {"+10", "10", 16, 16, true},
+ {"0x10", "16", 0, 16, true},
+ {in: "0x10", base: 16, ok: false},
+ {"-0x10", "-16", 0, -16, true},
+ {"+0x10", "16", 0, 16, true},
+ {"00", "0", 0, 0, true},
+ {"0", "0", 8, 0, true},
+ {"07", "7", 0, 7, true},
+ {"7", "7", 8, 7, true},
+ {"023", "19", 0, 19, true},
+ {"23", "23", 8, 19, true},
+ {"cafebabe", "cafebabe", 16, 0xcafebabe, true},
+ {"0b0", "0", 0, 0, true},
+ {"-111", "-111", 2, -7, true},
+ {"-0b111", "-7", 0, -7, true},
+ {"0b1001010111", "599", 0, 0x257, true},
+ {"1001010111", "1001010111", 2, 0x257, true},
+}
+
+func format(base int) string {
+ switch base {
+ case 2:
+ return "%b"
+ case 8:
+ return "%o"
+ case 16:
+ return "%x"
+ }
+ return "%d"
+}
+
+func TestGetString(t *testing.T) {
+ z := new(Int)
+ for i, test := range stringTests {
+ if !test.ok {
+ continue
+ }
+ z.SetInt64(test.val)
+
+ if test.base == 10 {
+ s := z.String()
+ if s != test.out {
+ t.Errorf("#%da got %s; want %s", i, s, test.out)
+ }
+ }
+
+ s := fmt.Sprintf(format(test.base), z)
+ if s != test.out {
+ t.Errorf("#%db got %s; want %s", i, s, test.out)
+ }
+ }
+}
+
+func TestSetString(t *testing.T) {
+ tmp := new(Int)
+ for i, test := range stringTests {
+ // initialize to a non-zero value so that issues with parsing
+ // 0 are detected
+ tmp.SetInt64(1234567890)
+ n1, ok1 := new(Int).SetString(test.in, test.base)
+ n2, ok2 := tmp.SetString(test.in, test.base)
+ expected := NewInt(test.val)
+ if ok1 != test.ok || ok2 != test.ok {
+ t.Errorf("#%d (input '%s') ok incorrect (should be %t)", i, test.in, test.ok)
+ continue
+ }
+ if !ok1 {
+ if n1 != nil {
+ t.Errorf("#%d (input '%s') n1 != nil", i, test.in)
+ }
+ continue
+ }
+ if !ok2 {
+ if n2 != nil {
+ t.Errorf("#%d (input '%s') n2 != nil", i, test.in)
+ }
+ continue
+ }
+
+ if ok1 && !isNormalized(n1) {
+ t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n1)
+ }
+ if ok2 && !isNormalized(n2) {
+ t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n2)
+ }
+
+ if n1.Cmp(expected) != 0 {
+ t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n1, test.val)
+ }
+ if n2.Cmp(expected) != 0 {
+ t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n2, test.val)
+ }
+ }
+}
+
+var formatTests = []struct {
+ input string
+ format string
+ output string
+}{
+ {"<nil>", "%x", "<nil>"},
+ {"<nil>", "%#x", "<nil>"},
+ {"<nil>", "%#y", "%!y(big.Int=<nil>)"},
+
+ {"10", "%b", "1010"},
+ {"10", "%o", "12"},
+ {"10", "%d", "10"},
+ {"10", "%v", "10"},
+ {"10", "%x", "a"},
+ {"10", "%X", "A"},
+ {"-10", "%X", "-A"},
+ {"10", "%y", "%!y(big.Int=10)"},
+ {"-10", "%y", "%!y(big.Int=-10)"},
+
+ {"10", "%#b", "1010"},
+ {"10", "%#o", "012"},
+ {"10", "%#d", "10"},
+ {"10", "%#v", "10"},
+ {"10", "%#x", "0xa"},
+ {"10", "%#X", "0XA"},
+ {"-10", "%#X", "-0XA"},
+ {"10", "%#y", "%!y(big.Int=10)"},
+ {"-10", "%#y", "%!y(big.Int=-10)"},
+
+ {"1234", "%d", "1234"},
+ {"1234", "%3d", "1234"},
+ {"1234", "%4d", "1234"},
+ {"-1234", "%d", "-1234"},
+ {"1234", "% 5d", " 1234"},
+ {"1234", "%+5d", "+1234"},
+ {"1234", "%-5d", "1234 "},
+ {"1234", "%x", "4d2"},
+ {"1234", "%X", "4D2"},
+ {"-1234", "%3x", "-4d2"},
+ {"-1234", "%4x", "-4d2"},
+ {"-1234", "%5x", " -4d2"},
+ {"-1234", "%-5x", "-4d2 "},
+ {"1234", "%03d", "1234"},
+ {"1234", "%04d", "1234"},
+ {"1234", "%05d", "01234"},
+ {"1234", "%06d", "001234"},
+ {"-1234", "%06d", "-01234"},
+ {"1234", "%+06d", "+01234"},
+ {"1234", "% 06d", " 01234"},
+ {"1234", "%-6d", "1234 "},
+ {"1234", "%-06d", "1234 "},
+ {"-1234", "%-06d", "-1234 "},
+
+ {"1234", "%.3d", "1234"},
+ {"1234", "%.4d", "1234"},
+ {"1234", "%.5d", "01234"},
+ {"1234", "%.6d", "001234"},
+ {"-1234", "%.3d", "-1234"},
+ {"-1234", "%.4d", "-1234"},
+ {"-1234", "%.5d", "-01234"},
+ {"-1234", "%.6d", "-001234"},
+
+ {"1234", "%8.3d", " 1234"},
+ {"1234", "%8.4d", " 1234"},
+ {"1234", "%8.5d", " 01234"},
+ {"1234", "%8.6d", " 001234"},
+ {"-1234", "%8.3d", " -1234"},
+ {"-1234", "%8.4d", " -1234"},
+ {"-1234", "%8.5d", " -01234"},
+ {"-1234", "%8.6d", " -001234"},
+
+ {"1234", "%+8.3d", " +1234"},
+ {"1234", "%+8.4d", " +1234"},
+ {"1234", "%+8.5d", " +01234"},
+ {"1234", "%+8.6d", " +001234"},
+ {"-1234", "%+8.3d", " -1234"},
+ {"-1234", "%+8.4d", " -1234"},
+ {"-1234", "%+8.5d", " -01234"},
+ {"-1234", "%+8.6d", " -001234"},
+
+ {"1234", "% 8.3d", " 1234"},
+ {"1234", "% 8.4d", " 1234"},
+ {"1234", "% 8.5d", " 01234"},
+ {"1234", "% 8.6d", " 001234"},
+ {"-1234", "% 8.3d", " -1234"},
+ {"-1234", "% 8.4d", " -1234"},
+ {"-1234", "% 8.5d", " -01234"},
+ {"-1234", "% 8.6d", " -001234"},
+
+ {"1234", "%.3x", "4d2"},
+ {"1234", "%.4x", "04d2"},
+ {"1234", "%.5x", "004d2"},
+ {"1234", "%.6x", "0004d2"},
+ {"-1234", "%.3x", "-4d2"},
+ {"-1234", "%.4x", "-04d2"},
+ {"-1234", "%.5x", "-004d2"},
+ {"-1234", "%.6x", "-0004d2"},
+
+ {"1234", "%8.3x", " 4d2"},
+ {"1234", "%8.4x", " 04d2"},
+ {"1234", "%8.5x", " 004d2"},
+ {"1234", "%8.6x", " 0004d2"},
+ {"-1234", "%8.3x", " -4d2"},
+ {"-1234", "%8.4x", " -04d2"},
+ {"-1234", "%8.5x", " -004d2"},
+ {"-1234", "%8.6x", " -0004d2"},
+
+ {"1234", "%+8.3x", " +4d2"},
+ {"1234", "%+8.4x", " +04d2"},
+ {"1234", "%+8.5x", " +004d2"},
+ {"1234", "%+8.6x", " +0004d2"},
+ {"-1234", "%+8.3x", " -4d2"},
+ {"-1234", "%+8.4x", " -04d2"},
+ {"-1234", "%+8.5x", " -004d2"},
+ {"-1234", "%+8.6x", " -0004d2"},
+
+ {"1234", "% 8.3x", " 4d2"},
+ {"1234", "% 8.4x", " 04d2"},
+ {"1234", "% 8.5x", " 004d2"},
+ {"1234", "% 8.6x", " 0004d2"},
+ {"1234", "% 8.7x", " 00004d2"},
+ {"1234", "% 8.8x", " 000004d2"},
+ {"-1234", "% 8.3x", " -4d2"},
+ {"-1234", "% 8.4x", " -04d2"},
+ {"-1234", "% 8.5x", " -004d2"},
+ {"-1234", "% 8.6x", " -0004d2"},
+ {"-1234", "% 8.7x", "-00004d2"},
+ {"-1234", "% 8.8x", "-000004d2"},
+
+ {"1234", "%-8.3d", "1234 "},
+ {"1234", "%-8.4d", "1234 "},
+ {"1234", "%-8.5d", "01234 "},
+ {"1234", "%-8.6d", "001234 "},
+ {"1234", "%-8.7d", "0001234 "},
+ {"1234", "%-8.8d", "00001234"},
+ {"-1234", "%-8.3d", "-1234 "},
+ {"-1234", "%-8.4d", "-1234 "},
+ {"-1234", "%-8.5d", "-01234 "},
+ {"-1234", "%-8.6d", "-001234 "},
+ {"-1234", "%-8.7d", "-0001234"},
+ {"-1234", "%-8.8d", "-00001234"},
+
+ {"16777215", "%b", "111111111111111111111111"}, // 2**24 - 1
+
+ {"0", "%.d", ""},
+ {"0", "%.0d", ""},
+ {"0", "%3.d", ""},
+}
+
+func TestFormat(t *testing.T) {
+ for i, test := range formatTests {
+ var x *Int
+ if test.input != "<nil>" {
+ var ok bool
+ x, ok = new(Int).SetString(test.input, 0)
+ if !ok {
+ t.Errorf("#%d failed reading input %s", i, test.input)
+ }
+ }
+ output := fmt.Sprintf(test.format, x)
+ if output != test.output {
+ t.Errorf("#%d got %q; want %q, {%q, %q, %q}", i, output, test.output, test.input, test.format, test.output)
+ }
+ }
+}
+
+var scanTests = []struct {
+ input string
+ format string
+ output string
+ remaining int
+}{
+ {"1010", "%b", "10", 0},
+ {"0b1010", "%v", "10", 0},
+ {"12", "%o", "10", 0},
+ {"012", "%v", "10", 0},
+ {"10", "%d", "10", 0},
+ {"10", "%v", "10", 0},
+ {"a", "%x", "10", 0},
+ {"0xa", "%v", "10", 0},
+ {"A", "%X", "10", 0},
+ {"-A", "%X", "-10", 0},
+ {"+0b1011001", "%v", "89", 0},
+ {"0xA", "%v", "10", 0},
+ {"0 ", "%v", "0", 1},
+ {"2+3", "%v", "2", 2},
+ {"0XABC 12", "%v", "2748", 3},
+}
+
+func TestScan(t *testing.T) {
+ var buf bytes.Buffer
+ for i, test := range scanTests {
+ x := new(Int)
+ buf.Reset()
+ buf.WriteString(test.input)
+ if _, err := fmt.Fscanf(&buf, test.format, x); err != nil {
+ t.Errorf("#%d error: %s", i, err)
+ }
+ if x.String() != test.output {
+ t.Errorf("#%d got %s; want %s", i, x.String(), test.output)
+ }
+ if buf.Len() != test.remaining {
+ t.Errorf("#%d got %d bytes remaining; want %d", i, buf.Len(), test.remaining)
+ }
+ }
+}
+
+// Examples from the Go Language Spec, section "Arithmetic operators"
+var divisionSignsTests = []struct {
+ x, y int64
+ q, r int64 // T-division
+ d, m int64 // Euclidian division
+}{
+ {5, 3, 1, 2, 1, 2},
+ {-5, 3, -1, -2, -2, 1},
+ {5, -3, -1, 2, -1, 2},
+ {-5, -3, 1, -2, 2, 1},
+ {1, 2, 0, 1, 0, 1},
+ {8, 4, 2, 0, 2, 0},
+}
+
+func TestDivisionSigns(t *testing.T) {
+ for i, test := range divisionSignsTests {
+ x := NewInt(test.x)
+ y := NewInt(test.y)
+ q := NewInt(test.q)
+ r := NewInt(test.r)
+ d := NewInt(test.d)
+ m := NewInt(test.m)
+
+ q1 := new(Int).Quo(x, y)
+ r1 := new(Int).Rem(x, y)
+ if !isNormalized(q1) {
+ t.Errorf("#%d Quo: %v is not normalized", i, *q1)
+ }
+ if !isNormalized(r1) {
+ t.Errorf("#%d Rem: %v is not normalized", i, *r1)
+ }
+ if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
+ t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
+ }
+
+ q2, r2 := new(Int).QuoRem(x, y, new(Int))
+ if !isNormalized(q2) {
+ t.Errorf("#%d Quo: %v is not normalized", i, *q2)
+ }
+ if !isNormalized(r2) {
+ t.Errorf("#%d Rem: %v is not normalized", i, *r2)
+ }
+ if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
+ t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
+ }
+
+ d1 := new(Int).Div(x, y)
+ m1 := new(Int).Mod(x, y)
+ if !isNormalized(d1) {
+ t.Errorf("#%d Div: %v is not normalized", i, *d1)
+ }
+ if !isNormalized(m1) {
+ t.Errorf("#%d Mod: %v is not normalized", i, *m1)
+ }
+ if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
+ t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
+ }
+
+ d2, m2 := new(Int).DivMod(x, y, new(Int))
+ if !isNormalized(d2) {
+ t.Errorf("#%d Div: %v is not normalized", i, *d2)
+ }
+ if !isNormalized(m2) {
+ t.Errorf("#%d Mod: %v is not normalized", i, *m2)
+ }
+ if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
+ t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
+ }
+ }
+}
+
+func checkSetBytes(b []byte) bool {
+ hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
+ hex2 := hex.EncodeToString(b)
+
+ for len(hex1) < len(hex2) {
+ hex1 = "0" + hex1
+ }
+
+ for len(hex1) > len(hex2) {
+ hex2 = "0" + hex2
+ }
+
+ return hex1 == hex2
+}
+
+func TestSetBytes(t *testing.T) {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+func checkBytes(b []byte) bool {
+ b2 := new(Int).SetBytes(b).Bytes()
+ return bytes.Equal(b, b2)
+}
+
+func TestBytes(t *testing.T) {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+func checkQuo(x, y []byte) bool {
+ u := new(Int).SetBytes(x)
+ v := new(Int).SetBytes(y)
+
+ if len(v.abs) == 0 {
+ return true
+ }
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(u, v, r)
+
+ if r.Cmp(v) >= 0 {
+ return false
+ }
+
+ uprime := new(Int).Set(q)
+ uprime.Mul(uprime, v)
+ uprime.Add(uprime, r)
+
+ return uprime.Cmp(u) == 0
+}
+
+var quoTests = []struct {
+ x, y string
+ q, r string
+}{
+ {
+ "476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
+ "9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
+ "50911",
+ "1",
+ },
+ {
+ "11510768301994997771168",
+ "1328165573307167369775",
+ "8",
+ "885443715537658812968",
+ },
+}
+
+func TestQuo(t *testing.T) {
+ if err := quick.Check(checkQuo, nil); err != nil {
+ t.Error(err)
+ }
+
+ for i, test := range quoTests {
+ x, _ := new(Int).SetString(test.x, 10)
+ y, _ := new(Int).SetString(test.y, 10)
+ expectedQ, _ := new(Int).SetString(test.q, 10)
+ expectedR, _ := new(Int).SetString(test.r, 10)
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(x, y, r)
+
+ if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
+ t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
+ }
+ }
+}
+
+func TestQuoStepD6(t *testing.T) {
+ // See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
+ // a code path which only triggers 1 in 10^{-19} cases.
+
+ u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
+ v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(u, v, r)
+ const expectedQ64 = "18446744073709551613"
+ const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
+ const expectedQ32 = "4294967293"
+ const expectedR32 = "39614081266355540837921718287"
+ if q.String() != expectedQ64 && q.String() != expectedQ32 ||
+ r.String() != expectedR64 && r.String() != expectedR32 {
+ t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
+ }
+}
+
+var bitLenTests = []struct {
+ in string
+ out int
+}{
+ {"-1", 1},
+ {"0", 0},
+ {"1", 1},
+ {"2", 2},
+ {"4", 3},
+ {"0xabc", 12},
+ {"0x8000", 16},
+ {"0x80000000", 32},
+ {"0x800000000000", 48},
+ {"0x8000000000000000", 64},
+ {"0x80000000000000000000", 80},
+ {"-0x4000000000000000000000", 87},
+}
+
+func TestBitLen(t *testing.T) {
+ for i, test := range bitLenTests {
+ x, ok := new(Int).SetString(test.in, 0)
+ if !ok {
+ t.Errorf("#%d test input invalid: %s", i, test.in)
+ continue
+ }
+
+ if n := x.BitLen(); n != test.out {
+ t.Errorf("#%d got %d want %d", i, n, test.out)
+ }
+ }
+}
+
+var expTests = []struct {
+ x, y, m string
+ out string
+}{
+ // y <= 0
+ {"0", "0", "", "1"},
+ {"1", "0", "", "1"},
+ {"-10", "0", "", "1"},
+ {"1234", "-1", "", "1"},
+
+ // m == 1
+ {"0", "0", "1", "0"},
+ {"1", "0", "1", "0"},
+ {"-10", "0", "1", "0"},
+ {"1234", "-1", "1", "0"},
+
+ // misc
+ {"5", "-7", "", "1"},
+ {"-5", "-7", "", "1"},
+ {"5", "0", "", "1"},
+ {"-5", "0", "", "1"},
+ {"5", "1", "", "5"},
+ {"-5", "1", "", "-5"},
+ {"-5", "1", "7", "2"},
+ {"-2", "3", "2", "0"},
+ {"5", "2", "", "25"},
+ {"1", "65537", "2", "1"},
+ {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+ {"0x8000000000000000", "2", "6719", "4944"},
+ {"0x8000000000000000", "3", "6719", "5447"},
+ {"0x8000000000000000", "1000", "6719", "1603"},
+ {"0x8000000000000000", "1000000", "6719", "3199"},
+ {"0x8000000000000000", "-1000000", "6719", "1"},
+ {
+ "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+ "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+ },
+ // test case for issue 8822
+ {
+ "-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
+ "0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+ "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
+ "21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+ },
+}
+
+func TestExp(t *testing.T) {
+ for i, test := range expTests {
+ x, ok1 := new(Int).SetString(test.x, 0)
+ y, ok2 := new(Int).SetString(test.y, 0)
+ out, ok3 := new(Int).SetString(test.out, 0)
+
+ var ok4 bool
+ var m *Int
+
+ if len(test.m) == 0 {
+ m, ok4 = nil, true
+ } else {
+ m, ok4 = new(Int).SetString(test.m, 0)
+ }
+
+ if !ok1 || !ok2 || !ok3 || !ok4 {
+ t.Errorf("#%d: error in input", i)
+ continue
+ }
+
+ z1 := new(Int).Exp(x, y, m)
+ if !isNormalized(z1) {
+ t.Errorf("#%d: %v is not normalized", i, *z1)
+ }
+ if z1.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z1, out)
+ }
+
+ if m == nil {
+ // The result should be the same as for m == 0;
+ // specifically, there should be no div-zero panic.
+ m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
+ z2 := new(Int).Exp(x, y, m)
+ if z2.Cmp(z1) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z2, z1)
+ }
+ }
+ }
+}
+
+func checkGcd(aBytes, bBytes []byte) bool {
+ x := new(Int)
+ y := new(Int)
+ a := new(Int).SetBytes(aBytes)
+ b := new(Int).SetBytes(bBytes)
+
+ d := new(Int).GCD(x, y, a, b)
+ x.Mul(x, a)
+ y.Mul(y, b)
+ x.Add(x, y)
+
+ return x.Cmp(d) == 0
+}
+
+var gcdTests = []struct {
+ d, x, y, a, b string
+}{
+ // a <= 0 || b <= 0
+ {"0", "0", "0", "0", "0"},
+ {"0", "0", "0", "0", "7"},
+ {"0", "0", "0", "11", "0"},
+ {"0", "0", "0", "-77", "35"},
+ {"0", "0", "0", "64515", "-24310"},
+ {"0", "0", "0", "-64515", "-24310"},
+
+ {"1", "-9", "47", "120", "23"},
+ {"7", "1", "-2", "77", "35"},
+ {"935", "-3", "8", "64515", "24310"},
+ {"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
+ {"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
+
+ // test early exit (after one Euclidean iteration) in binaryGCD
+ {"1", "", "", "1", "98920366548084643601728869055592650835572950932266967461790948584315647051443"},
+}
+
+func testGcd(t *testing.T, d, x, y, a, b *Int) {
+ var X *Int
+ if x != nil {
+ X = new(Int)
+ }
+ var Y *Int
+ if y != nil {
+ Y = new(Int)
+ }
+
+ D := new(Int).GCD(X, Y, a, b)
+ if D.Cmp(d) != 0 {
+ t.Errorf("GCD(%s, %s): got d = %s, want %s", a, b, D, d)
+ }
+ if x != nil && X.Cmp(x) != 0 {
+ t.Errorf("GCD(%s, %s): got x = %s, want %s", a, b, X, x)
+ }
+ if y != nil && Y.Cmp(y) != 0 {
+ t.Errorf("GCD(%s, %s): got y = %s, want %s", a, b, Y, y)
+ }
+
+ // binaryGCD requires a > 0 && b > 0
+ if a.Sign() <= 0 || b.Sign() <= 0 {
+ return
+ }
+
+ D.binaryGCD(a, b)
+ if D.Cmp(d) != 0 {
+ t.Errorf("binaryGcd(%s, %s): got d = %s, want %s", a, b, D, d)
+ }
+}
+
+func TestGcd(t *testing.T) {
+ for _, test := range gcdTests {
+ d, _ := new(Int).SetString(test.d, 0)
+ x, _ := new(Int).SetString(test.x, 0)
+ y, _ := new(Int).SetString(test.y, 0)
+ a, _ := new(Int).SetString(test.a, 0)
+ b, _ := new(Int).SetString(test.b, 0)
+
+ testGcd(t, d, nil, nil, a, b)
+ testGcd(t, d, x, nil, a, b)
+ testGcd(t, d, nil, y, a, b)
+ testGcd(t, d, x, y, a, b)
+ }
+
+ quick.Check(checkGcd, nil)
+}
+
+var primes = []string{
+ "2",
+ "3",
+ "5",
+ "7",
+ "11",
+
+ "13756265695458089029",
+ "13496181268022124907",
+ "10953742525620032441",
+ "17908251027575790097",
+
+ // http://code.google.com/p/go/issues/detail?id=638
+ "18699199384836356663",
+
+ "98920366548084643601728869055592650835572950932266967461790948584315647051443",
+ "94560208308847015747498523884063394671606671904944666360068158221458669711639",
+
+ // http://primes.utm.edu/lists/small/small3.html
+ "449417999055441493994709297093108513015373787049558499205492347871729927573118262811508386655998299074566974373711472560655026288668094291699357843464363003144674940345912431129144354948751003607115263071543163",
+ "230975859993204150666423538988557839555560243929065415434980904258310530753006723857139742334640122533598517597674807096648905501653461687601339782814316124971547968912893214002992086353183070342498989426570593",
+ "5521712099665906221540423207019333379125265462121169655563495403888449493493629943498064604536961775110765377745550377067893607246020694972959780839151452457728855382113555867743022746090187341871655890805971735385789993",
+ "203956878356401977405765866929034577280193993314348263094772646453283062722701277632936616063144088173312372882677123879538709400158306567338328279154499698366071906766440037074217117805690872792848149112022286332144876183376326512083574821647933992961249917319836219304274280243803104015000563790123",
+}
+
+var composites = []string{
+ "21284175091214687912771199898307297748211672914763848041968395774954376176754",
+ "6084766654921918907427900243509372380954290099172559290432744450051395395951",
+ "84594350493221918389213352992032324280367711247940675652888030554255915464401",
+ "82793403787388584738507275144194252681",
+}
+
+func TestProbablyPrime(t *testing.T) {
+ nreps := 20
+ if testing.Short() {
+ nreps = 1
+ }
+ for i, s := range primes {
+ p, _ := new(Int).SetString(s, 10)
+ if !p.ProbablyPrime(nreps) {
+ t.Errorf("#%d prime found to be non-prime (%s)", i, s)
+ }
+ }
+
+ for i, s := range composites {
+ c, _ := new(Int).SetString(s, 10)
+ if c.ProbablyPrime(nreps) {
+ t.Errorf("#%d composite found to be prime (%s)", i, s)
+ }
+ if testing.Short() {
+ break
+ }
+ }
+}
+
+type intShiftTest struct {
+ in string
+ shift uint
+ out string
+}
+
+var rshTests = []intShiftTest{
+ {"0", 0, "0"},
+ {"-0", 0, "0"},
+ {"0", 1, "0"},
+ {"0", 2, "0"},
+ {"1", 0, "1"},
+ {"1", 1, "0"},
+ {"1", 2, "0"},
+ {"2", 0, "2"},
+ {"2", 1, "1"},
+ {"-1", 0, "-1"},
+ {"-1", 1, "-1"},
+ {"-1", 10, "-1"},
+ {"-100", 2, "-25"},
+ {"-100", 3, "-13"},
+ {"-100", 100, "-1"},
+ {"4294967296", 0, "4294967296"},
+ {"4294967296", 1, "2147483648"},
+ {"4294967296", 2, "1073741824"},
+ {"18446744073709551616", 0, "18446744073709551616"},
+ {"18446744073709551616", 1, "9223372036854775808"},
+ {"18446744073709551616", 2, "4611686018427387904"},
+ {"18446744073709551616", 64, "1"},
+ {"340282366920938463463374607431768211456", 64, "18446744073709551616"},
+ {"340282366920938463463374607431768211456", 128, "1"},
+}
+
+func TestRsh(t *testing.T) {
+ for i, test := range rshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ out := new(Int).Rsh(in, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ }
+}
+
+func TestRshSelf(t *testing.T) {
+ for i, test := range rshTests {
+ z, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ z.Rsh(z, test.shift)
+
+ if !isNormalized(z) {
+ t.Errorf("#%d: %v is not normalized", i, *z)
+ }
+ if z.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z, expected)
+ }
+ }
+}
+
+var lshTests = []intShiftTest{
+ {"0", 0, "0"},
+ {"0", 1, "0"},
+ {"0", 2, "0"},
+ {"1", 0, "1"},
+ {"1", 1, "2"},
+ {"1", 2, "4"},
+ {"2", 0, "2"},
+ {"2", 1, "4"},
+ {"2", 2, "8"},
+ {"-87", 1, "-174"},
+ {"4294967296", 0, "4294967296"},
+ {"4294967296", 1, "8589934592"},
+ {"4294967296", 2, "17179869184"},
+ {"18446744073709551616", 0, "18446744073709551616"},
+ {"9223372036854775808", 1, "18446744073709551616"},
+ {"4611686018427387904", 2, "18446744073709551616"},
+ {"1", 64, "18446744073709551616"},
+ {"18446744073709551616", 64, "340282366920938463463374607431768211456"},
+ {"1", 128, "340282366920938463463374607431768211456"},
+}
+
+func TestLsh(t *testing.T) {
+ for i, test := range lshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ out := new(Int).Lsh(in, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ }
+}
+
+func TestLshSelf(t *testing.T) {
+ for i, test := range lshTests {
+ z, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ z.Lsh(z, test.shift)
+
+ if !isNormalized(z) {
+ t.Errorf("#%d: %v is not normalized", i, *z)
+ }
+ if z.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z, expected)
+ }
+ }
+}
+
+func TestLshRsh(t *testing.T) {
+ for i, test := range rshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ out := new(Int).Lsh(in, test.shift)
+ out = out.Rsh(out, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if in.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+ for i, test := range lshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ out := new(Int).Lsh(in, test.shift)
+ out.Rsh(out, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if in.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+}
+
+var int64Tests = []int64{
+ 0,
+ 1,
+ -1,
+ 4294967295,
+ -4294967295,
+ 4294967296,
+ -4294967296,
+ 9223372036854775807,
+ -9223372036854775807,
+ -9223372036854775808,
+}
+
+func TestInt64(t *testing.T) {
+ for i, testVal := range int64Tests {
+ in := NewInt(testVal)
+ out := in.Int64()
+
+ if out != testVal {
+ t.Errorf("#%d got %d want %d", i, out, testVal)
+ }
+ }
+}
+
+var uint64Tests = []uint64{
+ 0,
+ 1,
+ 4294967295,
+ 4294967296,
+ 8589934591,
+ 8589934592,
+ 9223372036854775807,
+ 9223372036854775808,
+ 18446744073709551615, // 1<<64 - 1
+}
+
+func TestUint64(t *testing.T) {
+ in := new(Int)
+ for i, testVal := range uint64Tests {
+ in.SetUint64(testVal)
+ out := in.Uint64()
+
+ if out != testVal {
+ t.Errorf("#%d got %d want %d", i, out, testVal)
+ }
+
+ str := fmt.Sprint(testVal)
+ strOut := in.String()
+ if strOut != str {
+ t.Errorf("#%d.String got %s want %s", i, strOut, str)
+ }
+ }
+}
+
+var bitwiseTests = []struct {
+ x, y string
+ and, or, xor, andNot string
+}{
+ {"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
+ {"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
+ {"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
+ {"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
+ {"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
+ {"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
+ {"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
+ {"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
+ {"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
+ {"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
+ {"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
+ {"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
+ {"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
+ {
+ "0x1000009dc6e3d9822cba04129bcbe3401",
+ "0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "0x1000001186210100001000009048c2001",
+ "0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+ "0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+ "0x8c40c2d8822caa04120b8321400",
+ },
+ {
+ "0x1000009dc6e3d9822cba04129bcbe3401",
+ "-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "0x8c40c2d8822caa04120b8321401",
+ "-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
+ "-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
+ "0x1000001186210100001000009048c2000",
+ },
+ {
+ "-0x1000009dc6e3d9822cba04129bcbe3401",
+ "-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+ "-0x1000001186210100001000009048c2001",
+ "0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+ "0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
+ },
+}
+
+type bitFun func(z, x, y *Int) *Int
+
+func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+ expected := new(Int)
+ expected.SetString(exp, 0)
+
+ out := f(new(Int), x, y)
+ if out.Cmp(expected) != 0 {
+ t.Errorf("%s: got %s want %s", msg, out, expected)
+ }
+}
+
+func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+ self := new(Int)
+ self.Set(x)
+ expected := new(Int)
+ expected.SetString(exp, 0)
+
+ self = f(self, self, y)
+ if self.Cmp(expected) != 0 {
+ t.Errorf("%s: got %s want %s", msg, self, expected)
+ }
+}
+
+func altBit(x *Int, i int) uint {
+ z := new(Int).Rsh(x, uint(i))
+ z = z.And(z, NewInt(1))
+ if z.Cmp(new(Int)) != 0 {
+ return 1
+ }
+ return 0
+}
+
+func altSetBit(z *Int, x *Int, i int, b uint) *Int {
+ one := NewInt(1)
+ m := one.Lsh(one, uint(i))
+ switch b {
+ case 1:
+ return z.Or(x, m)
+ case 0:
+ return z.AndNot(x, m)
+ }
+ panic("set bit is not 0 or 1")
+}
+
+func testBitset(t *testing.T, x *Int) {
+ n := x.BitLen()
+ z := new(Int).Set(x)
+ z1 := new(Int).Set(x)
+ for i := 0; i < n+10; i++ {
+ old := z.Bit(i)
+ old1 := altBit(z1, i)
+ if old != old1 {
+ t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
+ }
+ z := new(Int).SetBit(z, i, 1)
+ z1 := altSetBit(new(Int), z1, i, 1)
+ if z.Bit(i) == 0 {
+ t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
+ }
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
+ }
+ z.SetBit(z, i, 0)
+ altSetBit(z1, z1, i, 0)
+ if z.Bit(i) != 0 {
+ t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
+ }
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
+ }
+ altSetBit(z1, z1, i, old)
+ z.SetBit(z, i, old)
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
+ }
+ }
+ if z.Cmp(x) != 0 {
+ t.Errorf("bitset: got %s want %s", z, x)
+ }
+}
+
+var bitsetTests = []struct {
+ x string
+ i int
+ b uint
+}{
+ {"0", 0, 0},
+ {"0", 200, 0},
+ {"1", 0, 1},
+ {"1", 1, 0},
+ {"-1", 0, 1},
+ {"-1", 200, 1},
+ {"0x2000000000000000000000000000", 108, 0},
+ {"0x2000000000000000000000000000", 109, 1},
+ {"0x2000000000000000000000000000", 110, 0},
+ {"-0x2000000000000000000000000001", 108, 1},
+ {"-0x2000000000000000000000000001", 109, 0},
+ {"-0x2000000000000000000000000001", 110, 1},
+}
+
+func TestBitSet(t *testing.T) {
+ for _, test := range bitwiseTests {
+ x := new(Int)
+ x.SetString(test.x, 0)
+ testBitset(t, x)
+ x = new(Int)
+ x.SetString(test.y, 0)
+ testBitset(t, x)
+ }
+ for i, test := range bitsetTests {
+ x := new(Int)
+ x.SetString(test.x, 0)
+ b := x.Bit(test.i)
+ if b != test.b {
+ t.Errorf("#%d got %v want %v", i, b, test.b)
+ }
+ }
+ z := NewInt(1)
+ z.SetBit(NewInt(0), 2, 1)
+ if z.Cmp(NewInt(4)) != 0 {
+ t.Errorf("destination leaked into result; got %s want 4", z)
+ }
+}
+
+func BenchmarkBitset(b *testing.B) {
+ z := new(Int)
+ z.SetBit(z, 512, 1)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ z.SetBit(z, i&512, 1)
+ }
+}
+
+func BenchmarkBitsetNeg(b *testing.B) {
+ z := NewInt(-1)
+ z.SetBit(z, 512, 0)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ z.SetBit(z, i&512, 0)
+ }
+}
+
+func BenchmarkBitsetOrig(b *testing.B) {
+ z := new(Int)
+ altSetBit(z, z, 512, 1)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ altSetBit(z, z, i&512, 1)
+ }
+}
+
+func BenchmarkBitsetNegOrig(b *testing.B) {
+ z := NewInt(-1)
+ altSetBit(z, z, 512, 0)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ altSetBit(z, z, i&512, 0)
+ }
+}
+
+func TestBitwise(t *testing.T) {
+ x := new(Int)
+ y := new(Int)
+ for _, test := range bitwiseTests {
+ x.SetString(test.x, 0)
+ y.SetString(test.y, 0)
+
+ testBitFun(t, "and", (*Int).And, x, y, test.and)
+ testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
+ testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+ testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+ testBitFun(t, "or", (*Int).Or, x, y, test.or)
+ testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
+ testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
+ testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
+ }
+}
+
+var notTests = []struct {
+ in string
+ out string
+}{
+ {"0", "-1"},
+ {"1", "-2"},
+ {"7", "-8"},
+ {"0", "-1"},
+ {"-81910", "81909"},
+ {
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "-298472983472983471903246121093472394872319615612417471234712062",
+ },
+}
+
+func TestNot(t *testing.T) {
+ in := new(Int)
+ out := new(Int)
+ expected := new(Int)
+ for i, test := range notTests {
+ in.SetString(test.in, 10)
+ expected.SetString(test.out, 10)
+ out = out.Not(in)
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ out = out.Not(out)
+ if out.Cmp(in) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+}
+
+var modInverseTests = []struct {
+ element string
+ modulus string
+}{
+ {"1234567", "458948883992"},
+ {"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
+}
+
+func TestModInverse(t *testing.T) {
+ var element, modulus, gcd, inverse Int
+ one := NewInt(1)
+ for i, test := range modInverseTests {
+ (&element).SetString(test.element, 10)
+ (&modulus).SetString(test.modulus, 10)
+ (&inverse).ModInverse(&element, &modulus)
+ (&inverse).Mul(&inverse, &element)
+ (&inverse).Mod(&inverse, &modulus)
+ if (&inverse).Cmp(one) != 0 {
+ t.Errorf("#%d: failed (e·e^(-1)=%s)", i, &inverse)
+ }
+ }
+ // exhaustive test for small values
+ for n := 2; n < 100; n++ {
+ (&modulus).SetInt64(int64(n))
+ for x := 1; x < n; x++ {
+ (&element).SetInt64(int64(x))
+ (&gcd).GCD(nil, nil, &element, &modulus)
+ if (&gcd).Cmp(one) != 0 {
+ continue
+ }
+ (&inverse).ModInverse(&element, &modulus)
+ (&inverse).Mul(&inverse, &element)
+ (&inverse).Mod(&inverse, &modulus)
+ if (&inverse).Cmp(one) != 0 {
+ t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+ }
+ }
+ }
+}
+
+var encodingTests = []string{
+ "-539345864568634858364538753846587364875430589374589",
+ "-678645873",
+ "-100",
+ "-2",
+ "-1",
+ "0",
+ "1",
+ "2",
+ "10",
+ "42",
+ "1234567890",
+ "298472983472983471903246121093472394872319615612417471234712061",
+}
+
+func TestIntGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range encodingTests {
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ var tx Int
+ tx.SetString(test, 10)
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %s failed: %s", &tx, err)
+ }
+ var rx Int
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %s failed: %s", &tx, err)
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+// Sending a nil Int pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilIntInSlice(t *testing.T) {
+ buf := new(bytes.Buffer)
+ enc := gob.NewEncoder(buf)
+ dec := gob.NewDecoder(buf)
+
+ var in = make([]*Int, 1)
+ err := enc.Encode(&in)
+ if err != nil {
+ t.Errorf("gob encode failed: %q", err)
+ }
+ var out []*Int
+ err = dec.Decode(&out)
+ if err != nil {
+ t.Fatalf("gob decode failed: %q", err)
+ }
+ if len(out) != 1 {
+ t.Fatalf("wrong len; want 1 got %d", len(out))
+ }
+ var zero Int
+ if out[0].Cmp(&zero) != 0 {
+ t.Errorf("transmission of (*Int)(nill) failed: got %s want 0", out)
+ }
+}
+
+func TestIntJSONEncoding(t *testing.T) {
+ for _, test := range encodingTests {
+ var tx Int
+ tx.SetString(test, 10)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ }
+ var rx Int
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+var intVals = []string{
+ "-141592653589793238462643383279502884197169399375105820974944592307816406286",
+ "-1415926535897932384626433832795028841971",
+ "-141592653589793",
+ "-1",
+ "0",
+ "1",
+ "141592653589793",
+ "1415926535897932384626433832795028841971",
+ "141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+func TestIntJSONEncodingTextMarshaller(t *testing.T) {
+ for _, num := range intVals {
+ var tx Int
+ tx.SetString(num, 0)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+func TestIntXMLEncodingTextMarshaller(t *testing.T) {
+ for _, num := range intVals {
+ var tx Int
+ tx.SetString(num, 0)
+ b, err := xml.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := xml.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+func TestIssue2607(t *testing.T) {
+ // This code sequence used to hang.
+ n := NewInt(10)
+ n.Rand(rand.New(rand.NewSource(9)), n)
+}
diff --git a/src/math/big/nat.go b/src/math/big/nat.go
new file mode 100644
index 0000000..16a87f5
--- /dev/null
+++ b/src/math/big/nat.go
@@ -0,0 +1,1508 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package big implements multi-precision arithmetic (big numbers).
+// The following numeric types are supported:
+//
+// - Int signed integers
+// - Rat rational numbers
+//
+// Methods are typically of the form:
+//
+// func (z *Int) Op(x, y *Int) *Int (similar for *Rat)
+//
+// and implement operations z = x Op y with the result as receiver; if it
+// is one of the operands it may be overwritten (and its memory reused).
+// To enable chaining of operations, the result is also returned. Methods
+// returning a result other than *Int or *Rat take one of the operands as
+// the receiver.
+//
+package big
+
+// This file contains operations on unsigned multi-precision integers.
+// These are the building blocks for the operations on signed integers
+// and rationals.
+
+import (
+ "errors"
+ "io"
+ "math"
+ "math/rand"
+ "sync"
+)
+
+// An unsigned integer x of the form
+//
+// x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
+//
+// with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
+// with the digits x[i] as the slice elements.
+//
+// A number is normalized if the slice contains no leading 0 digits.
+// During arithmetic operations, denormalized values may occur but are
+// always normalized before returning the final result. The normalized
+// representation of 0 is the empty or nil slice (length = 0).
+//
+type nat []Word
+
+var (
+ natOne = nat{1}
+ natTwo = nat{2}
+ natTen = nat{10}
+)
+
+func (z nat) clear() {
+ for i := range z {
+ z[i] = 0
+ }
+}
+
+func (z nat) norm() nat {
+ i := len(z)
+ for i > 0 && z[i-1] == 0 {
+ i--
+ }
+ return z[0:i]
+}
+
+func (z nat) make(n int) nat {
+ if n <= cap(z) {
+ return z[0:n] // reuse z
+ }
+ // Choosing a good value for e has significant performance impact
+ // because it increases the chance that a value can be reused.
+ const e = 4 // extra capacity
+ return make(nat, n, n+e)
+}
+
+func (z nat) setWord(x Word) nat {
+ if x == 0 {
+ return z.make(0)
+ }
+ z = z.make(1)
+ z[0] = x
+ return z
+}
+
+func (z nat) setUint64(x uint64) nat {
+ // single-digit values
+ if w := Word(x); uint64(w) == x {
+ return z.setWord(w)
+ }
+
+ // compute number of words n required to represent x
+ n := 0
+ for t := x; t > 0; t >>= _W {
+ n++
+ }
+
+ // split x into n words
+ z = z.make(n)
+ for i := range z {
+ z[i] = Word(x & _M)
+ x >>= _W
+ }
+
+ return z
+}
+
+func (z nat) set(x nat) nat {
+ z = z.make(len(x))
+ copy(z, x)
+ return z
+}
+
+func (z nat) add(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ return z.add(y, x)
+ case m == 0:
+ // n == 0 because m >= n; result is 0
+ return z.make(0)
+ case n == 0:
+ // result is x
+ return z.set(x)
+ }
+ // m > 0
+
+ z = z.make(m + 1)
+ c := addVV(z[0:n], x, y)
+ if m > n {
+ c = addVW(z[n:m], x[n:], c)
+ }
+ z[m] = c
+
+ return z.norm()
+}
+
+func (z nat) sub(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ panic("underflow")
+ case m == 0:
+ // n == 0 because m >= n; result is 0
+ return z.make(0)
+ case n == 0:
+ // result is x
+ return z.set(x)
+ }
+ // m > 0
+
+ z = z.make(m)
+ c := subVV(z[0:n], x, y)
+ if m > n {
+ c = subVW(z[n:], x[n:], c)
+ }
+ if c != 0 {
+ panic("underflow")
+ }
+
+ return z.norm()
+}
+
+func (x nat) cmp(y nat) (r int) {
+ m := len(x)
+ n := len(y)
+ if m != n || m == 0 {
+ switch {
+ case m < n:
+ r = -1
+ case m > n:
+ r = 1
+ }
+ return
+ }
+
+ i := m - 1
+ for i > 0 && x[i] == y[i] {
+ i--
+ }
+
+ switch {
+ case x[i] < y[i]:
+ r = -1
+ case x[i] > y[i]:
+ r = 1
+ }
+ return
+}
+
+func (z nat) mulAddWW(x nat, y, r Word) nat {
+ m := len(x)
+ if m == 0 || y == 0 {
+ return z.setWord(r) // result is r
+ }
+ // m > 0
+
+ z = z.make(m + 1)
+ z[m] = mulAddVWW(z[0:m], x, y, r)
+
+ return z.norm()
+}
+
+// basicMul multiplies x and y and leaves the result in z.
+// The (non-normalized) result is placed in z[0 : len(x) + len(y)].
+func basicMul(z, x, y nat) {
+ z[0 : len(x)+len(y)].clear() // initialize z
+ for i, d := range y {
+ if d != 0 {
+ z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
+ }
+ }
+}
+
+// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
+// Factored out for readability - do not use outside karatsuba.
+func karatsubaAdd(z, x nat, n int) {
+ if c := addVV(z[0:n], z, x); c != 0 {
+ addVW(z[n:n+n>>1], z[n:], c)
+ }
+}
+
+// Like karatsubaAdd, but does subtract.
+func karatsubaSub(z, x nat, n int) {
+ if c := subVV(z[0:n], z, x); c != 0 {
+ subVW(z[n:n+n>>1], z[n:], c)
+ }
+}
+
+// Operands that are shorter than karatsubaThreshold are multiplied using
+// "grade school" multiplication; for longer operands the Karatsuba algorithm
+// is used.
+var karatsubaThreshold int = 40 // computed by calibrate.go
+
+// karatsuba multiplies x and y and leaves the result in z.
+// Both x and y must have the same length n and n must be a
+// power of 2. The result vector z must have len(z) >= 6*n.
+// The (non-normalized) result is placed in z[0 : 2*n].
+func karatsuba(z, x, y nat) {
+ n := len(y)
+
+ // Switch to basic multiplication if numbers are odd or small.
+ // (n is always even if karatsubaThreshold is even, but be
+ // conservative)
+ if n&1 != 0 || n < karatsubaThreshold || n < 2 {
+ basicMul(z, x, y)
+ return
+ }
+ // n&1 == 0 && n >= karatsubaThreshold && n >= 2
+
+ // Karatsuba multiplication is based on the observation that
+ // for two numbers x and y with:
+ //
+ // x = x1*b + x0
+ // y = y1*b + y0
+ //
+ // the product x*y can be obtained with 3 products z2, z1, z0
+ // instead of 4:
+ //
+ // x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
+ // = z2*b*b + z1*b + z0
+ //
+ // with:
+ //
+ // xd = x1 - x0
+ // yd = y0 - y1
+ //
+ // z1 = xd*yd + z2 + z0
+ // = (x1-x0)*(y0 - y1) + z2 + z0
+ // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
+ // = x1*y0 - z2 - z0 + x0*y1 + z2 + z0
+ // = x1*y0 + x0*y1
+
+ // split x, y into "digits"
+ n2 := n >> 1 // n2 >= 1
+ x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
+ y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
+
+ // z is used for the result and temporary storage:
+ //
+ // 6*n 5*n 4*n 3*n 2*n 1*n 0*n
+ // z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
+ //
+ // For each recursive call of karatsuba, an unused slice of
+ // z is passed in that has (at least) half the length of the
+ // caller's z.
+
+ // compute z0 and z2 with the result "in place" in z
+ karatsuba(z, x0, y0) // z0 = x0*y0
+ karatsuba(z[n:], x1, y1) // z2 = x1*y1
+
+ // compute xd (or the negative value if underflow occurs)
+ s := 1 // sign of product xd*yd
+ xd := z[2*n : 2*n+n2]
+ if subVV(xd, x1, x0) != 0 { // x1-x0
+ s = -s
+ subVV(xd, x0, x1) // x0-x1
+ }
+
+ // compute yd (or the negative value if underflow occurs)
+ yd := z[2*n+n2 : 3*n]
+ if subVV(yd, y0, y1) != 0 { // y0-y1
+ s = -s
+ subVV(yd, y1, y0) // y1-y0
+ }
+
+ // p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
+ // p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
+ p := z[n*3:]
+ karatsuba(p, xd, yd)
+
+ // save original z2:z0
+ // (ok to use upper half of z since we're done recursing)
+ r := z[n*4:]
+ copy(r, z[:n*2])
+
+ // add up all partial products
+ //
+ // 2*n n 0
+ // z = [ z2 | z0 ]
+ // + [ z0 ]
+ // + [ z2 ]
+ // + [ p ]
+ //
+ karatsubaAdd(z[n2:], r, n)
+ karatsubaAdd(z[n2:], r[n:], n)
+ if s > 0 {
+ karatsubaAdd(z[n2:], p, n)
+ } else {
+ karatsubaSub(z[n2:], p, n)
+ }
+}
+
+// alias returns true if x and y share the same base array.
+func alias(x, y nat) bool {
+ return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
+}
+
+// addAt implements z += x<<(_W*i); z must be long enough.
+// (we don't use nat.add because we need z to stay the same
+// slice, and we don't need to normalize z after each addition)
+func addAt(z, x nat, i int) {
+ if n := len(x); n > 0 {
+ if c := addVV(z[i:i+n], z[i:], x); c != 0 {
+ j := i + n
+ if j < len(z) {
+ addVW(z[j:], z[j:], c)
+ }
+ }
+ }
+}
+
+func max(x, y int) int {
+ if x > y {
+ return x
+ }
+ return y
+}
+
+// karatsubaLen computes an approximation to the maximum k <= n such that
+// k = p<<i for a number p <= karatsubaThreshold and an i >= 0. Thus, the
+// result is the largest number that can be divided repeatedly by 2 before
+// becoming about the value of karatsubaThreshold.
+func karatsubaLen(n int) int {
+ i := uint(0)
+ for n > karatsubaThreshold {
+ n >>= 1
+ i++
+ }
+ return n << i
+}
+
+func (z nat) mul(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ return z.mul(y, x)
+ case m == 0 || n == 0:
+ return z.make(0)
+ case n == 1:
+ return z.mulAddWW(x, y[0], 0)
+ }
+ // m >= n > 1
+
+ // determine if z can be reused
+ if alias(z, x) || alias(z, y) {
+ z = nil // z is an alias for x or y - cannot reuse
+ }
+
+ // use basic multiplication if the numbers are small
+ if n < karatsubaThreshold {
+ z = z.make(m + n)
+ basicMul(z, x, y)
+ return z.norm()
+ }
+ // m >= n && n >= karatsubaThreshold && n >= 2
+
+ // determine Karatsuba length k such that
+ //
+ // x = xh*b + x0 (0 <= x0 < b)
+ // y = yh*b + y0 (0 <= y0 < b)
+ // b = 1<<(_W*k) ("base" of digits xi, yi)
+ //
+ k := karatsubaLen(n)
+ // k <= n
+
+ // multiply x0 and y0 via Karatsuba
+ x0 := x[0:k] // x0 is not normalized
+ y0 := y[0:k] // y0 is not normalized
+ z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
+ karatsuba(z, x0, y0)
+ z = z[0 : m+n] // z has final length but may be incomplete
+ z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
+
+ // If xh != 0 or yh != 0, add the missing terms to z. For
+ //
+ // xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
+ // yh = y1*b (0 <= y1 < b)
+ //
+ // the missing terms are
+ //
+ // x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
+ //
+ // since all the yi for i > 1 are 0 by choice of k: If any of them
+ // were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
+ // be a larger valid threshold contradicting the assumption about k.
+ //
+ if k < n || m != n {
+ var t nat
+
+ // add x0*y1*b
+ x0 := x0.norm()
+ y1 := y[k:] // y1 is normalized because y is
+ t = t.mul(x0, y1) // update t so we don't lose t's underlying array
+ addAt(z, t, k)
+
+ // add xi*y0<<i, xi*y1*b<<(i+k)
+ y0 := y0.norm()
+ for i := k; i < len(x); i += k {
+ xi := x[i:]
+ if len(xi) > k {
+ xi = xi[:k]
+ }
+ xi = xi.norm()
+ t = t.mul(xi, y0)
+ addAt(z, t, i)
+ t = t.mul(xi, y1)
+ addAt(z, t, i+k)
+ }
+ }
+
+ return z.norm()
+}
+
+// mulRange computes the product of all the unsigned integers in the
+// range [a, b] inclusively. If a > b (empty range), the result is 1.
+func (z nat) mulRange(a, b uint64) nat {
+ switch {
+ case a == 0:
+ // cut long ranges short (optimization)
+ return z.setUint64(0)
+ case a > b:
+ return z.setUint64(1)
+ case a == b:
+ return z.setUint64(a)
+ case a+1 == b:
+ return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
+ }
+ m := (a + b) / 2
+ return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
+}
+
+// q = (x-r)/y, with 0 <= r < y
+func (z nat) divW(x nat, y Word) (q nat, r Word) {
+ m := len(x)
+ switch {
+ case y == 0:
+ panic("division by zero")
+ case y == 1:
+ q = z.set(x) // result is x
+ return
+ case m == 0:
+ q = z.make(0) // result is 0
+ return
+ }
+ // m > 0
+ z = z.make(m)
+ r = divWVW(z, 0, x, y)
+ q = z.norm()
+ return
+}
+
+func (z nat) div(z2, u, v nat) (q, r nat) {
+ if len(v) == 0 {
+ panic("division by zero")
+ }
+
+ if u.cmp(v) < 0 {
+ q = z.make(0)
+ r = z2.set(u)
+ return
+ }
+
+ if len(v) == 1 {
+ var r2 Word
+ q, r2 = z.divW(u, v[0])
+ r = z2.setWord(r2)
+ return
+ }
+
+ q, r = z.divLarge(z2, u, v)
+ return
+}
+
+// q = (uIn-r)/v, with 0 <= r < y
+// Uses z as storage for q, and u as storage for r if possible.
+// See Knuth, Volume 2, section 4.3.1, Algorithm D.
+// Preconditions:
+// len(v) >= 2
+// len(uIn) >= len(v)
+func (z nat) divLarge(u, uIn, v nat) (q, r nat) {
+ n := len(v)
+ m := len(uIn) - n
+
+ // determine if z can be reused
+ // TODO(gri) should find a better solution - this if statement
+ // is very costly (see e.g. time pidigits -s -n 10000)
+ if alias(z, uIn) || alias(z, v) {
+ z = nil // z is an alias for uIn or v - cannot reuse
+ }
+ q = z.make(m + 1)
+
+ qhatv := make(nat, n+1)
+ if alias(u, uIn) || alias(u, v) {
+ u = nil // u is an alias for uIn or v - cannot reuse
+ }
+ u = u.make(len(uIn) + 1)
+ u.clear()
+
+ // D1.
+ shift := leadingZeros(v[n-1])
+ if shift > 0 {
+ // do not modify v, it may be used by another goroutine simultaneously
+ v1 := make(nat, n)
+ shlVU(v1, v, shift)
+ v = v1
+ }
+ u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
+
+ // D2.
+ for j := m; j >= 0; j-- {
+ // D3.
+ qhat := Word(_M)
+ if u[j+n] != v[n-1] {
+ var rhat Word
+ qhat, rhat = divWW(u[j+n], u[j+n-1], v[n-1])
+
+ // x1 | x2 = q̂v_{n-2}
+ x1, x2 := mulWW(qhat, v[n-2])
+ // test if q̂v_{n-2} > br̂ + u_{j+n-2}
+ for greaterThan(x1, x2, rhat, u[j+n-2]) {
+ qhat--
+ prevRhat := rhat
+ rhat += v[n-1]
+ // v[n-1] >= 0, so this tests for overflow.
+ if rhat < prevRhat {
+ break
+ }
+ x1, x2 = mulWW(qhat, v[n-2])
+ }
+ }
+
+ // D4.
+ qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
+
+ c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
+ if c != 0 {
+ c := addVV(u[j:j+n], u[j:], v)
+ u[j+n] += c
+ qhat--
+ }
+
+ q[j] = qhat
+ }
+
+ q = q.norm()
+ shrVU(u, u, shift)
+ r = u.norm()
+
+ return q, r
+}
+
+// Length of x in bits. x must be normalized.
+func (x nat) bitLen() int {
+ if i := len(x) - 1; i >= 0 {
+ return i*_W + bitLen(x[i])
+ }
+ return 0
+}
+
+// MaxBase is the largest number base accepted for string conversions.
+const MaxBase = 'z' - 'a' + 10 + 1 // = hexValue('z') + 1
+
+func hexValue(ch rune) Word {
+ d := int(MaxBase + 1) // illegal base
+ switch {
+ case '0' <= ch && ch <= '9':
+ d = int(ch - '0')
+ case 'a' <= ch && ch <= 'z':
+ d = int(ch - 'a' + 10)
+ case 'A' <= ch && ch <= 'Z':
+ d = int(ch - 'A' + 10)
+ }
+ return Word(d)
+}
+
+// scan sets z to the natural number corresponding to the longest possible prefix
+// read from r representing an unsigned integer in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined. The syntax follows the syntax of
+// unsigned integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z nat) scan(r io.RuneScanner, base int) (nat, int, error) {
+ // reject illegal bases
+ if base < 0 || base == 1 || MaxBase < base {
+ return z, 0, errors.New("illegal number base")
+ }
+
+ // one char look-ahead
+ ch, _, err := r.ReadRune()
+ if err != nil {
+ return z, 0, err
+ }
+
+ // determine base if necessary
+ b := Word(base)
+ if base == 0 {
+ b = 10
+ if ch == '0' {
+ switch ch, _, err = r.ReadRune(); err {
+ case nil:
+ b = 8
+ switch ch {
+ case 'x', 'X':
+ b = 16
+ case 'b', 'B':
+ b = 2
+ }
+ if b == 2 || b == 16 {
+ if ch, _, err = r.ReadRune(); err != nil {
+ return z, 0, err
+ }
+ }
+ case io.EOF:
+ return z.make(0), 10, nil
+ default:
+ return z, 10, err
+ }
+ }
+ }
+
+ // convert string
+ // - group as many digits d as possible together into a "super-digit" dd with "super-base" bb
+ // - only when bb does not fit into a word anymore, do a full number mulAddWW using bb and dd
+ z = z.make(0)
+ bb := Word(1)
+ dd := Word(0)
+ for max := _M / b; ; {
+ d := hexValue(ch)
+ if d >= b {
+ r.UnreadRune() // ch does not belong to number anymore
+ break
+ }
+
+ if bb <= max {
+ bb *= b
+ dd = dd*b + d
+ } else {
+ // bb * b would overflow
+ z = z.mulAddWW(z, bb, dd)
+ bb = b
+ dd = d
+ }
+
+ if ch, _, err = r.ReadRune(); err != nil {
+ if err != io.EOF {
+ return z, int(b), err
+ }
+ break
+ }
+ }
+
+ switch {
+ case bb > 1:
+ // there was at least one mantissa digit
+ z = z.mulAddWW(z, bb, dd)
+ case base == 0 && b == 8:
+ // there was only the octal prefix 0 (possibly followed by digits > 7);
+ // return base 10, not 8
+ return z, 10, nil
+ case base != 0 || b != 8:
+ // there was neither a mantissa digit nor the octal prefix 0
+ return z, int(b), errors.New("syntax error scanning number")
+ }
+
+ return z.norm(), int(b), nil
+}
+
+// Character sets for string conversion.
+const (
+ lowercaseDigits = "0123456789abcdefghijklmnopqrstuvwxyz"
+ uppercaseDigits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+)
+
+// decimalString returns a decimal representation of x.
+// It calls x.string with the charset "0123456789".
+func (x nat) decimalString() string {
+ return x.string(lowercaseDigits[0:10])
+}
+
+// string converts x to a string using digits from a charset; a digit with
+// value d is represented by charset[d]. The conversion base is determined
+// by len(charset), which must be >= 2 and <= 256.
+func (x nat) string(charset string) string {
+ b := Word(len(charset))
+
+ // special cases
+ switch {
+ case b < 2 || MaxBase > 256:
+ panic("illegal base")
+ case len(x) == 0:
+ return string(charset[0])
+ }
+
+ // allocate buffer for conversion
+ i := int(float64(x.bitLen())/math.Log2(float64(b))) + 1 // off by one at most
+ s := make([]byte, i)
+
+ // convert power of two and non power of two bases separately
+ if b == b&-b {
+ // shift is base-b digit size in bits
+ shift := trailingZeroBits(b) // shift > 0 because b >= 2
+ mask := Word(1)<<shift - 1
+ w := x[0]
+ nbits := uint(_W) // number of unprocessed bits in w
+
+ // convert less-significant words
+ for k := 1; k < len(x); k++ {
+ // convert full digits
+ for nbits >= shift {
+ i--
+ s[i] = charset[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ // convert any partial leading digit and advance to next word
+ if nbits == 0 {
+ // no partial digit remaining, just advance
+ w = x[k]
+ nbits = _W
+ } else {
+ // partial digit in current (k-1) and next (k) word
+ w |= x[k] << nbits
+ i--
+ s[i] = charset[w&mask]
+
+ // advance
+ w = x[k] >> (shift - nbits)
+ nbits = _W - (shift - nbits)
+ }
+ }
+
+ // convert digits of most-significant word (omit leading zeros)
+ for nbits >= 0 && w != 0 {
+ i--
+ s[i] = charset[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ } else {
+ // determine "big base"; i.e., the largest possible value bb
+ // that is a power of base b and still fits into a Word
+ // (as in 10^19 for 19 decimal digits in a 64bit Word)
+ bb := b // big base is b**ndigits
+ ndigits := 1 // number of base b digits
+ for max := Word(_M / b); bb <= max; bb *= b {
+ ndigits++ // maximize ndigits where bb = b**ndigits, bb <= _M
+ }
+
+ // construct table of successive squares of bb*leafSize to use in subdivisions
+ // result (table != nil) <=> (len(x) > leafSize > 0)
+ table := divisors(len(x), b, ndigits, bb)
+
+ // preserve x, create local copy for use by convertWords
+ q := nat(nil).set(x)
+
+ // convert q to string s in base b
+ q.convertWords(s, charset, b, ndigits, bb, table)
+
+ // strip leading zeros
+ // (x != 0; thus s must contain at least one non-zero digit
+ // and the loop will terminate)
+ i = 0
+ for zero := charset[0]; s[i] == zero; {
+ i++
+ }
+ }
+
+ return string(s[i:])
+}
+
+// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
+// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
+// repeated nat/Word division.
+//
+// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
+// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
+// Recursive conversion divides q by its approximate square root, yielding two parts, each half
+// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
+// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
+// is made better by splitting the subblocks recursively. Best is to split blocks until one more
+// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
+// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
+// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
+// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
+// specific hardware.
+//
+func (q nat) convertWords(s []byte, charset string, b Word, ndigits int, bb Word, table []divisor) {
+ // split larger blocks recursively
+ if table != nil {
+ // len(q) > leafSize > 0
+ var r nat
+ index := len(table) - 1
+ for len(q) > leafSize {
+ // find divisor close to sqrt(q) if possible, but in any case < q
+ maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length
+ minLength := maxLength >> 1 // ~= log2 sqrt(q)
+ for index > 0 && table[index-1].nbits > minLength {
+ index-- // desired
+ }
+ if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
+ index--
+ if index < 0 {
+ panic("internal inconsistency")
+ }
+ }
+
+ // split q into the two digit number (q'*bbb + r) to form independent subblocks
+ q, r = q.div(r, q, table[index].bbb)
+
+ // convert subblocks and collect results in s[:h] and s[h:]
+ h := len(s) - table[index].ndigits
+ r.convertWords(s[h:], charset, b, ndigits, bb, table[0:index])
+ s = s[:h] // == q.convertWords(s, charset, b, ndigits, bb, table[0:index+1])
+ }
+ }
+
+ // having split any large blocks now process the remaining (small) block iteratively
+ i := len(s)
+ var r Word
+ if b == 10 {
+ // hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ // avoid % computation since r%10 == r - int(r/10)*10;
+ // this appears to be faster for BenchmarkString10000Base10
+ // and smaller strings (but a bit slower for larger ones)
+ t := r / 10
+ s[i] = charset[r-t<<3-t-t] // TODO(gri) replace w/ t*10 once compiler produces better code
+ r = t
+ }
+ }
+ } else {
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ s[i] = charset[r%b]
+ r /= b
+ }
+ }
+ }
+
+ // prepend high-order zeroes
+ zero := charset[0]
+ for i > 0 { // while need more leading zeroes
+ i--
+ s[i] = zero
+ }
+}
+
+// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
+// Benchmark and configure leafSize using: go test -bench="Leaf"
+// 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
+// 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
+var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
+
+type divisor struct {
+ bbb nat // divisor
+ nbits int // bit length of divisor (discounting leading zeroes) ~= log2(bbb)
+ ndigits int // digit length of divisor in terms of output base digits
+}
+
+var cacheBase10 struct {
+ sync.Mutex
+ table [64]divisor // cached divisors for base 10
+}
+
+// expWW computes x**y
+func (z nat) expWW(x, y Word) nat {
+ return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
+}
+
+// construct table of powers of bb*leafSize to use in subdivisions
+func divisors(m int, b Word, ndigits int, bb Word) []divisor {
+ // only compute table when recursive conversion is enabled and x is large
+ if leafSize == 0 || m <= leafSize {
+ return nil
+ }
+
+ // determine k where (bb**leafSize)**(2**k) >= sqrt(x)
+ k := 1
+ for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
+ k++
+ }
+
+ // reuse and extend existing table of divisors or create new table as appropriate
+ var table []divisor // for b == 10, table overlaps with cacheBase10.table
+ if b == 10 {
+ cacheBase10.Lock()
+ table = cacheBase10.table[0:k] // reuse old table for this conversion
+ } else {
+ table = make([]divisor, k) // create new table for this conversion
+ }
+
+ // extend table
+ if table[k-1].ndigits == 0 {
+ // add new entries as needed
+ var larger nat
+ for i := 0; i < k; i++ {
+ if table[i].ndigits == 0 {
+ if i == 0 {
+ table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
+ table[0].ndigits = ndigits * leafSize
+ } else {
+ table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
+ table[i].ndigits = 2 * table[i-1].ndigits
+ }
+
+ // optimization: exploit aggregated extra bits in macro blocks
+ larger = nat(nil).set(table[i].bbb)
+ for mulAddVWW(larger, larger, b, 0) == 0 {
+ table[i].bbb = table[i].bbb.set(larger)
+ table[i].ndigits++
+ }
+
+ table[i].nbits = table[i].bbb.bitLen()
+ }
+ }
+ }
+
+ if b == 10 {
+ cacheBase10.Unlock()
+ }
+
+ return table
+}
+
+const deBruijn32 = 0x077CB531
+
+var deBruijn32Lookup = []byte{
+ 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
+ 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
+}
+
+const deBruijn64 = 0x03f79d71b4ca8b09
+
+var deBruijn64Lookup = []byte{
+ 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
+ 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
+ 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
+ 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func trailingZeroBits(x Word) uint {
+ // x & -x leaves only the right-most bit set in the word. Let k be the
+ // index of that bit. Since only a single bit is set, the value is two
+ // to the power of k. Multiplying by a power of two is equivalent to
+ // left shifting, in this case by k bits. The de Bruijn constant is
+ // such that all six bit, consecutive substrings are distinct.
+ // Therefore, if we have a left shifted version of this constant we can
+ // find by how many bits it was shifted by looking at which six bit
+ // substring ended up at the top of the word.
+ // (Knuth, volume 4, section 7.3.1)
+ switch _W {
+ case 32:
+ return uint(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
+ case 64:
+ return uint(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
+ default:
+ panic("unknown word size")
+ }
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func (x nat) trailingZeroBits() uint {
+ if len(x) == 0 {
+ return 0
+ }
+ var i uint
+ for x[i] == 0 {
+ i++
+ }
+ // x[i] != 0
+ return i*_W + trailingZeroBits(x[i])
+}
+
+// z = x << s
+func (z nat) shl(x nat, s uint) nat {
+ m := len(x)
+ if m == 0 {
+ return z.make(0)
+ }
+ // m > 0
+
+ n := m + int(s/_W)
+ z = z.make(n + 1)
+ z[n] = shlVU(z[n-m:n], x, s%_W)
+ z[0 : n-m].clear()
+
+ return z.norm()
+}
+
+// z = x >> s
+func (z nat) shr(x nat, s uint) nat {
+ m := len(x)
+ n := m - int(s/_W)
+ if n <= 0 {
+ return z.make(0)
+ }
+ // n > 0
+
+ z = z.make(n)
+ shrVU(z, x[m-n:], s%_W)
+
+ return z.norm()
+}
+
+func (z nat) setBit(x nat, i uint, b uint) nat {
+ j := int(i / _W)
+ m := Word(1) << (i % _W)
+ n := len(x)
+ switch b {
+ case 0:
+ z = z.make(n)
+ copy(z, x)
+ if j >= n {
+ // no need to grow
+ return z
+ }
+ z[j] &^= m
+ return z.norm()
+ case 1:
+ if j >= n {
+ z = z.make(j + 1)
+ z[n:].clear()
+ } else {
+ z = z.make(n)
+ }
+ copy(z, x)
+ z[j] |= m
+ // no need to normalize
+ return z
+ }
+ panic("set bit is not 0 or 1")
+}
+
+func (z nat) bit(i uint) uint {
+ j := int(i / _W)
+ if j >= len(z) {
+ return 0
+ }
+ return uint(z[j] >> (i % _W) & 1)
+}
+
+func (z nat) and(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ if m > n {
+ m = n
+ }
+ // m <= n
+
+ z = z.make(m)
+ for i := 0; i < m; i++ {
+ z[i] = x[i] & y[i]
+ }
+
+ return z.norm()
+}
+
+func (z nat) andNot(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ if n > m {
+ n = m
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] &^ y[i]
+ }
+ copy(z[n:m], x[n:m])
+
+ return z.norm()
+}
+
+func (z nat) or(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ s := x
+ if m < n {
+ n, m = m, n
+ s = y
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] | y[i]
+ }
+ copy(z[n:m], s[n:m])
+
+ return z.norm()
+}
+
+func (z nat) xor(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ s := x
+ if m < n {
+ n, m = m, n
+ s = y
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] ^ y[i]
+ }
+ copy(z[n:m], s[n:m])
+
+ return z.norm()
+}
+
+// greaterThan returns true iff (x1<<_W + x2) > (y1<<_W + y2)
+func greaterThan(x1, x2, y1, y2 Word) bool {
+ return x1 > y1 || x1 == y1 && x2 > y2
+}
+
+// modW returns x % d.
+func (x nat) modW(d Word) (r Word) {
+ // TODO(agl): we don't actually need to store the q value.
+ var q nat
+ q = q.make(len(x))
+ return divWVW(q, 0, x, d)
+}
+
+// random creates a random integer in [0..limit), using the space in z if
+// possible. n is the bit length of limit.
+func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
+ if alias(z, limit) {
+ z = nil // z is an alias for limit - cannot reuse
+ }
+ z = z.make(len(limit))
+
+ bitLengthOfMSW := uint(n % _W)
+ if bitLengthOfMSW == 0 {
+ bitLengthOfMSW = _W
+ }
+ mask := Word((1 << bitLengthOfMSW) - 1)
+
+ for {
+ switch _W {
+ case 32:
+ for i := range z {
+ z[i] = Word(rand.Uint32())
+ }
+ case 64:
+ for i := range z {
+ z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
+ }
+ default:
+ panic("unknown word size")
+ }
+ z[len(limit)-1] &= mask
+ if z.cmp(limit) < 0 {
+ break
+ }
+ }
+
+ return z.norm()
+}
+
+// If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
+// otherwise it sets z to x**y. The result is the value of z.
+func (z nat) expNN(x, y, m nat) nat {
+ if alias(z, x) || alias(z, y) {
+ // We cannot allow in-place modification of x or y.
+ z = nil
+ }
+
+ // x**y mod 1 == 0
+ if len(m) == 1 && m[0] == 1 {
+ return z.setWord(0)
+ }
+ // m == 0 || m > 1
+
+ // x**0 == 1
+ if len(y) == 0 {
+ return z.setWord(1)
+ }
+ // y > 0
+
+ if len(m) != 0 {
+ // We likely end up being as long as the modulus.
+ z = z.make(len(m))
+ }
+ z = z.set(x)
+
+ // If the base is non-trivial and the exponent is large, we use
+ // 4-bit, windowed exponentiation. This involves precomputing 14 values
+ // (x^2...x^15) but then reduces the number of multiply-reduces by a
+ // third. Even for a 32-bit exponent, this reduces the number of
+ // operations.
+ if len(x) > 1 && len(y) > 1 && len(m) > 0 {
+ return z.expNNWindowed(x, y, m)
+ }
+
+ v := y[len(y)-1] // v > 0 because y is normalized and y > 0
+ shift := leadingZeros(v) + 1
+ v <<= shift
+ var q nat
+
+ const mask = 1 << (_W - 1)
+
+ // We walk through the bits of the exponent one by one. Each time we
+ // see a bit, we square, thus doubling the power. If the bit is a one,
+ // we also multiply by x, thus adding one to the power.
+
+ w := _W - int(shift)
+ // zz and r are used to avoid allocating in mul and div as
+ // otherwise the arguments would alias.
+ var zz, r nat
+ for j := 0; j < w; j++ {
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+
+ if v&mask != 0 {
+ zz = zz.mul(z, x)
+ zz, z = z, zz
+ }
+
+ if len(m) != 0 {
+ zz, r = zz.div(r, z, m)
+ zz, r, q, z = q, z, zz, r
+ }
+
+ v <<= 1
+ }
+
+ for i := len(y) - 2; i >= 0; i-- {
+ v = y[i]
+
+ for j := 0; j < _W; j++ {
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+
+ if v&mask != 0 {
+ zz = zz.mul(z, x)
+ zz, z = z, zz
+ }
+
+ if len(m) != 0 {
+ zz, r = zz.div(r, z, m)
+ zz, r, q, z = q, z, zz, r
+ }
+
+ v <<= 1
+ }
+ }
+
+ return z.norm()
+}
+
+// expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
+func (z nat) expNNWindowed(x, y, m nat) nat {
+ // zz and r are used to avoid allocating in mul and div as otherwise
+ // the arguments would alias.
+ var zz, r nat
+
+ const n = 4
+ // powers[i] contains x^i.
+ var powers [1 << n]nat
+ powers[0] = natOne
+ powers[1] = x
+ for i := 2; i < 1<<n; i += 2 {
+ p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
+ *p = p.mul(*p2, *p2)
+ zz, r = zz.div(r, *p, m)
+ *p, r = r, *p
+ *p1 = p1.mul(*p, x)
+ zz, r = zz.div(r, *p1, m)
+ *p1, r = r, *p1
+ }
+
+ z = z.setWord(1)
+
+ for i := len(y) - 1; i >= 0; i-- {
+ yi := y[i]
+ for j := 0; j < _W; j += n {
+ if i != len(y)-1 || j != 0 {
+ // Unrolled loop for significant performance
+ // gain. Use go test -bench=".*" in crypto/rsa
+ // to check performance before making changes.
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+ }
+
+ zz = zz.mul(z, powers[yi>>(_W-n)])
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ yi <<= n
+ }
+ }
+
+ return z.norm()
+}
+
+// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
+// If it returns true, n is prime with probability 1 - 1/4^reps.
+// If it returns false, n is not prime.
+func (n nat) probablyPrime(reps int) bool {
+ if len(n) == 0 {
+ return false
+ }
+
+ if len(n) == 1 {
+ if n[0] < 2 {
+ return false
+ }
+
+ if n[0]%2 == 0 {
+ return n[0] == 2
+ }
+
+ // We have to exclude these cases because we reject all
+ // multiples of these numbers below.
+ switch n[0] {
+ case 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53:
+ return true
+ }
+ }
+
+ const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29}
+ const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
+
+ var r Word
+ switch _W {
+ case 32:
+ r = n.modW(primesProduct32)
+ case 64:
+ r = n.modW(primesProduct64 & _M)
+ default:
+ panic("Unknown word size")
+ }
+
+ if r%3 == 0 || r%5 == 0 || r%7 == 0 || r%11 == 0 ||
+ r%13 == 0 || r%17 == 0 || r%19 == 0 || r%23 == 0 || r%29 == 0 {
+ return false
+ }
+
+ if _W == 64 && (r%31 == 0 || r%37 == 0 || r%41 == 0 ||
+ r%43 == 0 || r%47 == 0 || r%53 == 0) {
+ return false
+ }
+
+ nm1 := nat(nil).sub(n, natOne)
+ // determine q, k such that nm1 = q << k
+ k := nm1.trailingZeroBits()
+ q := nat(nil).shr(nm1, k)
+
+ nm3 := nat(nil).sub(nm1, natTwo)
+ rand := rand.New(rand.NewSource(int64(n[0])))
+
+ var x, y, quotient nat
+ nm3Len := nm3.bitLen()
+
+NextRandom:
+ for i := 0; i < reps; i++ {
+ x = x.random(rand, nm3, nm3Len)
+ x = x.add(x, natTwo)
+ y = y.expNN(x, q, n)
+ if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
+ continue
+ }
+ for j := uint(1); j < k; j++ {
+ y = y.mul(y, y)
+ quotient, y = quotient.div(y, y, n)
+ if y.cmp(nm1) == 0 {
+ continue NextRandom
+ }
+ if y.cmp(natOne) == 0 {
+ return false
+ }
+ }
+ return false
+ }
+
+ return true
+}
+
+// bytes writes the value of z into buf using big-endian encoding.
+// len(buf) must be >= len(z)*_S. The value of z is encoded in the
+// slice buf[i:]. The number i of unused bytes at the beginning of
+// buf is returned as result.
+func (z nat) bytes(buf []byte) (i int) {
+ i = len(buf)
+ for _, d := range z {
+ for j := 0; j < _S; j++ {
+ i--
+ buf[i] = byte(d)
+ d >>= 8
+ }
+ }
+
+ for i < len(buf) && buf[i] == 0 {
+ i++
+ }
+
+ return
+}
+
+// setBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z nat) setBytes(buf []byte) nat {
+ z = z.make((len(buf) + _S - 1) / _S)
+
+ k := 0
+ s := uint(0)
+ var d Word
+ for i := len(buf); i > 0; i-- {
+ d |= Word(buf[i-1]) << s
+ if s += 8; s == _S*8 {
+ z[k] = d
+ k++
+ s = 0
+ d = 0
+ }
+ }
+ if k < len(z) {
+ z[k] = d
+ }
+
+ return z.norm()
+}
diff --git a/src/math/big/nat_test.go b/src/math/big/nat_test.go
new file mode 100644
index 0000000..a2ae533
--- /dev/null
+++ b/src/math/big/nat_test.go
@@ -0,0 +1,771 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "io"
+ "runtime"
+ "strings"
+ "testing"
+)
+
+var cmpTests = []struct {
+ x, y nat
+ r int
+}{
+ {nil, nil, 0},
+ {nil, nat(nil), 0},
+ {nat(nil), nil, 0},
+ {nat(nil), nat(nil), 0},
+ {nat{0}, nat{0}, 0},
+ {nat{0}, nat{1}, -1},
+ {nat{1}, nat{0}, 1},
+ {nat{1}, nat{1}, 0},
+ {nat{0, _M}, nat{1}, 1},
+ {nat{1}, nat{0, _M}, -1},
+ {nat{1, _M}, nat{0, _M}, 1},
+ {nat{0, _M}, nat{1, _M}, -1},
+ {nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
+ {nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
+}
+
+func TestCmp(t *testing.T) {
+ for i, a := range cmpTests {
+ r := a.x.cmp(a.y)
+ if r != a.r {
+ t.Errorf("#%d got r = %v; want %v", i, r, a.r)
+ }
+ }
+}
+
+type funNN func(z, x, y nat) nat
+type argNN struct {
+ z, x, y nat
+}
+
+var sumNN = []argNN{
+ {},
+ {nat{1}, nil, nat{1}},
+ {nat{1111111110}, nat{123456789}, nat{987654321}},
+ {nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
+ {nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
+ {nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
+}
+
+var prodNN = []argNN{
+ {},
+ {nil, nil, nil},
+ {nil, nat{991}, nil},
+ {nat{991}, nat{991}, nat{1}},
+ {nat{991 * 991}, nat{991}, nat{991}},
+ {nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
+ {nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
+ {nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
+ // 3^100 * 3^28 = 3^128
+ {
+ natFromString("11790184577738583171520872861412518665678211592275841109096961"),
+ natFromString("515377520732011331036461129765621272702107522001"),
+ natFromString("22876792454961"),
+ },
+ // z = 111....1 (70000 digits)
+ // x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
+ // y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
+ {
+ natFromString(strings.Repeat("1", 70000)),
+ natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
+ natFromString(strings.Repeat("1", 700)),
+ },
+ // z = 111....1 (20000 digits)
+ // x = 10^10000 + 1
+ // y = 111....1 (10000 digits)
+ {
+ natFromString(strings.Repeat("1", 20000)),
+ natFromString("1" + strings.Repeat("0", 9999) + "1"),
+ natFromString(strings.Repeat("1", 10000)),
+ },
+}
+
+func natFromString(s string) nat {
+ x, _, err := nat(nil).scan(strings.NewReader(s), 0)
+ if err != nil {
+ panic(err)
+ }
+ return x
+}
+
+func TestSet(t *testing.T) {
+ for _, a := range sumNN {
+ z := nat(nil).set(a.z)
+ if z.cmp(a.z) != 0 {
+ t.Errorf("got z = %v; want %v", z, a.z)
+ }
+ }
+}
+
+func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
+ z := f(nil, a.x, a.y)
+ if z.cmp(a.z) != 0 {
+ t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
+ }
+}
+
+func TestFunNN(t *testing.T) {
+ for _, a := range sumNN {
+ arg := a
+ testFunNN(t, "add", nat.add, arg)
+
+ arg = argNN{a.z, a.y, a.x}
+ testFunNN(t, "add symmetric", nat.add, arg)
+
+ arg = argNN{a.x, a.z, a.y}
+ testFunNN(t, "sub", nat.sub, arg)
+
+ arg = argNN{a.y, a.z, a.x}
+ testFunNN(t, "sub symmetric", nat.sub, arg)
+ }
+
+ for _, a := range prodNN {
+ arg := a
+ testFunNN(t, "mul", nat.mul, arg)
+
+ arg = argNN{a.z, a.y, a.x}
+ testFunNN(t, "mul symmetric", nat.mul, arg)
+ }
+}
+
+var mulRangesN = []struct {
+ a, b uint64
+ prod string
+}{
+ {0, 0, "0"},
+ {1, 1, "1"},
+ {1, 2, "2"},
+ {1, 3, "6"},
+ {10, 10, "10"},
+ {0, 100, "0"},
+ {0, 1e9, "0"},
+ {1, 0, "1"}, // empty range
+ {100, 1, "1"}, // empty range
+ {1, 10, "3628800"}, // 10!
+ {1, 20, "2432902008176640000"}, // 20!
+ {1, 100,
+ "933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "51185210916864000000000000000000000000", // 100!
+ },
+}
+
+func TestMulRangeN(t *testing.T) {
+ for i, r := range mulRangesN {
+ prod := nat(nil).mulRange(r.a, r.b).decimalString()
+ if prod != r.prod {
+ t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+// allocBytes returns the number of bytes allocated by invoking f.
+func allocBytes(f func()) uint64 {
+ var stats runtime.MemStats
+ runtime.ReadMemStats(&stats)
+ t := stats.TotalAlloc
+ f()
+ runtime.ReadMemStats(&stats)
+ return stats.TotalAlloc - t
+}
+
+// TestMulUnbalanced tests that multiplying numbers of different lengths
+// does not cause deep recursion and in turn allocate too much memory.
+// Test case for issue 3807.
+func TestMulUnbalanced(t *testing.T) {
+ defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
+ x := rndNat(50000)
+ y := rndNat(40)
+ allocSize := allocBytes(func() {
+ nat(nil).mul(x, y)
+ })
+ inputSize := uint64(len(x)+len(y)) * _S
+ if ratio := allocSize / uint64(inputSize); ratio > 10 {
+ t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
+ }
+}
+
+func rndNat(n int) nat {
+ return nat(rndV(n)).norm()
+}
+
+func BenchmarkMul(b *testing.B) {
+ mulx := rndNat(1e4)
+ muly := rndNat(1e4)
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ var z nat
+ z.mul(mulx, muly)
+ }
+}
+
+func toString(x nat, charset string) string {
+ base := len(charset)
+
+ // special cases
+ switch {
+ case base < 2:
+ panic("illegal base")
+ case len(x) == 0:
+ return string(charset[0])
+ }
+
+ // allocate buffer for conversion
+ i := x.bitLen()/log2(Word(base)) + 1 // +1: round up
+ s := make([]byte, i)
+
+ // don't destroy x
+ q := nat(nil).set(x)
+
+ // convert
+ for len(q) > 0 {
+ i--
+ var r Word
+ q, r = q.divW(q, Word(base))
+ s[i] = charset[r]
+ }
+
+ return string(s[i:])
+}
+
+var strTests = []struct {
+ x nat // nat value to be converted
+ c string // conversion charset
+ s string // expected result
+}{
+ {nil, "01", "0"},
+ {nat{1}, "01", "1"},
+ {nat{0xc5}, "01", "11000101"},
+ {nat{03271}, lowercaseDigits[0:8], "3271"},
+ {nat{10}, lowercaseDigits[0:10], "10"},
+ {nat{1234567890}, uppercaseDigits[0:10], "1234567890"},
+ {nat{0xdeadbeef}, lowercaseDigits[0:16], "deadbeef"},
+ {nat{0xdeadbeef}, uppercaseDigits[0:16], "DEADBEEF"},
+ {nat{0x229be7}, lowercaseDigits[0:17], "1a2b3c"},
+ {nat{0x309663e6}, uppercaseDigits[0:32], "O9COV6"},
+}
+
+func TestString(t *testing.T) {
+ for _, a := range strTests {
+ s := a.x.string(a.c)
+ if s != a.s {
+ t.Errorf("string%+v\n\tgot s = %s; want %s", a, s, a.s)
+ }
+
+ x, b, err := nat(nil).scan(strings.NewReader(a.s), len(a.c))
+ if x.cmp(a.x) != 0 {
+ t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != len(a.c) {
+ t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, len(a.c))
+ }
+ if err != nil {
+ t.Errorf("scan%+v\n\tgot error = %s", a, err)
+ }
+ }
+}
+
+var natScanTests = []struct {
+ s string // string to be scanned
+ base int // input base
+ x nat // expected nat
+ b int // expected base
+ ok bool // expected success
+ next rune // next character (or 0, if at EOF)
+}{
+ // error: illegal base
+ {base: -1},
+ {base: 1},
+ {base: 37},
+
+ // error: no mantissa
+ {},
+ {s: "?"},
+ {base: 10},
+ {base: 36},
+ {s: "?", base: 10},
+ {s: "0x"},
+ {s: "345", base: 2},
+
+ // no errors
+ {"0", 0, nil, 10, true, 0},
+ {"0", 10, nil, 10, true, 0},
+ {"0", 36, nil, 36, true, 0},
+ {"1", 0, nat{1}, 10, true, 0},
+ {"1", 10, nat{1}, 10, true, 0},
+ {"0 ", 0, nil, 10, true, ' '},
+ {"08", 0, nil, 10, true, '8'},
+ {"018", 0, nat{1}, 8, true, '8'},
+ {"0b1", 0, nat{1}, 2, true, 0},
+ {"0b11000101", 0, nat{0xc5}, 2, true, 0},
+ {"03271", 0, nat{03271}, 8, true, 0},
+ {"10ab", 0, nat{10}, 10, true, 'a'},
+ {"1234567890", 0, nat{1234567890}, 10, true, 0},
+ {"xyz", 36, nat{(33*36+34)*36 + 35}, 36, true, 0},
+ {"xyz?", 36, nat{(33*36+34)*36 + 35}, 36, true, '?'},
+ {"0x", 16, nil, 16, true, 'x'},
+ {"0xdeadbeef", 0, nat{0xdeadbeef}, 16, true, 0},
+ {"0XDEADBEEF", 0, nat{0xdeadbeef}, 16, true, 0},
+}
+
+func TestScanBase(t *testing.T) {
+ for _, a := range natScanTests {
+ r := strings.NewReader(a.s)
+ x, b, err := nat(nil).scan(r, a.base)
+ if err == nil && !a.ok {
+ t.Errorf("scan%+v\n\texpected error", a)
+ }
+ if err != nil {
+ if a.ok {
+ t.Errorf("scan%+v\n\tgot error = %s", a, err)
+ }
+ continue
+ }
+ if x.cmp(a.x) != 0 {
+ t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != a.b {
+ t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, a.base)
+ }
+ next, _, err := r.ReadRune()
+ if err == io.EOF {
+ next = 0
+ err = nil
+ }
+ if err == nil && next != a.next {
+ t.Errorf("scan%+v\n\tgot next = %q; want %q", a, next, a.next)
+ }
+ }
+}
+
+var pi = "3" +
+ "14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651" +
+ "32823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461" +
+ "28475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920" +
+ "96282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179" +
+ "31051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798" +
+ "60943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901" +
+ "22495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837" +
+ "29780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083" +
+ "81420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909" +
+ "21642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151" +
+ "55748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035" +
+ "63707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104" +
+ "75216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992" +
+ "45863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818" +
+ "34797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548" +
+ "16136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179" +
+ "04946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886" +
+ "26945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645" +
+ "99581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745" +
+ "53050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382" +
+ "68683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244" +
+ "13654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767" +
+ "88952521385225499546667278239864565961163548862305774564980355936345681743241125150760694794510965960940252288" +
+ "79710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821" +
+ "68299894872265880485756401427047755513237964145152374623436454285844479526586782105114135473573952311342716610" +
+ "21359695362314429524849371871101457654035902799344037420073105785390621983874478084784896833214457138687519435" +
+ "06430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675" +
+ "14269123974894090718649423196156794520809514655022523160388193014209376213785595663893778708303906979207734672" +
+ "21825625996615014215030680384477345492026054146659252014974428507325186660021324340881907104863317346496514539" +
+ "05796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007" +
+ "23055876317635942187312514712053292819182618612586732157919841484882916447060957527069572209175671167229109816" +
+ "90915280173506712748583222871835209353965725121083579151369882091444210067510334671103141267111369908658516398" +
+ "31501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064" +
+ "20467525907091548141654985946163718027098199430992448895757128289059232332609729971208443357326548938239119325" +
+ "97463667305836041428138830320382490375898524374417029132765618093773444030707469211201913020330380197621101100" +
+ "44929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915" +
+ "44110104468232527162010526522721116603966655730925471105578537634668206531098965269186205647693125705863566201" +
+ "85581007293606598764861179104533488503461136576867532494416680396265797877185560845529654126654085306143444318" +
+ "58676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797" +
+ "27082668306343285878569830523580893306575740679545716377525420211495576158140025012622859413021647155097925923" +
+ "09907965473761255176567513575178296664547791745011299614890304639947132962107340437518957359614589019389713111" +
+ "79042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043" +
+ "06332175182979866223717215916077166925474873898665494945011465406284336639379003976926567214638530673609657120" +
+ "91807638327166416274888800786925602902284721040317211860820419000422966171196377921337575114959501566049631862" +
+ "94726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939" +
+ "78054193414473774418426312986080998886874132604721569516239658645730216315981931951673538129741677294786724229" +
+ "24654366800980676928238280689964004824354037014163149658979409243237896907069779422362508221688957383798623001" +
+ "59377647165122893578601588161755782973523344604281512627203734314653197777416031990665541876397929334419521541" +
+ "34189948544473456738316249934191318148092777710386387734317720754565453220777092120190516609628049092636019759" +
+ "88281613323166636528619326686336062735676303544776280350450777235547105859548702790814356240145171806246436267" +
+ "94561275318134078330336254232783944975382437205835311477119926063813346776879695970309833913077109870408591337"
+
+// Test case for BenchmarkScanPi.
+func TestScanPi(t *testing.T) {
+ var x nat
+ z, _, err := x.scan(strings.NewReader(pi), 10)
+ if err != nil {
+ t.Errorf("scanning pi: %s", err)
+ }
+ if s := z.decimalString(); s != pi {
+ t.Errorf("scanning pi: got %s", s)
+ }
+}
+
+func TestScanPiParallel(t *testing.T) {
+ const n = 2
+ c := make(chan int)
+ for i := 0; i < n; i++ {
+ go func() {
+ TestScanPi(t)
+ c <- 0
+ }()
+ }
+ for i := 0; i < n; i++ {
+ <-c
+ }
+}
+
+func BenchmarkScanPi(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ var x nat
+ x.scan(strings.NewReader(pi), 10)
+ }
+}
+
+func BenchmarkStringPiParallel(b *testing.B) {
+ var x nat
+ x, _, _ = x.scan(strings.NewReader(pi), 0)
+ if x.decimalString() != pi {
+ panic("benchmark incorrect: conversion failed")
+ }
+ b.RunParallel(func(pb *testing.PB) {
+ for pb.Next() {
+ x.decimalString()
+ }
+ })
+}
+
+func BenchmarkScan10Base2(b *testing.B) { ScanHelper(b, 2, 10, 10) }
+func BenchmarkScan100Base2(b *testing.B) { ScanHelper(b, 2, 10, 100) }
+func BenchmarkScan1000Base2(b *testing.B) { ScanHelper(b, 2, 10, 1000) }
+func BenchmarkScan10000Base2(b *testing.B) { ScanHelper(b, 2, 10, 10000) }
+func BenchmarkScan100000Base2(b *testing.B) { ScanHelper(b, 2, 10, 100000) }
+
+func BenchmarkScan10Base8(b *testing.B) { ScanHelper(b, 8, 10, 10) }
+func BenchmarkScan100Base8(b *testing.B) { ScanHelper(b, 8, 10, 100) }
+func BenchmarkScan1000Base8(b *testing.B) { ScanHelper(b, 8, 10, 1000) }
+func BenchmarkScan10000Base8(b *testing.B) { ScanHelper(b, 8, 10, 10000) }
+func BenchmarkScan100000Base8(b *testing.B) { ScanHelper(b, 8, 10, 100000) }
+
+func BenchmarkScan10Base10(b *testing.B) { ScanHelper(b, 10, 10, 10) }
+func BenchmarkScan100Base10(b *testing.B) { ScanHelper(b, 10, 10, 100) }
+func BenchmarkScan1000Base10(b *testing.B) { ScanHelper(b, 10, 10, 1000) }
+func BenchmarkScan10000Base10(b *testing.B) { ScanHelper(b, 10, 10, 10000) }
+func BenchmarkScan100000Base10(b *testing.B) { ScanHelper(b, 10, 10, 100000) }
+
+func BenchmarkScan10Base16(b *testing.B) { ScanHelper(b, 16, 10, 10) }
+func BenchmarkScan100Base16(b *testing.B) { ScanHelper(b, 16, 10, 100) }
+func BenchmarkScan1000Base16(b *testing.B) { ScanHelper(b, 16, 10, 1000) }
+func BenchmarkScan10000Base16(b *testing.B) { ScanHelper(b, 16, 10, 10000) }
+func BenchmarkScan100000Base16(b *testing.B) { ScanHelper(b, 16, 10, 100000) }
+
+func ScanHelper(b *testing.B, base int, x, y Word) {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(x, y)
+
+ var s string
+ s = z.string(lowercaseDigits[0:base])
+ if t := toString(z, lowercaseDigits[0:base]); t != s {
+ b.Fatalf("scanning: got %s; want %s", s, t)
+ }
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ z.scan(strings.NewReader(s), base)
+ }
+}
+
+func BenchmarkString10Base2(b *testing.B) { StringHelper(b, 2, 10, 10) }
+func BenchmarkString100Base2(b *testing.B) { StringHelper(b, 2, 10, 100) }
+func BenchmarkString1000Base2(b *testing.B) { StringHelper(b, 2, 10, 1000) }
+func BenchmarkString10000Base2(b *testing.B) { StringHelper(b, 2, 10, 10000) }
+func BenchmarkString100000Base2(b *testing.B) { StringHelper(b, 2, 10, 100000) }
+
+func BenchmarkString10Base8(b *testing.B) { StringHelper(b, 8, 10, 10) }
+func BenchmarkString100Base8(b *testing.B) { StringHelper(b, 8, 10, 100) }
+func BenchmarkString1000Base8(b *testing.B) { StringHelper(b, 8, 10, 1000) }
+func BenchmarkString10000Base8(b *testing.B) { StringHelper(b, 8, 10, 10000) }
+func BenchmarkString100000Base8(b *testing.B) { StringHelper(b, 8, 10, 100000) }
+
+func BenchmarkString10Base10(b *testing.B) { StringHelper(b, 10, 10, 10) }
+func BenchmarkString100Base10(b *testing.B) { StringHelper(b, 10, 10, 100) }
+func BenchmarkString1000Base10(b *testing.B) { StringHelper(b, 10, 10, 1000) }
+func BenchmarkString10000Base10(b *testing.B) { StringHelper(b, 10, 10, 10000) }
+func BenchmarkString100000Base10(b *testing.B) { StringHelper(b, 10, 10, 100000) }
+
+func BenchmarkString10Base16(b *testing.B) { StringHelper(b, 16, 10, 10) }
+func BenchmarkString100Base16(b *testing.B) { StringHelper(b, 16, 10, 100) }
+func BenchmarkString1000Base16(b *testing.B) { StringHelper(b, 16, 10, 1000) }
+func BenchmarkString10000Base16(b *testing.B) { StringHelper(b, 16, 10, 10000) }
+func BenchmarkString100000Base16(b *testing.B) { StringHelper(b, 16, 10, 100000) }
+
+func StringHelper(b *testing.B, base int, x, y Word) {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(x, y)
+ z.string(lowercaseDigits[0:base]) // warm divisor cache
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ _ = z.string(lowercaseDigits[0:base])
+ }
+}
+
+func BenchmarkLeafSize0(b *testing.B) { LeafSizeHelper(b, 10, 0) } // test without splitting
+func BenchmarkLeafSize1(b *testing.B) { LeafSizeHelper(b, 10, 1) }
+func BenchmarkLeafSize2(b *testing.B) { LeafSizeHelper(b, 10, 2) }
+func BenchmarkLeafSize3(b *testing.B) { LeafSizeHelper(b, 10, 3) }
+func BenchmarkLeafSize4(b *testing.B) { LeafSizeHelper(b, 10, 4) }
+func BenchmarkLeafSize5(b *testing.B) { LeafSizeHelper(b, 10, 5) }
+func BenchmarkLeafSize6(b *testing.B) { LeafSizeHelper(b, 10, 6) }
+func BenchmarkLeafSize7(b *testing.B) { LeafSizeHelper(b, 10, 7) }
+func BenchmarkLeafSize8(b *testing.B) { LeafSizeHelper(b, 10, 8) }
+func BenchmarkLeafSize9(b *testing.B) { LeafSizeHelper(b, 10, 9) }
+func BenchmarkLeafSize10(b *testing.B) { LeafSizeHelper(b, 10, 10) }
+func BenchmarkLeafSize11(b *testing.B) { LeafSizeHelper(b, 10, 11) }
+func BenchmarkLeafSize12(b *testing.B) { LeafSizeHelper(b, 10, 12) }
+func BenchmarkLeafSize13(b *testing.B) { LeafSizeHelper(b, 10, 13) }
+func BenchmarkLeafSize14(b *testing.B) { LeafSizeHelper(b, 10, 14) }
+func BenchmarkLeafSize15(b *testing.B) { LeafSizeHelper(b, 10, 15) }
+func BenchmarkLeafSize16(b *testing.B) { LeafSizeHelper(b, 10, 16) }
+func BenchmarkLeafSize32(b *testing.B) { LeafSizeHelper(b, 10, 32) } // try some large lengths
+func BenchmarkLeafSize64(b *testing.B) { LeafSizeHelper(b, 10, 64) }
+
+func LeafSizeHelper(b *testing.B, base Word, size int) {
+ b.StopTimer()
+ originalLeafSize := leafSize
+ resetTable(cacheBase10.table[:])
+ leafSize = size
+ b.StartTimer()
+
+ for d := 1; d <= 10000; d *= 10 {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(base, Word(d)) // build target number
+ _ = z.string(lowercaseDigits[0:base]) // warm divisor cache
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ _ = z.string(lowercaseDigits[0:base])
+ }
+ }
+
+ b.StopTimer()
+ resetTable(cacheBase10.table[:])
+ leafSize = originalLeafSize
+ b.StartTimer()
+}
+
+func resetTable(table []divisor) {
+ if table != nil && table[0].bbb != nil {
+ for i := 0; i < len(table); i++ {
+ table[i].bbb = nil
+ table[i].nbits = 0
+ table[i].ndigits = 0
+ }
+ }
+}
+
+func TestStringPowers(t *testing.T) {
+ var b, p Word
+ for b = 2; b <= 16; b++ {
+ for p = 0; p <= 512; p++ {
+ x := nat(nil).expWW(b, p)
+ xs := x.string(lowercaseDigits[0:b])
+ xs2 := toString(x, lowercaseDigits[0:b])
+ if xs != xs2 {
+ t.Errorf("failed at %d ** %d in base %d: %s != %s", b, p, b, xs, xs2)
+ }
+ }
+ if b >= 3 && testing.Short() {
+ break
+ }
+ }
+}
+
+func TestLeadingZeros(t *testing.T) {
+ var x Word = _B >> 1
+ for i := 0; i <= _W; i++ {
+ if int(leadingZeros(x)) != i {
+ t.Errorf("failed at %x: got %d want %d", x, leadingZeros(x), i)
+ }
+ x >>= 1
+ }
+}
+
+type shiftTest struct {
+ in nat
+ shift uint
+ out nat
+}
+
+var leftShiftTests = []shiftTest{
+ {nil, 0, nil},
+ {nil, 1, nil},
+ {natOne, 0, natOne},
+ {natOne, 1, natTwo},
+ {nat{1 << (_W - 1)}, 1, nat{0}},
+ {nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
+}
+
+func TestShiftLeft(t *testing.T) {
+ for i, test := range leftShiftTests {
+ var z nat
+ z = z.shl(test.in, test.shift)
+ for j, d := range test.out {
+ if j >= len(z) || z[j] != d {
+ t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+ break
+ }
+ }
+ }
+}
+
+var rightShiftTests = []shiftTest{
+ {nil, 0, nil},
+ {nil, 1, nil},
+ {natOne, 0, natOne},
+ {natOne, 1, nil},
+ {natTwo, 1, natOne},
+ {nat{0, 1}, 1, nat{1 << (_W - 1)}},
+ {nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
+}
+
+func TestShiftRight(t *testing.T) {
+ for i, test := range rightShiftTests {
+ var z nat
+ z = z.shr(test.in, test.shift)
+ for j, d := range test.out {
+ if j >= len(z) || z[j] != d {
+ t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+ break
+ }
+ }
+ }
+}
+
+type modWTest struct {
+ in string
+ dividend string
+ out string
+}
+
+var modWTests32 = []modWTest{
+ {"23492635982634928349238759823742", "252341", "220170"},
+}
+
+var modWTests64 = []modWTest{
+ {"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
+}
+
+func runModWTests(t *testing.T, tests []modWTest) {
+ for i, test := range tests {
+ in, _ := new(Int).SetString(test.in, 10)
+ d, _ := new(Int).SetString(test.dividend, 10)
+ out, _ := new(Int).SetString(test.out, 10)
+
+ r := in.abs.modW(d.abs[0])
+ if r != out.abs[0] {
+ t.Errorf("#%d failed: got %d want %s", i, r, out)
+ }
+ }
+}
+
+func TestModW(t *testing.T) {
+ if _W >= 32 {
+ runModWTests(t, modWTests32)
+ }
+ if _W >= 64 {
+ runModWTests(t, modWTests64)
+ }
+}
+
+func TestTrailingZeroBits(t *testing.T) {
+ x := Word(1)
+ for i := uint(0); i <= _W; i++ {
+ n := trailingZeroBits(x)
+ if n != i%_W {
+ t.Errorf("got trailingZeroBits(%#x) = %d; want %d", x, n, i%_W)
+ }
+ x <<= 1
+ }
+
+ y := nat(nil).set(natOne)
+ for i := uint(0); i <= 3*_W; i++ {
+ n := y.trailingZeroBits()
+ if n != i {
+ t.Errorf("got 0x%s.trailingZeroBits() = %d; want %d", y.string(lowercaseDigits[0:16]), n, i)
+ }
+ y = y.shl(y, 1)
+ }
+}
+
+var expNNTests = []struct {
+ x, y, m string
+ out string
+}{
+ {"0", "0", "0", "1"},
+ {"0", "0", "1", "0"},
+ {"1", "1", "1", "0"},
+ {"2", "1", "1", "0"},
+ {"2", "2", "1", "0"},
+ {"10", "100000000000", "1", "0"},
+ {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+ {"0x8000000000000000", "2", "6719", "4944"},
+ {"0x8000000000000000", "3", "6719", "5447"},
+ {"0x8000000000000000", "1000", "6719", "1603"},
+ {"0x8000000000000000", "1000000", "6719", "3199"},
+ {
+ "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+ "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+ },
+}
+
+func TestExpNN(t *testing.T) {
+ for i, test := range expNNTests {
+ x, _, _ := nat(nil).scan(strings.NewReader(test.x), 0)
+ y, _, _ := nat(nil).scan(strings.NewReader(test.y), 0)
+ out, _, _ := nat(nil).scan(strings.NewReader(test.out), 0)
+
+ var m nat
+
+ if len(test.m) > 0 {
+ m, _, _ = nat(nil).scan(strings.NewReader(test.m), 0)
+ }
+
+ z := nat(nil).expNN(x, y, m)
+ if z.cmp(out) != 0 {
+ t.Errorf("#%d got %s want %s", i, z.decimalString(), out.decimalString())
+ }
+ }
+}
+
+func ExpHelper(b *testing.B, x, y Word) {
+ var z nat
+ for i := 0; i < b.N; i++ {
+ z.expWW(x, y)
+ }
+}
+
+func BenchmarkExp3Power0x10(b *testing.B) { ExpHelper(b, 3, 0x10) }
+func BenchmarkExp3Power0x40(b *testing.B) { ExpHelper(b, 3, 0x40) }
+func BenchmarkExp3Power0x100(b *testing.B) { ExpHelper(b, 3, 0x100) }
+func BenchmarkExp3Power0x400(b *testing.B) { ExpHelper(b, 3, 0x400) }
+func BenchmarkExp3Power0x1000(b *testing.B) { ExpHelper(b, 3, 0x1000) }
+func BenchmarkExp3Power0x4000(b *testing.B) { ExpHelper(b, 3, 0x4000) }
+func BenchmarkExp3Power0x10000(b *testing.B) { ExpHelper(b, 3, 0x10000) }
+func BenchmarkExp3Power0x40000(b *testing.B) { ExpHelper(b, 3, 0x40000) }
+func BenchmarkExp3Power0x100000(b *testing.B) { ExpHelper(b, 3, 0x100000) }
+func BenchmarkExp3Power0x400000(b *testing.B) { ExpHelper(b, 3, 0x400000) }
diff --git a/src/math/big/rat.go b/src/math/big/rat.go
new file mode 100644
index 0000000..c5339fe
--- /dev/null
+++ b/src/math/big/rat.go
@@ -0,0 +1,716 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision rational numbers.
+
+package big
+
+import (
+ "encoding/binary"
+ "errors"
+ "fmt"
+ "math"
+ "strings"
+)
+
+// A Rat represents a quotient a/b of arbitrary precision.
+// The zero value for a Rat represents the value 0.
+type Rat struct {
+ // To make zero values for Rat work w/o initialization,
+ // a zero value of b (len(b) == 0) acts like b == 1.
+ // a.neg determines the sign of the Rat, b.neg is ignored.
+ a, b Int
+}
+
+// NewRat creates a new Rat with numerator a and denominator b.
+func NewRat(a, b int64) *Rat {
+ return new(Rat).SetFrac64(a, b)
+}
+
+// SetFloat64 sets z to exactly f and returns z.
+// If f is not finite, SetFloat returns nil.
+func (z *Rat) SetFloat64(f float64) *Rat {
+ const expMask = 1<<11 - 1
+ bits := math.Float64bits(f)
+ mantissa := bits & (1<<52 - 1)
+ exp := int((bits >> 52) & expMask)
+ switch exp {
+ case expMask: // non-finite
+ return nil
+ case 0: // denormal
+ exp -= 1022
+ default: // normal
+ mantissa |= 1 << 52
+ exp -= 1023
+ }
+
+ shift := 52 - exp
+
+ // Optimization (?): partially pre-normalise.
+ for mantissa&1 == 0 && shift > 0 {
+ mantissa >>= 1
+ shift--
+ }
+
+ z.a.SetUint64(mantissa)
+ z.a.neg = f < 0
+ z.b.Set(intOne)
+ if shift > 0 {
+ z.b.Lsh(&z.b, uint(shift))
+ } else {
+ z.a.Lsh(&z.a, uint(-shift))
+ }
+ return z.norm()
+}
+
+// quotToFloat32 returns the non-negative float32 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases. It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat32(a, b nat) (f float32, exact bool) {
+ const (
+ // float size in bits
+ Fsize = 32
+
+ // mantissa
+ Msize = 23
+ Msize1 = Msize + 1 // incl. implicit 1
+ Msize2 = Msize1 + 1
+
+ // exponent
+ Esize = Fsize - Msize1
+ Ebias = 1<<(Esize-1) - 1
+ Emin = 1 - Ebias
+ Emax = Ebias
+ )
+
+ // TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+ alen := a.bitLen()
+ if alen == 0 {
+ return 0, true
+ }
+ blen := b.bitLen()
+ if blen == 0 {
+ panic("division by zero")
+ }
+
+ // 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+ // (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+ // This is 2 or 3 more than the float32 mantissa field width of Msize:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in "normal" representation;
+ // - the low-order 1 will be used during rounding then discarded.
+ exp := alen - blen
+ var a2, b2 nat
+ a2 = a2.set(a)
+ b2 = b2.set(b)
+ if shift := Msize2 - exp; shift > 0 {
+ a2 = a2.shl(a2, uint(shift))
+ } else if shift < 0 {
+ b2 = b2.shl(b2, uint(-shift))
+ }
+
+ // 2. Compute quotient and remainder (q, r). NB: due to the
+ // extra shift, the low-order bit of q is logically the
+ // high-order bit of r.
+ var q nat
+ q, r := q.div(a2, a2, b2) // (recycle a2)
+ mantissa := low32(q)
+ haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+ // 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>Msize2 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>Msize1 != 1 {
+ panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+ }
+
+ // 4. Rounding.
+ if Emin-Msize <= exp && exp <= Emin {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint(Emin - (exp - 1)) // [1..Esize1)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = 2 - Ebias // == exp + shift
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<Msize2 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
+
+ f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
+ if math.IsInf(float64(f), 0) {
+ exact = false
+ }
+ return
+}
+
+// quotToFloat64 returns the non-negative float64 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases. It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat64(a, b nat) (f float64, exact bool) {
+ const (
+ // float size in bits
+ Fsize = 64
+
+ // mantissa
+ Msize = 52
+ Msize1 = Msize + 1 // incl. implicit 1
+ Msize2 = Msize1 + 1
+
+ // exponent
+ Esize = Fsize - Msize1
+ Ebias = 1<<(Esize-1) - 1
+ Emin = 1 - Ebias
+ Emax = Ebias
+ )
+
+ // TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+ alen := a.bitLen()
+ if alen == 0 {
+ return 0, true
+ }
+ blen := b.bitLen()
+ if blen == 0 {
+ panic("division by zero")
+ }
+
+ // 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+ // (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+ // This is 2 or 3 more than the float64 mantissa field width of Msize:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in "normal" representation;
+ // - the low-order 1 will be used during rounding then discarded.
+ exp := alen - blen
+ var a2, b2 nat
+ a2 = a2.set(a)
+ b2 = b2.set(b)
+ if shift := Msize2 - exp; shift > 0 {
+ a2 = a2.shl(a2, uint(shift))
+ } else if shift < 0 {
+ b2 = b2.shl(b2, uint(-shift))
+ }
+
+ // 2. Compute quotient and remainder (q, r). NB: due to the
+ // extra shift, the low-order bit of q is logically the
+ // high-order bit of r.
+ var q nat
+ q, r := q.div(a2, a2, b2) // (recycle a2)
+ mantissa := low64(q)
+ haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+ // 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>Msize2 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>Msize1 != 1 {
+ panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+ }
+
+ // 4. Rounding.
+ if Emin-Msize <= exp && exp <= Emin {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint(Emin - (exp - 1)) // [1..Esize1)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = 2 - Ebias // == exp + shift
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<Msize2 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
+
+ f = math.Ldexp(float64(mantissa), exp-Msize1)
+ if math.IsInf(f, 0) {
+ exact = false
+ }
+ return
+}
+
+// Float32 returns the nearest float32 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float32, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float32() (f float32, exact bool) {
+ b := x.b.abs
+ if len(b) == 0 {
+ b = b.set(natOne) // materialize denominator
+ }
+ f, exact = quotToFloat32(x.a.abs, b)
+ if x.a.neg {
+ f = -f
+ }
+ return
+}
+
+// Float64 returns the nearest float64 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float64, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float64() (f float64, exact bool) {
+ b := x.b.abs
+ if len(b) == 0 {
+ b = b.set(natOne) // materialize denominator
+ }
+ f, exact = quotToFloat64(x.a.abs, b)
+ if x.a.neg {
+ f = -f
+ }
+ return
+}
+
+// SetFrac sets z to a/b and returns z.
+func (z *Rat) SetFrac(a, b *Int) *Rat {
+ z.a.neg = a.neg != b.neg
+ babs := b.abs
+ if len(babs) == 0 {
+ panic("division by zero")
+ }
+ if &z.a == b || alias(z.a.abs, babs) {
+ babs = nat(nil).set(babs) // make a copy
+ }
+ z.a.abs = z.a.abs.set(a.abs)
+ z.b.abs = z.b.abs.set(babs)
+ return z.norm()
+}
+
+// SetFrac64 sets z to a/b and returns z.
+func (z *Rat) SetFrac64(a, b int64) *Rat {
+ z.a.SetInt64(a)
+ if b == 0 {
+ panic("division by zero")
+ }
+ if b < 0 {
+ b = -b
+ z.a.neg = !z.a.neg
+ }
+ z.b.abs = z.b.abs.setUint64(uint64(b))
+ return z.norm()
+}
+
+// SetInt sets z to x (by making a copy of x) and returns z.
+func (z *Rat) SetInt(x *Int) *Rat {
+ z.a.Set(x)
+ z.b.abs = z.b.abs.make(0)
+ return z
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Rat) SetInt64(x int64) *Rat {
+ z.a.SetInt64(x)
+ z.b.abs = z.b.abs.make(0)
+ return z
+}
+
+// Set sets z to x (by making a copy of x) and returns z.
+func (z *Rat) Set(x *Rat) *Rat {
+ if z != x {
+ z.a.Set(&x.a)
+ z.b.Set(&x.b)
+ }
+ return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Rat) Abs(x *Rat) *Rat {
+ z.Set(x)
+ z.a.neg = false
+ return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Rat) Neg(x *Rat) *Rat {
+ z.Set(x)
+ z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
+ return z
+}
+
+// Inv sets z to 1/x and returns z.
+func (z *Rat) Inv(x *Rat) *Rat {
+ if len(x.a.abs) == 0 {
+ panic("division by zero")
+ }
+ z.Set(x)
+ a := z.b.abs
+ if len(a) == 0 {
+ a = a.set(natOne) // materialize numerator
+ }
+ b := z.a.abs
+ if b.cmp(natOne) == 0 {
+ b = b.make(0) // normalize denominator
+ }
+ z.a.abs, z.b.abs = a, b // sign doesn't change
+ return z
+}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x == 0
+// +1 if x > 0
+//
+func (x *Rat) Sign() int {
+ return x.a.Sign()
+}
+
+// IsInt returns true if the denominator of x is 1.
+func (x *Rat) IsInt() bool {
+ return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
+}
+
+// Num returns the numerator of x; it may be <= 0.
+// The result is a reference to x's numerator; it
+// may change if a new value is assigned to x, and vice versa.
+// The sign of the numerator corresponds to the sign of x.
+func (x *Rat) Num() *Int {
+ return &x.a
+}
+
+// Denom returns the denominator of x; it is always > 0.
+// The result is a reference to x's denominator; it
+// may change if a new value is assigned to x, and vice versa.
+func (x *Rat) Denom() *Int {
+ x.b.neg = false // the result is always >= 0
+ if len(x.b.abs) == 0 {
+ x.b.abs = x.b.abs.set(natOne) // materialize denominator
+ }
+ return &x.b
+}
+
+func (z *Rat) norm() *Rat {
+ switch {
+ case len(z.a.abs) == 0:
+ // z == 0 - normalize sign and denominator
+ z.a.neg = false
+ z.b.abs = z.b.abs.make(0)
+ case len(z.b.abs) == 0:
+ // z is normalized int - nothing to do
+ case z.b.abs.cmp(natOne) == 0:
+ // z is int - normalize denominator
+ z.b.abs = z.b.abs.make(0)
+ default:
+ neg := z.a.neg
+ z.a.neg = false
+ z.b.neg = false
+ if f := NewInt(0).binaryGCD(&z.a, &z.b); f.Cmp(intOne) != 0 {
+ z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
+ z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
+ if z.b.abs.cmp(natOne) == 0 {
+ // z is int - normalize denominator
+ z.b.abs = z.b.abs.make(0)
+ }
+ }
+ z.a.neg = neg
+ }
+ return z
+}
+
+// mulDenom sets z to the denominator product x*y (by taking into
+// account that 0 values for x or y must be interpreted as 1) and
+// returns z.
+func mulDenom(z, x, y nat) nat {
+ switch {
+ case len(x) == 0:
+ return z.set(y)
+ case len(y) == 0:
+ return z.set(x)
+ }
+ return z.mul(x, y)
+}
+
+// scaleDenom computes x*f.
+// If f == 0 (zero value of denominator), the result is (a copy of) x.
+func scaleDenom(x *Int, f nat) *Int {
+ var z Int
+ if len(f) == 0 {
+ return z.Set(x)
+ }
+ z.abs = z.abs.mul(x.abs, f)
+ z.neg = x.neg
+ return &z
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y
+// +1 if x > y
+//
+func (x *Rat) Cmp(y *Rat) int {
+ return scaleDenom(&x.a, y.b.abs).Cmp(scaleDenom(&y.a, x.b.abs))
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Rat) Add(x, y *Rat) *Rat {
+ a1 := scaleDenom(&x.a, y.b.abs)
+ a2 := scaleDenom(&y.a, x.b.abs)
+ z.a.Add(a1, a2)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Rat) Sub(x, y *Rat) *Rat {
+ a1 := scaleDenom(&x.a, y.b.abs)
+ a2 := scaleDenom(&y.a, x.b.abs)
+ z.a.Sub(a1, a2)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Rat) Mul(x, y *Rat) *Rat {
+ z.a.Mul(&x.a, &y.a)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Quo sets z to the quotient x/y and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+func (z *Rat) Quo(x, y *Rat) *Rat {
+ if len(y.a.abs) == 0 {
+ panic("division by zero")
+ }
+ a := scaleDenom(&x.a, y.b.abs)
+ b := scaleDenom(&y.a, x.b.abs)
+ z.a.abs = a.abs
+ z.b.abs = b.abs
+ z.a.neg = a.neg != b.neg
+ return z.norm()
+}
+
+func ratTok(ch rune) bool {
+ return strings.IndexRune("+-/0123456789.eE", ch) >= 0
+}
+
+// Scan is a support routine for fmt.Scanner. It accepts the formats
+// 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.
+func (z *Rat) Scan(s fmt.ScanState, ch rune) error {
+ tok, err := s.Token(true, ratTok)
+ if err != nil {
+ return err
+ }
+ if strings.IndexRune("efgEFGv", ch) < 0 {
+ return errors.New("Rat.Scan: invalid verb")
+ }
+ if _, ok := z.SetString(string(tok)); !ok {
+ return errors.New("Rat.Scan: invalid syntax")
+ }
+ return nil
+}
+
+// SetString sets z to the value of s and returns z and a boolean indicating
+// success. s can be given as a fraction "a/b" or as a floating-point number
+// optionally followed by an exponent. If the operation failed, the value of
+// z is undefined but the returned value is nil.
+func (z *Rat) SetString(s string) (*Rat, bool) {
+ if len(s) == 0 {
+ return nil, false
+ }
+
+ // check for a quotient
+ sep := strings.Index(s, "/")
+ if sep >= 0 {
+ if _, ok := z.a.SetString(s[0:sep], 10); !ok {
+ return nil, false
+ }
+ s = s[sep+1:]
+ var err error
+ if z.b.abs, _, err = z.b.abs.scan(strings.NewReader(s), 10); err != nil {
+ return nil, false
+ }
+ if len(z.b.abs) == 0 {
+ return nil, false
+ }
+ return z.norm(), true
+ }
+
+ // check for a decimal point
+ sep = strings.Index(s, ".")
+ // check for an exponent
+ e := strings.IndexAny(s, "eE")
+ var exp Int
+ if e >= 0 {
+ if e < sep {
+ // The E must come after the decimal point.
+ return nil, false
+ }
+ if _, ok := exp.SetString(s[e+1:], 10); !ok {
+ return nil, false
+ }
+ s = s[0:e]
+ }
+ if sep >= 0 {
+ s = s[0:sep] + s[sep+1:]
+ exp.Sub(&exp, NewInt(int64(len(s)-sep)))
+ }
+
+ if _, ok := z.a.SetString(s, 10); !ok {
+ return nil, false
+ }
+ powTen := nat(nil).expNN(natTen, exp.abs, nil)
+ if exp.neg {
+ z.b.abs = powTen
+ z.norm()
+ } else {
+ z.a.abs = z.a.abs.mul(z.a.abs, powTen)
+ z.b.abs = z.b.abs.make(0)
+ }
+
+ return z, true
+}
+
+// String returns a string representation of x in the form "a/b" (even if b == 1).
+func (x *Rat) String() string {
+ s := "/1"
+ if len(x.b.abs) != 0 {
+ s = "/" + x.b.abs.decimalString()
+ }
+ return x.a.String() + s
+}
+
+// RatString returns a string representation of x in the form "a/b" if b != 1,
+// and in the form "a" if b == 1.
+func (x *Rat) RatString() string {
+ if x.IsInt() {
+ return x.a.String()
+ }
+ return x.String()
+}
+
+// FloatString returns a string representation of x in decimal form with prec
+// digits of precision after the decimal point and the last digit rounded.
+func (x *Rat) FloatString(prec int) string {
+ if x.IsInt() {
+ s := x.a.String()
+ if prec > 0 {
+ s += "." + strings.Repeat("0", prec)
+ }
+ return s
+ }
+ // x.b.abs != 0
+
+ q, r := nat(nil).div(nat(nil), x.a.abs, x.b.abs)
+
+ p := natOne
+ if prec > 0 {
+ p = nat(nil).expNN(natTen, nat(nil).setUint64(uint64(prec)), nil)
+ }
+
+ r = r.mul(r, p)
+ r, r2 := r.div(nat(nil), r, x.b.abs)
+
+ // see if we need to round up
+ r2 = r2.add(r2, r2)
+ if x.b.abs.cmp(r2) <= 0 {
+ r = r.add(r, natOne)
+ if r.cmp(p) >= 0 {
+ q = nat(nil).add(q, natOne)
+ r = nat(nil).sub(r, p)
+ }
+ }
+
+ s := q.decimalString()
+ if x.a.neg {
+ s = "-" + s
+ }
+
+ if prec > 0 {
+ rs := r.decimalString()
+ leadingZeros := prec - len(rs)
+ s += "." + strings.Repeat("0", leadingZeros) + rs
+ }
+
+ return s
+}
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const ratGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Rat) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+ buf := make([]byte, 1+4+(len(x.a.abs)+len(x.b.abs))*_S) // extra bytes for version and sign bit (1), and numerator length (4)
+ i := x.b.abs.bytes(buf)
+ j := x.a.abs.bytes(buf[0:i])
+ n := i - j
+ if int(uint32(n)) != n {
+ // this should never happen
+ return nil, errors.New("Rat.GobEncode: numerator too large")
+ }
+ binary.BigEndian.PutUint32(buf[j-4:j], uint32(n))
+ j -= 1 + 4
+ b := ratGobVersion << 1 // make space for sign bit
+ if x.a.neg {
+ b |= 1
+ }
+ buf[j] = b
+ return buf[j:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Rat) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Rat{}
+ return nil
+ }
+ b := buf[0]
+ if b>>1 != ratGobVersion {
+ return errors.New(fmt.Sprintf("Rat.GobDecode: encoding version %d not supported", b>>1))
+ }
+ const j = 1 + 4
+ i := j + binary.BigEndian.Uint32(buf[j-4:j])
+ z.a.neg = b&1 != 0
+ z.a.abs = z.a.abs.setBytes(buf[j:i])
+ z.b.abs = z.b.abs.setBytes(buf[i:])
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+func (r *Rat) MarshalText() (text []byte, err error) {
+ return []byte(r.RatString()), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+func (r *Rat) UnmarshalText(text []byte) error {
+ if _, ok := r.SetString(string(text)); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Rat", text)
+ }
+ return nil
+}
diff --git a/src/math/big/rat_test.go b/src/math/big/rat_test.go
new file mode 100644
index 0000000..5dbbb35
--- /dev/null
+++ b/src/math/big/rat_test.go
@@ -0,0 +1,1160 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/json"
+ "encoding/xml"
+ "fmt"
+ "math"
+ "strconv"
+ "strings"
+ "testing"
+)
+
+func TestZeroRat(t *testing.T) {
+ var x, y, z Rat
+ y.SetFrac64(0, 42)
+
+ if x.Cmp(&y) != 0 {
+ t.Errorf("x and y should be both equal and zero")
+ }
+
+ if s := x.String(); s != "0/1" {
+ t.Errorf("got x = %s, want 0/1", s)
+ }
+
+ if s := x.RatString(); s != "0" {
+ t.Errorf("got x = %s, want 0", s)
+ }
+
+ z.Add(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x+y = %s, want 0", s)
+ }
+
+ z.Sub(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x-y = %s, want 0", s)
+ }
+
+ z.Mul(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x*y = %s, want 0", s)
+ }
+
+ // check for division by zero
+ defer func() {
+ if s := recover(); s == nil || s.(string) != "division by zero" {
+ panic(s)
+ }
+ }()
+ z.Quo(&x, &y)
+}
+
+var setStringTests = []struct {
+ in, out string
+ ok bool
+}{
+ {"0", "0", true},
+ {"-0", "0", true},
+ {"1", "1", true},
+ {"-1", "-1", true},
+ {"1.", "1", true},
+ {"1e0", "1", true},
+ {"1.e1", "10", true},
+ {in: "1e", ok: false},
+ {in: "1.e", ok: false},
+ {in: "1e+14e-5", ok: false},
+ {in: "1e4.5", ok: false},
+ {in: "r", ok: false},
+ {in: "a/b", ok: false},
+ {in: "a.b", ok: false},
+ {"-0.1", "-1/10", true},
+ {"-.1", "-1/10", true},
+ {"2/4", "1/2", true},
+ {".25", "1/4", true},
+ {"-1/5", "-1/5", true},
+ {"8129567.7690E14", "812956776900000000000", true},
+ {"78189e+4", "781890000", true},
+ {"553019.8935e+8", "55301989350000", true},
+ {"98765432109876543210987654321e-10", "98765432109876543210987654321/10000000000", true},
+ {"9877861857500000E-7", "3951144743/4", true},
+ {"2169378.417e-3", "2169378417/1000000", true},
+ {"884243222337379604041632732738665534", "884243222337379604041632732738665534", true},
+ {"53/70893980658822810696", "53/70893980658822810696", true},
+ {"106/141787961317645621392", "53/70893980658822810696", true},
+ {"204211327800791583.81095", "4084226556015831676219/20000", true},
+ {in: "1/0", ok: false},
+}
+
+func TestRatSetString(t *testing.T) {
+ for i, test := range setStringTests {
+ x, ok := new(Rat).SetString(test.in)
+
+ if ok {
+ if !test.ok {
+ t.Errorf("#%d SetString(%q) expected failure", i, test.in)
+ } else if x.RatString() != test.out {
+ t.Errorf("#%d SetString(%q) got %s want %s", i, test.in, x.RatString(), test.out)
+ }
+ } else if x != nil {
+ t.Errorf("#%d SetString(%q) got %p want nil", i, test.in, x)
+ }
+ }
+}
+
+func TestRatScan(t *testing.T) {
+ var buf bytes.Buffer
+ for i, test := range setStringTests {
+ x := new(Rat)
+ buf.Reset()
+ buf.WriteString(test.in)
+
+ _, err := fmt.Fscanf(&buf, "%v", x)
+ if err == nil != test.ok {
+ if test.ok {
+ t.Errorf("#%d error: %s", i, err)
+ } else {
+ t.Errorf("#%d expected error", i)
+ }
+ continue
+ }
+ if err == nil && x.RatString() != test.out {
+ t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+ }
+ }
+}
+
+var floatStringTests = []struct {
+ in string
+ prec int
+ out string
+}{
+ {"0", 0, "0"},
+ {"0", 4, "0.0000"},
+ {"1", 0, "1"},
+ {"1", 2, "1.00"},
+ {"-1", 0, "-1"},
+ {".25", 2, "0.25"},
+ {".25", 1, "0.3"},
+ {".25", 3, "0.250"},
+ {"-1/3", 3, "-0.333"},
+ {"-2/3", 4, "-0.6667"},
+ {"0.96", 1, "1.0"},
+ {"0.999", 2, "1.00"},
+ {"0.9", 0, "1"},
+ {".25", -1, "0"},
+ {".55", -1, "1"},
+}
+
+func TestFloatString(t *testing.T) {
+ for i, test := range floatStringTests {
+ x, _ := new(Rat).SetString(test.in)
+
+ if x.FloatString(test.prec) != test.out {
+ t.Errorf("#%d got %s want %s", i, x.FloatString(test.prec), test.out)
+ }
+ }
+}
+
+func TestRatSign(t *testing.T) {
+ zero := NewRat(0, 1)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ s := x.Sign()
+ e := x.Cmp(zero)
+ if s != e {
+ t.Errorf("got %d; want %d for z = %v", s, e, &x)
+ }
+ }
+}
+
+var ratCmpTests = []struct {
+ rat1, rat2 string
+ out int
+}{
+ {"0", "0/1", 0},
+ {"1/1", "1", 0},
+ {"-1", "-2/2", 0},
+ {"1", "0", 1},
+ {"0/1", "1/1", -1},
+ {"-5/1434770811533343057144", "-5/1434770811533343057145", -1},
+ {"49832350382626108453/8964749413", "49832350382626108454/8964749413", -1},
+ {"-37414950961700930/7204075375675961", "37414950961700930/7204075375675961", -1},
+ {"37414950961700930/7204075375675961", "74829901923401860/14408150751351922", 0},
+}
+
+func TestRatCmp(t *testing.T) {
+ for i, test := range ratCmpTests {
+ x, _ := new(Rat).SetString(test.rat1)
+ y, _ := new(Rat).SetString(test.rat2)
+
+ out := x.Cmp(y)
+ if out != test.out {
+ t.Errorf("#%d got out = %v; want %v", i, out, test.out)
+ }
+ }
+}
+
+func TestIsInt(t *testing.T) {
+ one := NewInt(1)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ i := x.IsInt()
+ e := x.Denom().Cmp(one) == 0
+ if i != e {
+ t.Errorf("got IsInt(%v) == %v; want %v", x, i, e)
+ }
+ }
+}
+
+func TestRatAbs(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ e := new(Rat).Set(x)
+ if e.Cmp(zero) < 0 {
+ e.Sub(zero, e)
+ }
+ z := new(Rat).Abs(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Abs(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+func TestRatNeg(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ e := new(Rat).Sub(zero, x)
+ z := new(Rat).Neg(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Neg(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+func TestRatInv(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ if x.Cmp(zero) == 0 {
+ continue // avoid division by zero
+ }
+ e := new(Rat).SetFrac(x.Denom(), x.Num())
+ z := new(Rat).Inv(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Inv(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+type ratBinFun func(z, x, y *Rat) *Rat
+type ratBinArg struct {
+ x, y, z string
+}
+
+func testRatBin(t *testing.T, i int, name string, f ratBinFun, a ratBinArg) {
+ x, _ := new(Rat).SetString(a.x)
+ y, _ := new(Rat).SetString(a.y)
+ z, _ := new(Rat).SetString(a.z)
+ out := f(new(Rat), x, y)
+
+ if out.Cmp(z) != 0 {
+ t.Errorf("%s #%d got %s want %s", name, i, out, z)
+ }
+}
+
+var ratBinTests = []struct {
+ x, y string
+ sum, prod string
+}{
+ {"0", "0", "0", "0"},
+ {"0", "1", "1", "0"},
+ {"-1", "0", "-1", "0"},
+ {"-1", "1", "0", "-1"},
+ {"1", "1", "2", "1"},
+ {"1/2", "1/2", "1", "1/4"},
+ {"1/4", "1/3", "7/12", "1/12"},
+ {"2/5", "-14/3", "-64/15", "-28/15"},
+ {"4707/49292519774798173060", "-3367/70976135186689855734", "84058377121001851123459/1749296273614329067191168098769082663020", "-1760941/388732505247628681598037355282018369560"},
+ {"-61204110018146728334/3", "-31052192278051565633/2", "-215564796870448153567/6", "950260896245257153059642991192710872711/3"},
+ {"-854857841473707320655/4237645934602118692642972629634714039", "-18/31750379913563777419", "-27/133467566250814981", "15387441146526731771790/134546868362786310073779084329032722548987800600710485341"},
+ {"618575745270541348005638912139/19198433543745179392300736", "-19948846211000086/637313996471", "27674141753240653/30123979153216", "-6169936206128396568797607742807090270137721977/6117715203873571641674006593837351328"},
+ {"-3/26206484091896184128", "5/2848423294177090248", "15310893822118706237/9330894968229805033368778458685147968", "-5/24882386581946146755650075889827061248"},
+ {"26946729/330400702820", "41563965/225583428284", "1238218672302860271/4658307703098666660055", "224002580204097/14906584649915733312176"},
+ {"-8259900599013409474/7", "-84829337473700364773/56707961321161574960", "-468402123685491748914621885145127724451/396955729248131024720", "350340947706464153265156004876107029701/198477864624065512360"},
+ {"575775209696864/1320203974639986246357", "29/712593081308", "410331716733912717985762465/940768218243776489278275419794956", "808/45524274987585732633"},
+ {"1786597389946320496771/2066653520653241", "6269770/1992362624741777", "3559549865190272133656109052308126637/4117523232840525481453983149257", "8967230/3296219033"},
+ {"-36459180403360509753/32150500941194292113930", "9381566963714/9633539", "301622077145533298008420642898530153/309723104686531919656937098270", "-3784609207827/3426986245"},
+}
+
+func TestRatBin(t *testing.T) {
+ for i, test := range ratBinTests {
+ arg := ratBinArg{test.x, test.y, test.sum}
+ testRatBin(t, i, "Add", (*Rat).Add, arg)
+
+ arg = ratBinArg{test.y, test.x, test.sum}
+ testRatBin(t, i, "Add symmetric", (*Rat).Add, arg)
+
+ arg = ratBinArg{test.sum, test.x, test.y}
+ testRatBin(t, i, "Sub", (*Rat).Sub, arg)
+
+ arg = ratBinArg{test.sum, test.y, test.x}
+ testRatBin(t, i, "Sub symmetric", (*Rat).Sub, arg)
+
+ arg = ratBinArg{test.x, test.y, test.prod}
+ testRatBin(t, i, "Mul", (*Rat).Mul, arg)
+
+ arg = ratBinArg{test.y, test.x, test.prod}
+ testRatBin(t, i, "Mul symmetric", (*Rat).Mul, arg)
+
+ if test.x != "0" {
+ arg = ratBinArg{test.prod, test.x, test.y}
+ testRatBin(t, i, "Quo", (*Rat).Quo, arg)
+ }
+
+ if test.y != "0" {
+ arg = ratBinArg{test.prod, test.y, test.x}
+ testRatBin(t, i, "Quo symmetric", (*Rat).Quo, arg)
+ }
+ }
+}
+
+func TestIssue820(t *testing.T) {
+ x := NewRat(3, 1)
+ y := NewRat(2, 1)
+ z := y.Quo(x, y)
+ q := NewRat(3, 2)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+
+ y = NewRat(3, 1)
+ x = NewRat(2, 1)
+ z = y.Quo(x, y)
+ q = NewRat(2, 3)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+
+ x = NewRat(3, 1)
+ z = x.Quo(x, x)
+ q = NewRat(3, 3)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+}
+
+var setFrac64Tests = []struct {
+ a, b int64
+ out string
+}{
+ {0, 1, "0"},
+ {0, -1, "0"},
+ {1, 1, "1"},
+ {-1, 1, "-1"},
+ {1, -1, "-1"},
+ {-1, -1, "1"},
+ {-9223372036854775808, -9223372036854775808, "1"},
+}
+
+func TestRatSetFrac64Rat(t *testing.T) {
+ for i, test := range setFrac64Tests {
+ x := new(Rat).SetFrac64(test.a, test.b)
+ if x.RatString() != test.out {
+ t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+ }
+ }
+}
+
+func TestRatGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range encodingTests {
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ var tx Rat
+ tx.SetString(test + ".14159265")
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %s failed: %s", &tx, err)
+ }
+ var rx Rat
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %s failed: %s", &tx, err)
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+// Sending a nil Rat pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilRatInSlice(t *testing.T) {
+ buf := new(bytes.Buffer)
+ enc := gob.NewEncoder(buf)
+ dec := gob.NewDecoder(buf)
+
+ var in = make([]*Rat, 1)
+ err := enc.Encode(&in)
+ if err != nil {
+ t.Errorf("gob encode failed: %q", err)
+ }
+ var out []*Rat
+ err = dec.Decode(&out)
+ if err != nil {
+ t.Fatalf("gob decode failed: %q", err)
+ }
+ if len(out) != 1 {
+ t.Fatalf("wrong len; want 1 got %d", len(out))
+ }
+ var zero Rat
+ if out[0].Cmp(&zero) != 0 {
+ t.Errorf("transmission of (*Int)(nill) failed: got %s want 0", out)
+ }
+}
+
+var ratNums = []string{
+ "-141592653589793238462643383279502884197169399375105820974944592307816406286",
+ "-1415926535897932384626433832795028841971",
+ "-141592653589793",
+ "-1",
+ "0",
+ "1",
+ "141592653589793",
+ "1415926535897932384626433832795028841971",
+ "141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+var ratDenoms = []string{
+ "1",
+ "718281828459045",
+ "7182818284590452353602874713526624977572",
+ "718281828459045235360287471352662497757247093699959574966967627724076630353",
+}
+
+func TestRatJSONEncoding(t *testing.T) {
+ for _, num := range ratNums {
+ for _, denom := range ratDenoms {
+ var tx Rat
+ tx.SetString(num + "/" + denom)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+func TestRatXMLEncoding(t *testing.T) {
+ for _, num := range ratNums {
+ for _, denom := range ratDenoms {
+ var tx Rat
+ tx.SetString(num + "/" + denom)
+ b, err := xml.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := xml.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+func TestIssue2379(t *testing.T) {
+ // 1) no aliasing
+ q := NewRat(3, 2)
+ x := new(Rat)
+ x.SetFrac(NewInt(3), NewInt(2))
+ if x.Cmp(q) != 0 {
+ t.Errorf("1) got %s want %s", x, q)
+ }
+
+ // 2) aliasing of numerator
+ x = NewRat(2, 3)
+ x.SetFrac(NewInt(3), x.Num())
+ if x.Cmp(q) != 0 {
+ t.Errorf("2) got %s want %s", x, q)
+ }
+
+ // 3) aliasing of denominator
+ x = NewRat(2, 3)
+ x.SetFrac(x.Denom(), NewInt(2))
+ if x.Cmp(q) != 0 {
+ t.Errorf("3) got %s want %s", x, q)
+ }
+
+ // 4) aliasing of numerator and denominator
+ x = NewRat(2, 3)
+ x.SetFrac(x.Denom(), x.Num())
+ if x.Cmp(q) != 0 {
+ t.Errorf("4) got %s want %s", x, q)
+ }
+
+ // 5) numerator and denominator are the same
+ q = NewRat(1, 1)
+ x = new(Rat)
+ n := NewInt(7)
+ x.SetFrac(n, n)
+ if x.Cmp(q) != 0 {
+ t.Errorf("5) got %s want %s", x, q)
+ }
+}
+
+func TestIssue3521(t *testing.T) {
+ a := new(Int)
+ b := new(Int)
+ a.SetString("64375784358435883458348587", 0)
+ b.SetString("4789759874531", 0)
+
+ // 0) a raw zero value has 1 as denominator
+ zero := new(Rat)
+ one := NewInt(1)
+ if zero.Denom().Cmp(one) != 0 {
+ t.Errorf("0) got %s want %s", zero.Denom(), one)
+ }
+
+ // 1a) a zero value remains zero independent of denominator
+ x := new(Rat)
+ x.Denom().Set(new(Int).Neg(b))
+ if x.Cmp(zero) != 0 {
+ t.Errorf("1a) got %s want %s", x, zero)
+ }
+
+ // 1b) a zero value may have a denominator != 0 and != 1
+ x.Num().Set(a)
+ qab := new(Rat).SetFrac(a, b)
+ if x.Cmp(qab) != 0 {
+ t.Errorf("1b) got %s want %s", x, qab)
+ }
+
+ // 2a) an integral value becomes a fraction depending on denominator
+ x.SetFrac64(10, 2)
+ x.Denom().SetInt64(3)
+ q53 := NewRat(5, 3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("2a) got %s want %s", x, q53)
+ }
+
+ // 2b) an integral value becomes a fraction depending on denominator
+ x = NewRat(10, 2)
+ x.Denom().SetInt64(3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("2b) got %s want %s", x, q53)
+ }
+
+ // 3) changing the numerator/denominator of a Rat changes the Rat
+ x.SetFrac(a, b)
+ a = x.Num()
+ b = x.Denom()
+ a.SetInt64(5)
+ b.SetInt64(3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("3) got %s want %s", x, q53)
+ }
+}
+
+// Test inputs to Rat.SetString. The prefix "long:" causes the test
+// to be skipped in --test.short mode. (The threshold is about 500us.)
+var float64inputs = []string{
+ // Constants plundered from strconv/testfp.txt.
+
+ // Table 1: Stress Inputs for Conversion to 53-bit Binary, < 1/2 ULP
+ "5e+125",
+ "69e+267",
+ "999e-026",
+ "7861e-034",
+ "75569e-254",
+ "928609e-261",
+ "9210917e+080",
+ "84863171e+114",
+ "653777767e+273",
+ "5232604057e-298",
+ "27235667517e-109",
+ "653532977297e-123",
+ "3142213164987e-294",
+ "46202199371337e-072",
+ "231010996856685e-073",
+ "9324754620109615e+212",
+ "78459735791271921e+049",
+ "272104041512242479e+200",
+ "6802601037806061975e+198",
+ "20505426358836677347e-221",
+ "836168422905420598437e-234",
+ "4891559871276714924261e+222",
+
+ // Table 2: Stress Inputs for Conversion to 53-bit Binary, > 1/2 ULP
+ "9e-265",
+ "85e-037",
+ "623e+100",
+ "3571e+263",
+ "81661e+153",
+ "920657e-023",
+ "4603285e-024",
+ "87575437e-309",
+ "245540327e+122",
+ "6138508175e+120",
+ "83356057653e+193",
+ "619534293513e+124",
+ "2335141086879e+218",
+ "36167929443327e-159",
+ "609610927149051e-255",
+ "3743626360493413e-165",
+ "94080055902682397e-242",
+ "899810892172646163e+283",
+ "7120190517612959703e+120",
+ "25188282901709339043e-252",
+ "308984926168550152811e-052",
+ "6372891218502368041059e+064",
+
+ // Table 14: Stress Inputs for Conversion to 24-bit Binary, <1/2 ULP
+ "5e-20",
+ "67e+14",
+ "985e+15",
+ "7693e-42",
+ "55895e-16",
+ "996622e-44",
+ "7038531e-32",
+ "60419369e-46",
+ "702990899e-20",
+ "6930161142e-48",
+ "25933168707e+13",
+ "596428896559e+20",
+
+ // Table 15: Stress Inputs for Conversion to 24-bit Binary, >1/2 ULP
+ "3e-23",
+ "57e+18",
+ "789e-35",
+ "2539e-18",
+ "76173e+28",
+ "887745e-11",
+ "5382571e-37",
+ "82381273e-35",
+ "750486563e-38",
+ "3752432815e-39",
+ "75224575729e-45",
+ "459926601011e+15",
+
+ // Constants plundered from strconv/atof_test.go.
+
+ "0",
+ "1",
+ "+1",
+ "1e23",
+ "1E23",
+ "100000000000000000000000",
+ "1e-100",
+ "123456700",
+ "99999999999999974834176",
+ "100000000000000000000001",
+ "100000000000000008388608",
+ "100000000000000016777215",
+ "100000000000000016777216",
+ "-1",
+ "-0.1",
+ "-0", // NB: exception made for this input
+ "1e-20",
+ "625e-3",
+
+ // largest float64
+ "1.7976931348623157e308",
+ "-1.7976931348623157e308",
+ // next float64 - too large
+ "1.7976931348623159e308",
+ "-1.7976931348623159e308",
+ // the border is ...158079
+ // borderline - okay
+ "1.7976931348623158e308",
+ "-1.7976931348623158e308",
+ // borderline - too large
+ "1.797693134862315808e308",
+ "-1.797693134862315808e308",
+
+ // a little too large
+ "1e308",
+ "2e308",
+ "1e309",
+
+ // way too large
+ "1e310",
+ "-1e310",
+ "1e400",
+ "-1e400",
+ "long:1e400000",
+ "long:-1e400000",
+
+ // denormalized
+ "1e-305",
+ "1e-306",
+ "1e-307",
+ "1e-308",
+ "1e-309",
+ "1e-310",
+ "1e-322",
+ // smallest denormal
+ "5e-324",
+ "4e-324",
+ "3e-324",
+ // too small
+ "2e-324",
+ // way too small
+ "1e-350",
+ "long:1e-400000",
+ // way too small, negative
+ "-1e-350",
+ "long:-1e-400000",
+
+ // try to overflow exponent
+ // [Disabled: too slow and memory-hungry with rationals.]
+ // "1e-4294967296",
+ // "1e+4294967296",
+ // "1e-18446744073709551616",
+ // "1e+18446744073709551616",
+
+ // http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ "2.2250738585072012e-308",
+ // http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ "2.2250738585072011e-308",
+
+ // A very large number (initially wrongly parsed by the fast algorithm).
+ "4.630813248087435e+307",
+
+ // A different kind of very large number.
+ "22.222222222222222",
+ "long:2." + strings.Repeat("2", 4000) + "e+1",
+
+ // Exactly halfway between 1 and math.Nextafter(1, 2).
+ // Round to even (down).
+ "1.00000000000000011102230246251565404236316680908203125",
+ // Slightly lower; still round down.
+ "1.00000000000000011102230246251565404236316680908203124",
+ // Slightly higher; round up.
+ "1.00000000000000011102230246251565404236316680908203126",
+ // Slightly higher, but you have to read all the way to the end.
+ "long:1.00000000000000011102230246251565404236316680908203125" + strings.Repeat("0", 10000) + "1",
+
+ // Smallest denormal, 2^(-1022-52)
+ "4.940656458412465441765687928682213723651e-324",
+ // Half of smallest denormal, 2^(-1022-53)
+ "2.470328229206232720882843964341106861825e-324",
+ // A little more than the exact half of smallest denormal
+ // 2^-1075 + 2^-1100. (Rounds to 1p-1074.)
+ "2.470328302827751011111470718709768633275e-324",
+ // The exact halfway between smallest normal and largest denormal:
+ // 2^-1022 - 2^-1075. (Rounds to 2^-1022.)
+ "2.225073858507201136057409796709131975935e-308",
+
+ "1152921504606846975", // 1<<60 - 1
+ "-1152921504606846975", // -(1<<60 - 1)
+ "1152921504606846977", // 1<<60 + 1
+ "-1152921504606846977", // -(1<<60 + 1)
+
+ "1/3",
+}
+
+// isFinite reports whether f represents a finite rational value.
+// It is equivalent to !math.IsNan(f) && !math.IsInf(f, 0).
+func isFinite(f float64) bool {
+ return math.Abs(f) <= math.MaxFloat64
+}
+
+func TestFloat32SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ if strings.HasPrefix(input, "long:") {
+ if testing.Short() {
+ continue
+ }
+ input = input[len("long:"):]
+ }
+
+ r, ok := new(Rat).SetString(input)
+ if !ok {
+ t.Errorf("Rat.SetString(%q) failed", input)
+ continue
+ }
+ f, exact := r.Float32()
+
+ // 1. Check string -> Rat -> float32 conversions are
+ // consistent with strconv.ParseFloat.
+ // Skip this check if the input uses "a/b" rational syntax.
+ if !strings.Contains(input, "/") {
+ e64, _ := strconv.ParseFloat(input, 32)
+ e := float32(e64)
+
+ // Careful: negative Rats too small for
+ // float64 become -0, but Rat obviously cannot
+ // preserve the sign from SetString("-0").
+ switch {
+ case math.Float32bits(e) == math.Float32bits(f):
+ // Ok: bitwise equal.
+ case f == 0 && r.Num().BitLen() == 0:
+ // Ok: Rat(0) is equivalent to both +/- float64(0).
+ default:
+ t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+ }
+ }
+
+ if !isFinite(float64(f)) {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox32(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip32(t, f)
+
+ // 4. Check exactness using slow algorithm.
+ if wasExact := new(Rat).SetFloat64(float64(f)).Cmp(r) == 0; wasExact != exact {
+ t.Errorf("Rat.SetString(%q).Float32().exact = %t, want %t", input, exact, wasExact)
+ }
+ }
+}
+
+func TestFloat64SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ if strings.HasPrefix(input, "long:") {
+ if testing.Short() {
+ continue
+ }
+ input = input[len("long:"):]
+ }
+
+ r, ok := new(Rat).SetString(input)
+ if !ok {
+ t.Errorf("Rat.SetString(%q) failed", input)
+ continue
+ }
+ f, exact := r.Float64()
+
+ // 1. Check string -> Rat -> float64 conversions are
+ // consistent with strconv.ParseFloat.
+ // Skip this check if the input uses "a/b" rational syntax.
+ if !strings.Contains(input, "/") {
+ e, _ := strconv.ParseFloat(input, 64)
+
+ // Careful: negative Rats too small for
+ // float64 become -0, but Rat obviously cannot
+ // preserve the sign from SetString("-0").
+ switch {
+ case math.Float64bits(e) == math.Float64bits(f):
+ // Ok: bitwise equal.
+ case f == 0 && r.Num().BitLen() == 0:
+ // Ok: Rat(0) is equivalent to both +/- float64(0).
+ default:
+ t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+ }
+ }
+
+ if !isFinite(f) {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox64(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip64(t, f)
+
+ // 4. Check exactness using slow algorithm.
+ if wasExact := new(Rat).SetFloat64(f).Cmp(r) == 0; wasExact != exact {
+ t.Errorf("Rat.SetString(%q).Float64().exact = %t, want %t", input, exact, wasExact)
+ }
+ }
+}
+
+func TestFloat32Distribution(t *testing.T) {
+ // Generate a distribution of (sign, mantissa, exp) values
+ // broader than the float32 range, and check Rat.Float32()
+ // always picks the closest float32 approximation.
+ var add = []int64{
+ 0,
+ 1,
+ 3,
+ 5,
+ 7,
+ 9,
+ 11,
+ }
+ var winc, einc = uint64(1), 1 // soak test (~1.5s on x86-64)
+ if testing.Short() {
+ winc, einc = 5, 15 // quick test (~60ms on x86-64)
+ }
+
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 30; wid += winc {
+ b := 1<<wid + a
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -150; exp < 150; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float32()
+
+ if !checkIsBestApprox32(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+ b, exp, f, f, math.Ldexp(float64(b), exp), r)
+ }
+
+ checkNonLossyRoundtrip32(t, f)
+ }
+ }
+ }
+ }
+}
+
+func TestFloat64Distribution(t *testing.T) {
+ // Generate a distribution of (sign, mantissa, exp) values
+ // broader than the float64 range, and check Rat.Float64()
+ // always picks the closest float64 approximation.
+ var add = []int64{
+ 0,
+ 1,
+ 3,
+ 5,
+ 7,
+ 9,
+ 11,
+ }
+ var winc, einc = uint64(1), 1 // soak test (~75s on x86-64)
+ if testing.Short() {
+ winc, einc = 10, 500 // quick test (~12ms on x86-64)
+ }
+
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 60; wid += winc {
+ b := 1<<wid + a
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -1100; exp < 1100; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float64()
+
+ if !checkIsBestApprox64(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+ b, exp, f, f, math.Ldexp(float64(b), exp), r)
+ }
+
+ checkNonLossyRoundtrip64(t, f)
+ }
+ }
+ }
+ }
+}
+
+// TestSetFloat64NonFinite checks that SetFloat64 of a non-finite value
+// returns nil.
+func TestSetFloat64NonFinite(t *testing.T) {
+ for _, f := range []float64{math.NaN(), math.Inf(+1), math.Inf(-1)} {
+ var r Rat
+ if r2 := r.SetFloat64(f); r2 != nil {
+ t.Errorf("SetFloat64(%g) was %v, want nil", f, r2)
+ }
+ }
+}
+
+// checkNonLossyRoundtrip32 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip32(t *testing.T, f float32) {
+ if !isFinite(float64(f)) {
+ return
+ }
+ r := new(Rat).SetFloat64(float64(f))
+ if r == nil {
+ t.Errorf("Rat.SetFloat64(float64(%g) (%b)) == nil", f, f)
+ return
+ }
+ f2, exact := r.Float32()
+ if f != f2 || !exact {
+ t.Errorf("Rat.SetFloat64(float64(%g)).Float32() = %g (%b), %v, want %g (%b), %v; delta = %b",
+ f, f2, f2, exact, f, f, true, f2-f)
+ }
+}
+
+// checkNonLossyRoundtrip64 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip64(t *testing.T, f float64) {
+ if !isFinite(f) {
+ return
+ }
+ r := new(Rat).SetFloat64(f)
+ if r == nil {
+ t.Errorf("Rat.SetFloat64(%g (%b)) == nil", f, f)
+ return
+ }
+ f2, exact := r.Float64()
+ if f != f2 || !exact {
+ t.Errorf("Rat.SetFloat64(%g).Float64() = %g (%b), %v, want %g (%b), %v; delta = %b",
+ f, f2, f2, exact, f, f, true, f2-f)
+ }
+}
+
+// delta returns the absolute difference between r and f.
+func delta(r *Rat, f float64) *Rat {
+ d := new(Rat).Sub(r, new(Rat).SetFloat64(f))
+ return d.Abs(d)
+}
+
+// checkIsBestApprox32 checks that f is the best possible float32
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox32(t *testing.T, f float32, r *Rat) bool {
+ if math.Abs(float64(f)) >= math.MaxFloat32 {
+ // Cannot check +Inf, -Inf, nor the float next to them (MaxFloat32).
+ // But we have tests for these special cases.
+ return true
+ }
+
+ // r must be strictly between f0 and f1, the floats bracketing f.
+ f0 := math.Nextafter32(f, float32(math.Inf(-1)))
+ f1 := math.Nextafter32(f, float32(math.Inf(+1)))
+
+ // For f to be correct, r must be closer to f than to f0 or f1.
+ df := delta(r, float64(f))
+ df0 := delta(r, float64(f0))
+ df1 := delta(r, float64(f1))
+ if df.Cmp(df0) > 0 {
+ t.Errorf("Rat(%v).Float32() = %g (%b), but previous float32 %g (%b) is closer", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) > 0 {
+ t.Errorf("Rat(%v).Float32() = %g (%b), but next float32 %g (%b) is closer", r, f, f, f1, f1)
+ return false
+ }
+ if df.Cmp(df0) == 0 && !isEven32(f) {
+ t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) == 0 && !isEven32(f) {
+ t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+ return false
+ }
+ return true
+}
+
+// checkIsBestApprox64 checks that f is the best possible float64
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox64(t *testing.T, f float64, r *Rat) bool {
+ if math.Abs(f) >= math.MaxFloat64 {
+ // Cannot check +Inf, -Inf, nor the float next to them (MaxFloat64).
+ // But we have tests for these special cases.
+ return true
+ }
+
+ // r must be strictly between f0 and f1, the floats bracketing f.
+ f0 := math.Nextafter(f, math.Inf(-1))
+ f1 := math.Nextafter(f, math.Inf(+1))
+
+ // For f to be correct, r must be closer to f than to f0 or f1.
+ df := delta(r, f)
+ df0 := delta(r, f0)
+ df1 := delta(r, f1)
+ if df.Cmp(df0) > 0 {
+ t.Errorf("Rat(%v).Float64() = %g (%b), but previous float64 %g (%b) is closer", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) > 0 {
+ t.Errorf("Rat(%v).Float64() = %g (%b), but next float64 %g (%b) is closer", r, f, f, f1, f1)
+ return false
+ }
+ if df.Cmp(df0) == 0 && !isEven64(f) {
+ t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) == 0 && !isEven64(f) {
+ t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+ return false
+ }
+ return true
+}
+
+func isEven32(f float32) bool { return math.Float32bits(f)&1 == 0 }
+func isEven64(f float64) bool { return math.Float64bits(f)&1 == 0 }
+
+func TestIsFinite(t *testing.T) {
+ finites := []float64{
+ 1.0 / 3,
+ 4891559871276714924261e+222,
+ math.MaxFloat64,
+ math.SmallestNonzeroFloat64,
+ -math.MaxFloat64,
+ -math.SmallestNonzeroFloat64,
+ }
+ for _, f := range finites {
+ if !isFinite(f) {
+ t.Errorf("!IsFinite(%g (%b))", f, f)
+ }
+ }
+ nonfinites := []float64{
+ math.NaN(),
+ math.Inf(-1),
+ math.Inf(+1),
+ }
+ for _, f := range nonfinites {
+ if isFinite(f) {
+ t.Errorf("IsFinite(%g, (%b))", f, f)
+ }
+ }
+}
diff --git a/src/math/bits.go b/src/math/bits.go
new file mode 100644
index 0000000..d85ee9c
--- /dev/null
+++ b/src/math/bits.go
@@ -0,0 +1,59 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+const (
+ uvnan = 0x7FF8000000000001
+ uvinf = 0x7FF0000000000000
+ uvneginf = 0xFFF0000000000000
+ mask = 0x7FF
+ shift = 64 - 11 - 1
+ bias = 1023
+)
+
+// Inf returns positive infinity if sign >= 0, negative infinity if sign < 0.
+func Inf(sign int) float64 {
+ var v uint64
+ if sign >= 0 {
+ v = uvinf
+ } else {
+ v = uvneginf
+ }
+ return Float64frombits(v)
+}
+
+// NaN returns an IEEE 754 ``not-a-number'' value.
+func NaN() float64 { return Float64frombits(uvnan) }
+
+// IsNaN reports whether f is an IEEE 754 ``not-a-number'' value.
+func IsNaN(f float64) (is bool) {
+ // IEEE 754 says that only NaNs satisfy f != f.
+ // To avoid the floating-point hardware, could use:
+ // x := Float64bits(f);
+ // return uint32(x>>shift)&mask == mask && x != uvinf && x != uvneginf
+ return f != f
+}
+
+// IsInf reports whether f is an infinity, according to sign.
+// If sign > 0, IsInf reports whether f is positive infinity.
+// If sign < 0, IsInf reports whether f is negative infinity.
+// If sign == 0, IsInf reports whether f is either infinity.
+func IsInf(f float64, sign int) bool {
+ // Test for infinity by comparing against maximum float.
+ // To avoid the floating-point hardware, could use:
+ // x := Float64bits(f);
+ // return sign >= 0 && x == uvinf || sign <= 0 && x == uvneginf;
+ return sign >= 0 && f > MaxFloat64 || sign <= 0 && f < -MaxFloat64
+}
+
+// normalize returns a normal number y and exponent exp
+// satisfying x == y × 2**exp. It assumes x is finite and non-zero.
+func normalize(x float64) (y float64, exp int) {
+ const SmallestNormal = 2.2250738585072014e-308 // 2**-1022
+ if Abs(x) < SmallestNormal {
+ return x * (1 << 52), -52
+ }
+ return x, 0
+}
diff --git a/src/math/cbrt.go b/src/math/cbrt.go
new file mode 100644
index 0000000..272e309
--- /dev/null
+++ b/src/math/cbrt.go
@@ -0,0 +1,76 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ The algorithm is based in part on "Optimal Partitioning of
+ Newton's Method for Calculating Roots", by Gunter Meinardus
+ and G. D. Taylor, Mathematics of Computation © 1980 American
+ Mathematical Society.
+ (http://www.jstor.org/stable/2006387?seq=9, accessed 11-Feb-2010)
+*/
+
+// Cbrt returns the cube root of x.
+//
+// Special cases are:
+// Cbrt(±0) = ±0
+// Cbrt(±Inf) = ±Inf
+// Cbrt(NaN) = NaN
+func Cbrt(x float64) float64 {
+ const (
+ A1 = 1.662848358e-01
+ A2 = 1.096040958e+00
+ A3 = 4.105032829e-01
+ A4 = 5.649335816e-01
+ B1 = 2.639607233e-01
+ B2 = 8.699282849e-01
+ B3 = 1.629083358e-01
+ B4 = 2.824667908e-01
+ C1 = 4.190115298e-01
+ C2 = 6.904625373e-01
+ C3 = 6.46502159e-02
+ C4 = 1.412333954e-01
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x) || IsInf(x, 0):
+ return x
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ // Reduce argument and estimate cube root
+ f, e := Frexp(x) // 0.5 <= f < 1.0
+ m := e % 3
+ if m > 0 {
+ m -= 3
+ e -= m // e is multiple of 3
+ }
+ switch m {
+ case 0: // 0.5 <= f < 1.0
+ f = A1*f + A2 - A3/(A4+f)
+ case -1:
+ f *= 0.5 // 0.25 <= f < 0.5
+ f = B1*f + B2 - B3/(B4+f)
+ default: // m == -2
+ f *= 0.25 // 0.125 <= f < 0.25
+ f = C1*f + C2 - C3/(C4+f)
+ }
+ y := Ldexp(f, e/3) // e/3 = exponent of cube root
+
+ // Iterate
+ s := y * y * y
+ t := s + x
+ y *= (t + x) / (s + t)
+ // Reiterate
+ s = (y*y*y - x) / x
+ y -= y * (((14.0/81.0)*s-(2.0/9.0))*s + (1.0 / 3.0)) * s
+ if sign {
+ y = -y
+ }
+ return y
+}
diff --git a/src/math/cmplx/abs.go b/src/math/cmplx/abs.go
new file mode 100644
index 0000000..f3cd107
--- /dev/null
+++ b/src/math/cmplx/abs.go
@@ -0,0 +1,12 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package cmplx provides basic constants and mathematical functions for
+// complex numbers.
+package cmplx
+
+import "math"
+
+// Abs returns the absolute value (also called the modulus) of x.
+func Abs(x complex128) float64 { return math.Hypot(real(x), imag(x)) }
diff --git a/src/math/cmplx/asin.go b/src/math/cmplx/asin.go
new file mode 100644
index 0000000..61880a2
--- /dev/null
+++ b/src/math/cmplx/asin.go
@@ -0,0 +1,170 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex circular arc sine
+//
+// DESCRIPTION:
+//
+// Inverse complex sine:
+// 2
+// w = -i clog( iz + csqrt( 1 - z ) ).
+//
+// casin(z) = -i casinh(iz)
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 10100 2.1e-15 3.4e-16
+// IEEE -10,+10 30000 2.2e-14 2.7e-15
+// Larger relative error can be observed for z near zero.
+// Also tested by csin(casin(z)) = z.
+
+// Asin returns the inverse sine of x.
+func Asin(x complex128) complex128 {
+ if imag(x) == 0 {
+ if math.Abs(real(x)) > 1 {
+ return complex(math.Pi/2, 0) // DOMAIN error
+ }
+ return complex(math.Asin(real(x)), 0)
+ }
+ ct := complex(-imag(x), real(x)) // i * x
+ xx := x * x
+ x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
+ x2 := Sqrt(x1) // x2 = sqrt(1 - x*x)
+ w := Log(ct + x2)
+ return complex(imag(w), -real(w)) // -i * w
+}
+
+// Asinh returns the inverse hyperbolic sine of x.
+func Asinh(x complex128) complex128 {
+ // TODO check range
+ if imag(x) == 0 {
+ if math.Abs(real(x)) > 1 {
+ return complex(math.Pi/2, 0) // DOMAIN error
+ }
+ return complex(math.Asinh(real(x)), 0)
+ }
+ xx := x * x
+ x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
+ return Log(x + Sqrt(x1)) // log(x + sqrt(1 + x*x))
+}
+
+// Complex circular arc cosine
+//
+// DESCRIPTION:
+//
+// w = arccos z = PI/2 - arcsin z.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 5200 1.6e-15 2.8e-16
+// IEEE -10,+10 30000 1.8e-14 2.2e-15
+
+// Acos returns the inverse cosine of x.
+func Acos(x complex128) complex128 {
+ w := Asin(x)
+ return complex(math.Pi/2-real(w), -imag(w))
+}
+
+// Acosh returns the inverse hyperbolic cosine of x.
+func Acosh(x complex128) complex128 {
+ w := Acos(x)
+ if imag(w) <= 0 {
+ return complex(-imag(w), real(w)) // i * w
+ }
+ return complex(imag(w), -real(w)) // -i * w
+}
+
+// Complex circular arc tangent
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+// 1 ( 2x )
+// Re w = - arctan(-----------) + k PI
+// 2 ( 2 2)
+// (1 - x - y )
+//
+// ( 2 2)
+// 1 (x + (y+1) )
+// Im w = - log(------------)
+// 4 ( 2 2)
+// (x + (y-1) )
+//
+// Where k is an arbitrary integer.
+//
+// catan(z) = -i catanh(iz).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 5900 1.3e-16 7.8e-18
+// IEEE -10,+10 30000 2.3e-15 8.5e-17
+// The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
+// had peak relative error 1.5e-16, rms relative error
+// 2.9e-17. See also clog().
+
+// Atan returns the inverse tangent of x.
+func Atan(x complex128) complex128 {
+ if real(x) == 0 && imag(x) > 1 {
+ return NaN()
+ }
+
+ x2 := real(x) * real(x)
+ a := 1 - x2 - imag(x)*imag(x)
+ if a == 0 {
+ return NaN()
+ }
+ t := 0.5 * math.Atan2(2*real(x), a)
+ w := reducePi(t)
+
+ t = imag(x) - 1
+ b := x2 + t*t
+ if b == 0 {
+ return NaN()
+ }
+ t = imag(x) + 1
+ c := (x2 + t*t) / b
+ return complex(w, 0.25*math.Log(c))
+}
+
+// Atanh returns the inverse hyperbolic tangent of x.
+func Atanh(x complex128) complex128 {
+ z := complex(-imag(x), real(x)) // z = i * x
+ z = Atan(z)
+ return complex(imag(z), -real(z)) // z = -i * z
+}
diff --git a/src/math/cmplx/cmath_test.go b/src/math/cmplx/cmath_test.go
new file mode 100644
index 0000000..f285646
--- /dev/null
+++ b/src/math/cmplx/cmath_test.go
@@ -0,0 +1,866 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import (
+ "math"
+ "testing"
+)
+
+var vc26 = []complex128{
+ (4.97901192488367350108546816 + 7.73887247457810456552351752i),
+ (7.73887247457810456552351752 - 0.27688005719200159404635997i),
+ (-0.27688005719200159404635997 - 5.01060361827107492160848778i),
+ (-5.01060361827107492160848778 + 9.63629370719841737980004837i),
+ (9.63629370719841737980004837 + 2.92637723924396464525443662i),
+ (2.92637723924396464525443662 + 5.22908343145930665230025625i),
+ (5.22908343145930665230025625 + 2.72793991043601025126008608i),
+ (2.72793991043601025126008608 + 1.82530809168085506044576505i),
+ (1.82530809168085506044576505 - 8.68592476857560136238589621i),
+ (-8.68592476857560136238589621 + 4.97901192488367350108546816i),
+}
+var vc = []complex128{
+ (4.9790119248836735e+00 + 7.7388724745781045e+00i),
+ (7.7388724745781045e+00 - 2.7688005719200159e-01i),
+ (-2.7688005719200159e-01 - 5.0106036182710749e+00i),
+ (-5.0106036182710749e+00 + 9.6362937071984173e+00i),
+ (9.6362937071984173e+00 + 2.9263772392439646e+00i),
+ (2.9263772392439646e+00 + 5.2290834314593066e+00i),
+ (5.2290834314593066e+00 + 2.7279399104360102e+00i),
+ (2.7279399104360102e+00 + 1.8253080916808550e+00i),
+ (1.8253080916808550e+00 - 8.6859247685756013e+00i),
+ (-8.6859247685756013e+00 + 4.9790119248836735e+00i),
+}
+
+// The expected results below were computed by the high precision calculators
+// at http://keisan.casio.com/. More exact input values (array vc[], above)
+// were obtained by printing them with "%.26f". The answers were calculated
+// to 26 digits (by using the "Digit number" drop-down control of each
+// calculator).
+
+var abs = []float64{
+ 9.2022120669932650313380972e+00,
+ 7.7438239742296106616261394e+00,
+ 5.0182478202557746902556648e+00,
+ 1.0861137372799545160704002e+01,
+ 1.0070841084922199607011905e+01,
+ 5.9922447613166942183705192e+00,
+ 5.8978784056736762299945176e+00,
+ 3.2822866700678709020367184e+00,
+ 8.8756430028990417290744307e+00,
+ 1.0011785496777731986390856e+01,
+}
+
+var acos = []complex128{
+ (1.0017679804707456328694569 - 2.9138232718554953784519807i),
+ (0.03606427612041407369636057 + 2.7358584434576260925091256i),
+ (1.6249365462333796703711823 + 2.3159537454335901187730929i),
+ (2.0485650849650740120660391 - 3.0795576791204117911123886i),
+ (0.29621132089073067282488147 - 3.0007392508200622519398814i),
+ (1.0664555914934156601503632 - 2.4872865024796011364747111i),
+ (0.48681307452231387690013905 - 2.463655912283054555225301i),
+ (0.6116977071277574248407752 - 1.8734458851737055262693056i),
+ (1.3649311280370181331184214 + 2.8793528632328795424123832i),
+ (2.6189310485682988308904501 - 2.9956543302898767795858704i),
+}
+var acosh = []complex128{
+ (2.9138232718554953784519807 + 1.0017679804707456328694569i),
+ (2.7358584434576260925091256 - 0.03606427612041407369636057i),
+ (2.3159537454335901187730929 - 1.6249365462333796703711823i),
+ (3.0795576791204117911123886 + 2.0485650849650740120660391i),
+ (3.0007392508200622519398814 + 0.29621132089073067282488147i),
+ (2.4872865024796011364747111 + 1.0664555914934156601503632i),
+ (2.463655912283054555225301 + 0.48681307452231387690013905i),
+ (1.8734458851737055262693056 + 0.6116977071277574248407752i),
+ (2.8793528632328795424123832 - 1.3649311280370181331184214i),
+ (2.9956543302898767795858704 + 2.6189310485682988308904501i),
+}
+var asin = []complex128{
+ (0.56902834632415098636186476 + 2.9138232718554953784519807i),
+ (1.5347320506744825455349611 - 2.7358584434576260925091256i),
+ (-0.054140219438483051139860579 - 2.3159537454335901187730929i),
+ (-0.47776875817017739283471738 + 3.0795576791204117911123886i),
+ (1.2745850059041659464064402 + 3.0007392508200622519398814i),
+ (0.50434073530148095908095852 + 2.4872865024796011364747111i),
+ (1.0839832522725827423311826 + 2.463655912283054555225301i),
+ (0.9590986196671391943905465 + 1.8734458851737055262693056i),
+ (0.20586519875787848611290031 - 2.8793528632328795424123832i),
+ (-1.0481347217734022116591284 + 2.9956543302898767795858704i),
+}
+var asinh = []complex128{
+ (2.9113760469415295679342185 + 0.99639459545704326759805893i),
+ (2.7441755423994259061579029 - 0.035468308789000500601119392i),
+ (-2.2962136462520690506126678 - 1.5144663565690151885726707i),
+ (-3.0771233459295725965402455 + 1.0895577967194013849422294i),
+ (3.0048366100923647417557027 + 0.29346979169819220036454168i),
+ (2.4800059370795363157364643 + 1.0545868606049165710424232i),
+ (2.4718773838309585611141821 + 0.47502344364250803363708842i),
+ (1.8910743588080159144378396 + 0.56882925572563602341139174i),
+ (2.8735426423367341878069406 - 1.362376149648891420997548i),
+ (-2.9981750586172477217567878 + 0.5183571985225367505624207i),
+}
+var atan = []complex128{
+ (1.5115747079332741358607654 + 0.091324403603954494382276776i),
+ (1.4424504323482602560806727 - 0.0045416132642803911503770933i),
+ (-1.5593488703630532674484026 - 0.20163295409248362456446431i),
+ (-1.5280619472445889867794105 + 0.081721556230672003746956324i),
+ (1.4759909163240799678221039 + 0.028602969320691644358773586i),
+ (1.4877353772046548932715555 + 0.14566877153207281663773599i),
+ (1.4206983927779191889826 + 0.076830486127880702249439993i),
+ (1.3162236060498933364869556 + 0.16031313000467530644933363i),
+ (1.5473450684303703578810093 - 0.11064907507939082484935782i),
+ (-1.4841462340185253987375812 + 0.049341850305024399493142411i),
+}
+var atanh = []complex128{
+ (0.058375027938968509064640438 + 1.4793488495105334458167782i),
+ (0.12977343497790381229915667 - 1.5661009410463561327262499i),
+ (-0.010576456067347252072200088 - 1.3743698658402284549750563i),
+ (-0.042218595678688358882784918 + 1.4891433968166405606692604i),
+ (0.095218997991316722061828397 + 1.5416884098777110330499698i),
+ (0.079965459366890323857556487 + 1.4252510353873192700350435i),
+ (0.15051245471980726221708301 + 1.4907432533016303804884461i),
+ (0.25082072933993987714470373 + 1.392057665392187516442986i),
+ (0.022896108815797135846276662 - 1.4609224989282864208963021i),
+ (-0.08665624101841876130537396 + 1.5207902036935093480142159i),
+}
+var conj = []complex128{
+ (4.9790119248836735e+00 - 7.7388724745781045e+00i),
+ (7.7388724745781045e+00 + 2.7688005719200159e-01i),
+ (-2.7688005719200159e-01 + 5.0106036182710749e+00i),
+ (-5.0106036182710749e+00 - 9.6362937071984173e+00i),
+ (9.6362937071984173e+00 - 2.9263772392439646e+00i),
+ (2.9263772392439646e+00 - 5.2290834314593066e+00i),
+ (5.2290834314593066e+00 - 2.7279399104360102e+00i),
+ (2.7279399104360102e+00 - 1.8253080916808550e+00i),
+ (1.8253080916808550e+00 + 8.6859247685756013e+00i),
+ (-8.6859247685756013e+00 - 4.9790119248836735e+00i),
+}
+var cos = []complex128{
+ (3.024540920601483938336569e+02 + 1.1073797572517071650045357e+03i),
+ (1.192858682649064973252758e-01 + 2.7857554122333065540970207e-01i),
+ (7.2144394304528306603857962e+01 - 2.0500129667076044169954205e+01i),
+ (2.24921952538403984190541e+03 - 7.317363745602773587049329e+03i),
+ (-9.148222970032421760015498e+00 + 1.953124661113563541862227e+00i),
+ (-9.116081175857732248227078e+01 - 1.992669213569952232487371e+01i),
+ (3.795639179042704640002918e+00 + 6.623513350981458399309662e+00i),
+ (-2.9144840732498869560679084e+00 - 1.214620271628002917638748e+00i),
+ (-7.45123482501299743872481e+02 + 2.8641692314488080814066734e+03i),
+ (-5.371977967039319076416747e+01 + 4.893348341339375830564624e+01i),
+}
+var cosh = []complex128{
+ (8.34638383523018249366948e+00 + 7.2181057886425846415112064e+01i),
+ (1.10421967379919366952251e+03 - 3.1379638689277575379469861e+02i),
+ (3.051485206773701584738512e-01 - 2.6805384730105297848044485e-01i),
+ (-7.33294728684187933370938e+01 + 1.574445942284918251038144e+01i),
+ (-7.478643293945957535757355e+03 + 1.6348382209913353929473321e+03i),
+ (4.622316522966235701630926e+00 - 8.088695185566375256093098e+00i),
+ (-8.544333183278877406197712e+01 + 3.7505836120128166455231717e+01i),
+ (-1.934457815021493925115198e+00 + 7.3725859611767228178358673e+00i),
+ (-2.352958770061749348353548e+00 - 2.034982010440878358915409e+00i),
+ (7.79756457532134748165069e+02 + 2.8549350716819176560377717e+03i),
+}
+var exp = []complex128{
+ (1.669197736864670815125146e+01 + 1.4436895109507663689174096e+02i),
+ (2.2084389286252583447276212e+03 - 6.2759289284909211238261917e+02i),
+ (2.227538273122775173434327e-01 + 7.2468284028334191250470034e-01i),
+ (-6.5182985958153548997881627e-03 - 1.39965837915193860879044e-03i),
+ (-1.4957286524084015746110777e+04 + 3.269676455931135688988042e+03i),
+ (9.218158701983105935659273e+00 - 1.6223985291084956009304582e+01i),
+ (-1.7088175716853040841444505e+02 + 7.501382609870410713795546e+01i),
+ (-3.852461315830959613132505e+00 + 1.4808420423156073221970892e+01i),
+ (-4.586775503301407379786695e+00 - 4.178501081246873415144744e+00i),
+ (4.451337963005453491095747e-05 - 1.62977574205442915935263e-04i),
+}
+var log = []complex128{
+ (2.2194438972179194425697051e+00 + 9.9909115046919291062461269e-01i),
+ (2.0468956191154167256337289e+00 - 3.5762575021856971295156489e-02i),
+ (1.6130808329853860438751244e+00 - 1.6259990074019058442232221e+00i),
+ (2.3851910394823008710032651e+00 + 2.0502936359659111755031062e+00i),
+ (2.3096442270679923004800651e+00 + 2.9483213155446756211881774e-01i),
+ (1.7904660933974656106951860e+00 + 1.0605860367252556281902109e+00i),
+ (1.7745926939841751666177512e+00 + 4.8084556083358307819310911e-01i),
+ (1.1885403350045342425648780e+00 + 5.8969634164776659423195222e-01i),
+ (2.1833107837679082586772505e+00 - 1.3636647724582455028314573e+00i),
+ (2.3037629487273259170991671e+00 + 2.6210913895386013290915234e+00i),
+}
+var log10 = []complex128{
+ (9.6389223745559042474184943e-01 + 4.338997735671419492599631e-01i),
+ (8.8895547241376579493490892e-01 - 1.5531488990643548254864806e-02i),
+ (7.0055210462945412305244578e-01 - 7.0616239649481243222248404e-01i),
+ (1.0358753067322445311676952e+00 + 8.9043121238134980156490909e-01i),
+ (1.003065742975330237172029e+00 + 1.2804396782187887479857811e-01i),
+ (7.7758954439739162532085157e-01 + 4.6060666333341810869055108e-01i),
+ (7.7069581462315327037689152e-01 + 2.0882857371769952195512475e-01i),
+ (5.1617650901191156135137239e-01 + 2.5610186717615977620363299e-01i),
+ (9.4819982567026639742663212e-01 - 5.9223208584446952284914289e-01i),
+ (1.0005115362454417135973429e+00 + 1.1383255270407412817250921e+00i),
+}
+
+type ff struct {
+ r, theta float64
+}
+
+var polar = []ff{
+ {9.2022120669932650313380972e+00, 9.9909115046919291062461269e-01},
+ {7.7438239742296106616261394e+00, -3.5762575021856971295156489e-02},
+ {5.0182478202557746902556648e+00, -1.6259990074019058442232221e+00},
+ {1.0861137372799545160704002e+01, 2.0502936359659111755031062e+00},
+ {1.0070841084922199607011905e+01, 2.9483213155446756211881774e-01},
+ {5.9922447613166942183705192e+00, 1.0605860367252556281902109e+00},
+ {5.8978784056736762299945176e+00, 4.8084556083358307819310911e-01},
+ {3.2822866700678709020367184e+00, 5.8969634164776659423195222e-01},
+ {8.8756430028990417290744307e+00, -1.3636647724582455028314573e+00},
+ {1.0011785496777731986390856e+01, 2.6210913895386013290915234e+00},
+}
+var pow = []complex128{
+ (-2.499956739197529585028819e+00 + 1.759751724335650228957144e+00i),
+ (7.357094338218116311191939e+04 - 5.089973412479151648145882e+04i),
+ (1.320777296067768517259592e+01 - 3.165621914333901498921986e+01i),
+ (-3.123287828297300934072149e-07 - 1.9849567521490553032502223E-7i),
+ (8.0622651468477229614813e+04 - 7.80028727944573092944363e+04i),
+ (-1.0268824572103165858577141e+00 - 4.716844738244989776610672e-01i),
+ (-4.35953819012244175753187e+01 + 2.2036445974645306917648585e+02i),
+ (8.3556092283250594950239e-01 - 1.2261571947167240272593282e+01i),
+ (1.582292972120769306069625e+03 + 1.273564263524278244782512e+04i),
+ (6.592208301642122149025369e-08 + 2.584887236651661903526389e-08i),
+}
+var sin = []complex128{
+ (-1.1073801774240233539648544e+03 + 3.024539773002502192425231e+02i),
+ (1.0317037521400759359744682e+00 - 3.2208979799929570242818e-02i),
+ (-2.0501952097271429804261058e+01 - 7.2137981348240798841800967e+01i),
+ (7.3173638080346338642193078e+03 + 2.249219506193664342566248e+03i),
+ (-1.964375633631808177565226e+00 - 9.0958264713870404464159683e+00i),
+ (1.992783647158514838337674e+01 - 9.11555769410191350416942e+01i),
+ (-6.680335650741921444300349e+00 + 3.763353833142432513086117e+00i),
+ (1.2794028166657459148245993e+00 - 2.7669092099795781155109602e+00i),
+ (2.8641693949535259594188879e+03 + 7.451234399649871202841615e+02i),
+ (-4.893811726244659135553033e+01 - 5.371469305562194635957655e+01i),
+}
+var sinh = []complex128{
+ (8.34559353341652565758198e+00 + 7.2187893208650790476628899e+01i),
+ (1.1042192548260646752051112e+03 - 3.1379650595631635858792056e+02i),
+ (-8.239469336509264113041849e-02 + 9.9273668758439489098514519e-01i),
+ (7.332295456982297798219401e+01 - 1.574585908122833444899023e+01i),
+ (-7.4786432301380582103534216e+03 + 1.63483823493980029604071e+03i),
+ (4.595842179016870234028347e+00 - 8.135290105518580753211484e+00i),
+ (-8.543842533574163435246793e+01 + 3.750798997857594068272375e+01i),
+ (-1.918003500809465688017307e+00 + 7.4358344619793504041350251e+00i),
+ (-2.233816733239658031433147e+00 - 2.143519070805995056229335e+00i),
+ (-7.797564130187551181105341e+02 - 2.8549352346594918614806877e+03i),
+}
+var sqrt = []complex128{
+ (2.6628203086086130543813948e+00 + 1.4531345674282185229796902e+00i),
+ (2.7823278427251986247149295e+00 - 4.9756907317005224529115567e-02i),
+ (1.5397025302089642757361015e+00 - 1.6271336573016637535695727e+00i),
+ (1.7103411581506875260277898e+00 + 2.8170677122737589676157029e+00i),
+ (3.1390392472953103383607947e+00 + 4.6612625849858653248980849e-01i),
+ (2.1117080764822417640789287e+00 + 1.2381170223514273234967850e+00i),
+ (2.3587032281672256703926939e+00 + 5.7827111903257349935720172e-01i),
+ (1.7335262588873410476661577e+00 + 5.2647258220721269141550382e-01i),
+ (2.3131094974708716531499282e+00 - 1.8775429304303785570775490e+00i),
+ (8.1420535745048086240947359e-01 + 3.0575897587277248522656113e+00i),
+}
+var tan = []complex128{
+ (-1.928757919086441129134525e-07 + 1.0000003267499169073251826e+00i),
+ (1.242412685364183792138948e+00 - 3.17149693883133370106696e+00i),
+ (-4.6745126251587795225571826e-05 - 9.9992439225263959286114298e-01i),
+ (4.792363401193648192887116e-09 + 1.0000000070589333451557723e+00i),
+ (2.345740824080089140287315e-03 + 9.947733046570988661022763e-01i),
+ (-2.396030789494815566088809e-05 + 9.9994781345418591429826779e-01i),
+ (-7.370204836644931340905303e-03 + 1.0043553413417138987717748e+00i),
+ (-3.691803847992048527007457e-02 + 9.6475071993469548066328894e-01i),
+ (-2.781955256713729368401878e-08 - 1.000000049848910609006646e+00i),
+ (9.4281590064030478879791249e-05 + 9.9999119340863718183758545e-01i),
+}
+var tanh = []complex128{
+ (1.0000921981225144748819918e+00 + 2.160986245871518020231507e-05i),
+ (9.9999967727531993209562591e-01 - 1.9953763222959658873657676e-07i),
+ (-1.765485739548037260789686e+00 + 1.7024216325552852445168471e+00i),
+ (-9.999189442732736452807108e-01 + 3.64906070494473701938098e-05i),
+ (9.9999999224622333738729767e-01 - 3.560088949517914774813046e-09i),
+ (1.0029324933367326862499343e+00 - 4.948790309797102353137528e-03i),
+ (9.9996113064788012488693567e-01 - 4.226995742097032481451259e-05i),
+ (1.0074784189316340029873945e+00 - 4.194050814891697808029407e-03i),
+ (9.9385534229718327109131502e-01 + 5.144217985914355502713437e-02i),
+ (-1.0000000491604982429364892e+00 - 2.901873195374433112227349e-08i),
+}
+
+// special cases
+var vcAbsSC = []complex128{
+ NaN(),
+}
+var absSC = []float64{
+ math.NaN(),
+}
+var vcAcosSC = []complex128{
+ NaN(),
+}
+var acosSC = []complex128{
+ NaN(),
+}
+var vcAcoshSC = []complex128{
+ NaN(),
+}
+var acoshSC = []complex128{
+ NaN(),
+}
+var vcAsinSC = []complex128{
+ NaN(),
+}
+var asinSC = []complex128{
+ NaN(),
+}
+var vcAsinhSC = []complex128{
+ NaN(),
+}
+var asinhSC = []complex128{
+ NaN(),
+}
+var vcAtanSC = []complex128{
+ NaN(),
+}
+var atanSC = []complex128{
+ NaN(),
+}
+var vcAtanhSC = []complex128{
+ NaN(),
+}
+var atanhSC = []complex128{
+ NaN(),
+}
+var vcConjSC = []complex128{
+ NaN(),
+}
+var conjSC = []complex128{
+ NaN(),
+}
+var vcCosSC = []complex128{
+ NaN(),
+}
+var cosSC = []complex128{
+ NaN(),
+}
+var vcCoshSC = []complex128{
+ NaN(),
+}
+var coshSC = []complex128{
+ NaN(),
+}
+var vcExpSC = []complex128{
+ NaN(),
+}
+var expSC = []complex128{
+ NaN(),
+}
+var vcIsNaNSC = []complex128{
+ complex(math.Inf(-1), math.Inf(-1)),
+ complex(math.Inf(-1), math.NaN()),
+ complex(math.NaN(), math.Inf(-1)),
+ complex(0, math.NaN()),
+ complex(math.NaN(), 0),
+ complex(math.Inf(1), math.Inf(1)),
+ complex(math.Inf(1), math.NaN()),
+ complex(math.NaN(), math.Inf(1)),
+ complex(math.NaN(), math.NaN()),
+}
+var isNaNSC = []bool{
+ false,
+ false,
+ false,
+ true,
+ true,
+ false,
+ false,
+ false,
+ true,
+}
+var vcLogSC = []complex128{
+ NaN(),
+}
+var logSC = []complex128{
+ NaN(),
+}
+var vcLog10SC = []complex128{
+ NaN(),
+}
+var log10SC = []complex128{
+ NaN(),
+}
+var vcPolarSC = []complex128{
+ NaN(),
+}
+var polarSC = []ff{
+ {math.NaN(), math.NaN()},
+}
+var vcPowSC = [][2]complex128{
+ {NaN(), NaN()},
+}
+var powSC = []complex128{
+ NaN(),
+}
+var vcSinSC = []complex128{
+ NaN(),
+}
+var sinSC = []complex128{
+ NaN(),
+}
+var vcSinhSC = []complex128{
+ NaN(),
+}
+var sinhSC = []complex128{
+ NaN(),
+}
+var vcSqrtSC = []complex128{
+ NaN(),
+}
+var sqrtSC = []complex128{
+ NaN(),
+}
+var vcTanSC = []complex128{
+ NaN(),
+}
+var tanSC = []complex128{
+ NaN(),
+}
+var vcTanhSC = []complex128{
+ NaN(),
+}
+var tanhSC = []complex128{
+ NaN(),
+}
+
+// functions borrowed from pkg/math/all_test.go
+func tolerance(a, b, e float64) bool {
+ d := a - b
+ if d < 0 {
+ d = -d
+ }
+
+ if a != 0 {
+ e = e * a
+ if e < 0 {
+ e = -e
+ }
+ }
+ return d < e
+}
+func soclose(a, b, e float64) bool { return tolerance(a, b, e) }
+func veryclose(a, b float64) bool { return tolerance(a, b, 4e-16) }
+func alike(a, b float64) bool {
+ switch {
+ case a != a && b != b: // math.IsNaN(a) && math.IsNaN(b):
+ return true
+ case a == b:
+ return math.Signbit(a) == math.Signbit(b)
+ }
+ return false
+}
+
+func cTolerance(a, b complex128, e float64) bool {
+ d := Abs(a - b)
+ if a != 0 {
+ e = e * Abs(a)
+ if e < 0 {
+ e = -e
+ }
+ }
+ return d < e
+}
+func cSoclose(a, b complex128, e float64) bool { return cTolerance(a, b, e) }
+func cVeryclose(a, b complex128) bool { return cTolerance(a, b, 4e-16) }
+func cAlike(a, b complex128) bool {
+ switch {
+ case IsNaN(a) && IsNaN(b):
+ return true
+ case a == b:
+ return math.Signbit(real(a)) == math.Signbit(real(b)) && math.Signbit(imag(a)) == math.Signbit(imag(b))
+ }
+ return false
+}
+
+func TestAbs(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Abs(vc[i]); !veryclose(abs[i], f) {
+ t.Errorf("Abs(%g) = %g, want %g", vc[i], f, abs[i])
+ }
+ }
+ for i := 0; i < len(vcAbsSC); i++ {
+ if f := Abs(vcAbsSC[i]); !alike(absSC[i], f) {
+ t.Errorf("Abs(%g) = %g, want %g", vcAbsSC[i], f, absSC[i])
+ }
+ }
+}
+func TestAcos(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Acos(vc[i]); !cSoclose(acos[i], f, 1e-14) {
+ t.Errorf("Acos(%g) = %g, want %g", vc[i], f, acos[i])
+ }
+ }
+ for i := 0; i < len(vcAcosSC); i++ {
+ if f := Acos(vcAcosSC[i]); !cAlike(acosSC[i], f) {
+ t.Errorf("Acos(%g) = %g, want %g", vcAcosSC[i], f, acosSC[i])
+ }
+ }
+}
+func TestAcosh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Acosh(vc[i]); !cSoclose(acosh[i], f, 1e-14) {
+ t.Errorf("Acosh(%g) = %g, want %g", vc[i], f, acosh[i])
+ }
+ }
+ for i := 0; i < len(vcAcoshSC); i++ {
+ if f := Acosh(vcAcoshSC[i]); !cAlike(acoshSC[i], f) {
+ t.Errorf("Acosh(%g) = %g, want %g", vcAcoshSC[i], f, acoshSC[i])
+ }
+ }
+}
+func TestAsin(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Asin(vc[i]); !cSoclose(asin[i], f, 1e-14) {
+ t.Errorf("Asin(%g) = %g, want %g", vc[i], f, asin[i])
+ }
+ }
+ for i := 0; i < len(vcAsinSC); i++ {
+ if f := Asin(vcAsinSC[i]); !cAlike(asinSC[i], f) {
+ t.Errorf("Asin(%g) = %g, want %g", vcAsinSC[i], f, asinSC[i])
+ }
+ }
+}
+func TestAsinh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Asinh(vc[i]); !cSoclose(asinh[i], f, 4e-15) {
+ t.Errorf("Asinh(%g) = %g, want %g", vc[i], f, asinh[i])
+ }
+ }
+ for i := 0; i < len(vcAsinhSC); i++ {
+ if f := Asinh(vcAsinhSC[i]); !cAlike(asinhSC[i], f) {
+ t.Errorf("Asinh(%g) = %g, want %g", vcAsinhSC[i], f, asinhSC[i])
+ }
+ }
+}
+func TestAtan(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Atan(vc[i]); !cVeryclose(atan[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vc[i], f, atan[i])
+ }
+ }
+ for i := 0; i < len(vcAtanSC); i++ {
+ if f := Atan(vcAtanSC[i]); !cAlike(atanSC[i], f) {
+ t.Errorf("Atan(%g) = %g, want %g", vcAtanSC[i], f, atanSC[i])
+ }
+ }
+}
+func TestAtanh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Atanh(vc[i]); !cVeryclose(atanh[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", vc[i], f, atanh[i])
+ }
+ }
+ for i := 0; i < len(vcAtanhSC); i++ {
+ if f := Atanh(vcAtanhSC[i]); !cAlike(atanhSC[i], f) {
+ t.Errorf("Atanh(%g) = %g, want %g", vcAtanhSC[i], f, atanhSC[i])
+ }
+ }
+}
+func TestConj(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Conj(vc[i]); !cVeryclose(conj[i], f) {
+ t.Errorf("Conj(%g) = %g, want %g", vc[i], f, conj[i])
+ }
+ }
+ for i := 0; i < len(vcConjSC); i++ {
+ if f := Conj(vcConjSC[i]); !cAlike(conjSC[i], f) {
+ t.Errorf("Conj(%g) = %g, want %g", vcConjSC[i], f, conjSC[i])
+ }
+ }
+}
+func TestCos(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Cos(vc[i]); !cSoclose(cos[i], f, 3e-15) {
+ t.Errorf("Cos(%g) = %g, want %g", vc[i], f, cos[i])
+ }
+ }
+ for i := 0; i < len(vcCosSC); i++ {
+ if f := Cos(vcCosSC[i]); !cAlike(cosSC[i], f) {
+ t.Errorf("Cos(%g) = %g, want %g", vcCosSC[i], f, cosSC[i])
+ }
+ }
+}
+func TestCosh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Cosh(vc[i]); !cSoclose(cosh[i], f, 2e-15) {
+ t.Errorf("Cosh(%g) = %g, want %g", vc[i], f, cosh[i])
+ }
+ }
+ for i := 0; i < len(vcCoshSC); i++ {
+ if f := Cosh(vcCoshSC[i]); !cAlike(coshSC[i], f) {
+ t.Errorf("Cosh(%g) = %g, want %g", vcCoshSC[i], f, coshSC[i])
+ }
+ }
+}
+func TestExp(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Exp(vc[i]); !cSoclose(exp[i], f, 1e-15) {
+ t.Errorf("Exp(%g) = %g, want %g", vc[i], f, exp[i])
+ }
+ }
+ for i := 0; i < len(vcExpSC); i++ {
+ if f := Exp(vcExpSC[i]); !cAlike(expSC[i], f) {
+ t.Errorf("Exp(%g) = %g, want %g", vcExpSC[i], f, expSC[i])
+ }
+ }
+}
+func TestIsNaN(t *testing.T) {
+ for i := 0; i < len(vcIsNaNSC); i++ {
+ if f := IsNaN(vcIsNaNSC[i]); isNaNSC[i] != f {
+ t.Errorf("IsNaN(%v) = %v, want %v", vcIsNaNSC[i], f, isNaNSC[i])
+ }
+ }
+}
+func TestLog(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Log(vc[i]); !cVeryclose(log[i], f) {
+ t.Errorf("Log(%g) = %g, want %g", vc[i], f, log[i])
+ }
+ }
+ for i := 0; i < len(vcLogSC); i++ {
+ if f := Log(vcLogSC[i]); !cAlike(logSC[i], f) {
+ t.Errorf("Log(%g) = %g, want %g", vcLogSC[i], f, logSC[i])
+ }
+ }
+}
+func TestLog10(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Log10(vc[i]); !cVeryclose(log10[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", vc[i], f, log10[i])
+ }
+ }
+ for i := 0; i < len(vcLog10SC); i++ {
+ if f := Log10(vcLog10SC[i]); !cAlike(log10SC[i], f) {
+ t.Errorf("Log10(%g) = %g, want %g", vcLog10SC[i], f, log10SC[i])
+ }
+ }
+}
+func TestPolar(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if r, theta := Polar(vc[i]); !veryclose(polar[i].r, r) && !veryclose(polar[i].theta, theta) {
+ t.Errorf("Polar(%g) = %g, %g want %g, %g", vc[i], r, theta, polar[i].r, polar[i].theta)
+ }
+ }
+ for i := 0; i < len(vcPolarSC); i++ {
+ if r, theta := Polar(vcPolarSC[i]); !alike(polarSC[i].r, r) && !alike(polarSC[i].theta, theta) {
+ t.Errorf("Polar(%g) = %g, %g, want %g, %g", vcPolarSC[i], r, theta, polarSC[i].r, polarSC[i].theta)
+ }
+ }
+}
+func TestPow(t *testing.T) {
+ // Special cases for Pow(0, c).
+ var zero = complex(0, 0)
+ zeroPowers := [][2]complex128{
+ {0, 1 + 0i},
+ {1.5, 0 + 0i},
+ {-1.5, complex(math.Inf(0), 0)},
+ {-1.5 + 1.5i, Inf()},
+ }
+ for _, zp := range zeroPowers {
+ if f := Pow(zero, zp[0]); f != zp[1] {
+ t.Errorf("Pow(%g, %g) = %g, want %g", zero, zp[0], f, zp[1])
+ }
+ }
+ var a = complex(3.0, 3.0)
+ for i := 0; i < len(vc); i++ {
+ if f := Pow(a, vc[i]); !cSoclose(pow[i], f, 4e-15) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", a, vc[i], f, pow[i])
+ }
+ }
+ for i := 0; i < len(vcPowSC); i++ {
+ if f := Pow(vcPowSC[i][0], vcPowSC[i][0]); !cAlike(powSC[i], f) {
+ t.Errorf("Pow(%g, %g) = %g, want %g", vcPowSC[i][0], vcPowSC[i][0], f, powSC[i])
+ }
+ }
+}
+func TestRect(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Rect(polar[i].r, polar[i].theta); !cVeryclose(vc[i], f) {
+ t.Errorf("Rect(%g, %g) = %g want %g", polar[i].r, polar[i].theta, f, vc[i])
+ }
+ }
+ for i := 0; i < len(vcPolarSC); i++ {
+ if f := Rect(polarSC[i].r, polarSC[i].theta); !cAlike(vcPolarSC[i], f) {
+ t.Errorf("Rect(%g, %g) = %g, want %g", polarSC[i].r, polarSC[i].theta, f, vcPolarSC[i])
+ }
+ }
+}
+func TestSin(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Sin(vc[i]); !cSoclose(sin[i], f, 2e-15) {
+ t.Errorf("Sin(%g) = %g, want %g", vc[i], f, sin[i])
+ }
+ }
+ for i := 0; i < len(vcSinSC); i++ {
+ if f := Sin(vcSinSC[i]); !cAlike(sinSC[i], f) {
+ t.Errorf("Sin(%g) = %g, want %g", vcSinSC[i], f, sinSC[i])
+ }
+ }
+}
+func TestSinh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Sinh(vc[i]); !cSoclose(sinh[i], f, 2e-15) {
+ t.Errorf("Sinh(%g) = %g, want %g", vc[i], f, sinh[i])
+ }
+ }
+ for i := 0; i < len(vcSinhSC); i++ {
+ if f := Sinh(vcSinhSC[i]); !cAlike(sinhSC[i], f) {
+ t.Errorf("Sinh(%g) = %g, want %g", vcSinhSC[i], f, sinhSC[i])
+ }
+ }
+}
+func TestSqrt(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Sqrt(vc[i]); !cVeryclose(sqrt[i], f) {
+ t.Errorf("Sqrt(%g) = %g, want %g", vc[i], f, sqrt[i])
+ }
+ }
+ for i := 0; i < len(vcSqrtSC); i++ {
+ if f := Sqrt(vcSqrtSC[i]); !cAlike(sqrtSC[i], f) {
+ t.Errorf("Sqrt(%g) = %g, want %g", vcSqrtSC[i], f, sqrtSC[i])
+ }
+ }
+}
+func TestTan(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Tan(vc[i]); !cSoclose(tan[i], f, 3e-15) {
+ t.Errorf("Tan(%g) = %g, want %g", vc[i], f, tan[i])
+ }
+ }
+ for i := 0; i < len(vcTanSC); i++ {
+ if f := Tan(vcTanSC[i]); !cAlike(tanSC[i], f) {
+ t.Errorf("Tan(%g) = %g, want %g", vcTanSC[i], f, tanSC[i])
+ }
+ }
+}
+func TestTanh(t *testing.T) {
+ for i := 0; i < len(vc); i++ {
+ if f := Tanh(vc[i]); !cSoclose(tanh[i], f, 2e-15) {
+ t.Errorf("Tanh(%g) = %g, want %g", vc[i], f, tanh[i])
+ }
+ }
+ for i := 0; i < len(vcTanhSC); i++ {
+ if f := Tanh(vcTanhSC[i]); !cAlike(tanhSC[i], f) {
+ t.Errorf("Tanh(%g) = %g, want %g", vcTanhSC[i], f, tanhSC[i])
+ }
+ }
+}
+
+func BenchmarkAbs(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Abs(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAcos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Acos(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAcosh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Acosh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAsin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Asin(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAsinh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Asinh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAtan(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atan(complex(2.5, 3.5))
+ }
+}
+func BenchmarkAtanh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Atanh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkConj(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Conj(complex(2.5, 3.5))
+ }
+}
+func BenchmarkCos(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cos(complex(2.5, 3.5))
+ }
+}
+func BenchmarkCosh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Cosh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkExp(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Exp(complex(2.5, 3.5))
+ }
+}
+func BenchmarkLog(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log(complex(2.5, 3.5))
+ }
+}
+func BenchmarkLog10(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Log10(complex(2.5, 3.5))
+ }
+}
+func BenchmarkPhase(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Phase(complex(2.5, 3.5))
+ }
+}
+func BenchmarkPolar(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Polar(complex(2.5, 3.5))
+ }
+}
+func BenchmarkPow(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Pow(complex(2.5, 3.5), complex(2.5, 3.5))
+ }
+}
+func BenchmarkRect(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Rect(2.5, 1.5)
+ }
+}
+func BenchmarkSin(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sin(complex(2.5, 3.5))
+ }
+}
+func BenchmarkSinh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sinh(complex(2.5, 3.5))
+ }
+}
+func BenchmarkSqrt(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Sqrt(complex(2.5, 3.5))
+ }
+}
+func BenchmarkTan(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Tan(complex(2.5, 3.5))
+ }
+}
+func BenchmarkTanh(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Tanh(complex(2.5, 3.5))
+ }
+}
diff --git a/src/math/cmplx/conj.go b/src/math/cmplx/conj.go
new file mode 100644
index 0000000..34a4277
--- /dev/null
+++ b/src/math/cmplx/conj.go
@@ -0,0 +1,8 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+// Conj returns the complex conjugate of x.
+func Conj(x complex128) complex128 { return complex(real(x), -imag(x)) }
diff --git a/src/math/cmplx/exp.go b/src/math/cmplx/exp.go
new file mode 100644
index 0000000..485ed2c
--- /dev/null
+++ b/src/math/cmplx/exp.go
@@ -0,0 +1,55 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex exponential function
+//
+// DESCRIPTION:
+//
+// Returns the complex exponential of the complex argument z.
+//
+// If
+// z = x + iy,
+// r = exp(x),
+// then
+// w = r cos y + i r sin y.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 8700 3.7e-17 1.1e-17
+// IEEE -10,+10 30000 3.0e-16 8.7e-17
+
+// Exp returns e**x, the base-e exponential of x.
+func Exp(x complex128) complex128 {
+ r := math.Exp(real(x))
+ s, c := math.Sincos(imag(x))
+ return complex(r*c, r*s)
+}
diff --git a/src/math/cmplx/isinf.go b/src/math/cmplx/isinf.go
new file mode 100644
index 0000000..d5a65b4
--- /dev/null
+++ b/src/math/cmplx/isinf.go
@@ -0,0 +1,21 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// IsInf returns true if either real(x) or imag(x) is an infinity.
+func IsInf(x complex128) bool {
+ if math.IsInf(real(x), 0) || math.IsInf(imag(x), 0) {
+ return true
+ }
+ return false
+}
+
+// Inf returns a complex infinity, complex(+Inf, +Inf).
+func Inf() complex128 {
+ inf := math.Inf(1)
+ return complex(inf, inf)
+}
diff --git a/src/math/cmplx/isnan.go b/src/math/cmplx/isnan.go
new file mode 100644
index 0000000..05d0cce
--- /dev/null
+++ b/src/math/cmplx/isnan.go
@@ -0,0 +1,25 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// IsNaN returns true if either real(x) or imag(x) is NaN
+// and neither is an infinity.
+func IsNaN(x complex128) bool {
+ switch {
+ case math.IsInf(real(x), 0) || math.IsInf(imag(x), 0):
+ return false
+ case math.IsNaN(real(x)) || math.IsNaN(imag(x)):
+ return true
+ }
+ return false
+}
+
+// NaN returns a complex ``not-a-number'' value.
+func NaN() complex128 {
+ nan := math.NaN()
+ return complex(nan, nan)
+}
diff --git a/src/math/cmplx/log.go b/src/math/cmplx/log.go
new file mode 100644
index 0000000..881a064
--- /dev/null
+++ b/src/math/cmplx/log.go
@@ -0,0 +1,64 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex natural logarithm
+//
+// DESCRIPTION:
+//
+// Returns complex logarithm to the base e (2.718...) of
+// the complex argument z.
+//
+// If
+// z = x + iy, r = sqrt( x**2 + y**2 ),
+// then
+// w = log(r) + i arctan(y/x).
+//
+// The arctangent ranges from -PI to +PI.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 7000 8.5e-17 1.9e-17
+// IEEE -10,+10 30000 5.0e-15 1.1e-16
+//
+// Larger relative error can be observed for z near 1 +i0.
+// In IEEE arithmetic the peak absolute error is 5.2e-16, rms
+// absolute error 1.0e-16.
+
+// Log returns the natural logarithm of x.
+func Log(x complex128) complex128 {
+ return complex(math.Log(Abs(x)), Phase(x))
+}
+
+// Log10 returns the decimal logarithm of x.
+func Log10(x complex128) complex128 {
+ return math.Log10E * Log(x)
+}
diff --git a/src/math/cmplx/phase.go b/src/math/cmplx/phase.go
new file mode 100644
index 0000000..03cece8
--- /dev/null
+++ b/src/math/cmplx/phase.go
@@ -0,0 +1,11 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// Phase returns the phase (also called the argument) of x.
+// The returned value is in the range [-Pi, Pi].
+func Phase(x complex128) float64 { return math.Atan2(imag(x), real(x)) }
diff --git a/src/math/cmplx/polar.go b/src/math/cmplx/polar.go
new file mode 100644
index 0000000..9b192bc
--- /dev/null
+++ b/src/math/cmplx/polar.go
@@ -0,0 +1,12 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+// Polar returns the absolute value r and phase θ of x,
+// such that x = r * e**θi.
+// The phase is in the range [-Pi, Pi].
+func Polar(x complex128) (r, θ float64) {
+ return Abs(x), Phase(x)
+}
diff --git a/src/math/cmplx/pow.go b/src/math/cmplx/pow.go
new file mode 100644
index 0000000..1630b87
--- /dev/null
+++ b/src/math/cmplx/pow.go
@@ -0,0 +1,78 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex power function
+//
+// DESCRIPTION:
+//
+// Raises complex A to the complex Zth power.
+// Definition is per AMS55 # 4.2.8,
+// analytically equivalent to cpow(a,z) = cexp(z clog(a)).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 9.4e-15 1.5e-15
+
+// Pow returns x**y, the base-x exponential of y.
+// For generalized compatibility with math.Pow:
+// Pow(0, ±0) returns 1+0i
+// Pow(0, c) for real(c)<0 returns Inf+0i if imag(c) is zero, otherwise Inf+Inf i.
+func Pow(x, y complex128) complex128 {
+ if x == 0 { // Guaranteed also true for x == -0.
+ r, i := real(y), imag(y)
+ switch {
+ case r == 0:
+ return 1
+ case r < 0:
+ if i == 0 {
+ return complex(math.Inf(1), 0)
+ }
+ return Inf()
+ case r > 0:
+ return 0
+ }
+ panic("not reached")
+ }
+ modulus := Abs(x)
+ if modulus == 0 {
+ return complex(0, 0)
+ }
+ r := math.Pow(modulus, real(y))
+ arg := Phase(x)
+ theta := real(y) * arg
+ if imag(y) != 0 {
+ r *= math.Exp(-imag(y) * arg)
+ theta += imag(y) * math.Log(modulus)
+ }
+ s, c := math.Sincos(theta)
+ return complex(r*c, r*s)
+}
diff --git a/src/math/cmplx/rect.go b/src/math/cmplx/rect.go
new file mode 100644
index 0000000..bf94d78
--- /dev/null
+++ b/src/math/cmplx/rect.go
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// Rect returns the complex number x with polar coordinates r, θ.
+func Rect(r, θ float64) complex128 {
+ s, c := math.Sincos(θ)
+ return complex(r*c, r*s)
+}
diff --git a/src/math/cmplx/sin.go b/src/math/cmplx/sin.go
new file mode 100644
index 0000000..2c57536
--- /dev/null
+++ b/src/math/cmplx/sin.go
@@ -0,0 +1,132 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex circular sine
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// w = sin x cosh y + i cos x sinh y.
+//
+// csin(z) = -i csinh(iz).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 8400 5.3e-17 1.3e-17
+// IEEE -10,+10 30000 3.8e-16 1.0e-16
+// Also tested by csin(casin(z)) = z.
+
+// Sin returns the sine of x.
+func Sin(x complex128) complex128 {
+ s, c := math.Sincos(real(x))
+ sh, ch := sinhcosh(imag(x))
+ return complex(s*ch, c*sh)
+}
+
+// Complex hyperbolic sine
+//
+// DESCRIPTION:
+//
+// csinh z = (cexp(z) - cexp(-z))/2
+// = sinh x * cos y + i cosh x * sin y .
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 3.1e-16 8.2e-17
+
+// Sinh returns the hyperbolic sine of x.
+func Sinh(x complex128) complex128 {
+ s, c := math.Sincos(imag(x))
+ sh, ch := sinhcosh(real(x))
+ return complex(c*sh, s*ch)
+}
+
+// Complex circular cosine
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// w = cos x cosh y - i sin x sinh y.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 8400 4.5e-17 1.3e-17
+// IEEE -10,+10 30000 3.8e-16 1.0e-16
+
+// Cos returns the cosine of x.
+func Cos(x complex128) complex128 {
+ s, c := math.Sincos(real(x))
+ sh, ch := sinhcosh(imag(x))
+ return complex(c*ch, -s*sh)
+}
+
+// Complex hyperbolic cosine
+//
+// DESCRIPTION:
+//
+// ccosh(z) = cosh x cos y + i sinh x sin y .
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 2.9e-16 8.1e-17
+
+// Cosh returns the hyperbolic cosine of x.
+func Cosh(x complex128) complex128 {
+ s, c := math.Sincos(imag(x))
+ sh, ch := sinhcosh(real(x))
+ return complex(c*ch, s*sh)
+}
+
+// calculate sinh and cosh
+func sinhcosh(x float64) (sh, ch float64) {
+ if math.Abs(x) <= 0.5 {
+ return math.Sinh(x), math.Cosh(x)
+ }
+ e := math.Exp(x)
+ ei := 0.5 / e
+ e *= 0.5
+ return e - ei, e + ei
+}
diff --git a/src/math/cmplx/sqrt.go b/src/math/cmplx/sqrt.go
new file mode 100644
index 0000000..4ef6807
--- /dev/null
+++ b/src/math/cmplx/sqrt.go
@@ -0,0 +1,104 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex square root
+//
+// DESCRIPTION:
+//
+// If z = x + iy, r = |z|, then
+//
+// 1/2
+// Re w = [ (r + x)/2 ] ,
+//
+// 1/2
+// Im w = [ (r - x)/2 ] .
+//
+// Cancellation error in r-x or r+x is avoided by using the
+// identity 2 Re w Im w = y.
+//
+// Note that -w is also a square root of z. The root chosen
+// is always in the right half plane and Im w has the same sign as y.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 25000 3.2e-17 9.6e-18
+// IEEE -10,+10 1,000,000 2.9e-16 6.1e-17
+
+// Sqrt returns the square root of x.
+// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
+func Sqrt(x complex128) complex128 {
+ if imag(x) == 0 {
+ if real(x) == 0 {
+ return complex(0, 0)
+ }
+ if real(x) < 0 {
+ return complex(0, math.Sqrt(-real(x)))
+ }
+ return complex(math.Sqrt(real(x)), 0)
+ }
+ if real(x) == 0 {
+ if imag(x) < 0 {
+ r := math.Sqrt(-0.5 * imag(x))
+ return complex(r, -r)
+ }
+ r := math.Sqrt(0.5 * imag(x))
+ return complex(r, r)
+ }
+ a := real(x)
+ b := imag(x)
+ var scale float64
+ // Rescale to avoid internal overflow or underflow.
+ if math.Abs(a) > 4 || math.Abs(b) > 4 {
+ a *= 0.25
+ b *= 0.25
+ scale = 2
+ } else {
+ a *= 1.8014398509481984e16 // 2**54
+ b *= 1.8014398509481984e16
+ scale = 7.450580596923828125e-9 // 2**-27
+ }
+ r := math.Hypot(a, b)
+ var t float64
+ if a > 0 {
+ t = math.Sqrt(0.5*r + 0.5*a)
+ r = scale * math.Abs((0.5*b)/t)
+ t *= scale
+ } else {
+ r = math.Sqrt(0.5*r - 0.5*a)
+ t = scale * math.Abs((0.5*b)/r)
+ r *= scale
+ }
+ if b < 0 {
+ return complex(t, -r)
+ }
+ return complex(t, r)
+}
diff --git a/src/math/cmplx/tan.go b/src/math/cmplx/tan.go
new file mode 100644
index 0000000..9485315
--- /dev/null
+++ b/src/math/cmplx/tan.go
@@ -0,0 +1,184 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package cmplx
+
+import "math"
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
+// The go code is a simplified version of the original C.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// Complex circular tangent
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// sin 2x + i sinh 2y
+// w = --------------------.
+// cos 2x + cosh 2y
+//
+// On the real axis the denominator is zero at odd multiples
+// of PI/2. The denominator is evaluated by its Taylor
+// series near these points.
+//
+// ctan(z) = -i ctanh(iz).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 5200 7.1e-17 1.6e-17
+// IEEE -10,+10 30000 7.2e-16 1.2e-16
+// Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
+
+// Tan returns the tangent of x.
+func Tan(x complex128) complex128 {
+ d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
+ if math.Abs(d) < 0.25 {
+ d = tanSeries(x)
+ }
+ if d == 0 {
+ return Inf()
+ }
+ return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
+}
+
+// Complex hyperbolic tangent
+//
+// DESCRIPTION:
+//
+// tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -10,+10 30000 1.7e-14 2.4e-16
+
+// Tanh returns the hyperbolic tangent of x.
+func Tanh(x complex128) complex128 {
+ d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
+ if d == 0 {
+ return Inf()
+ }
+ return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
+}
+
+// Program to subtract nearest integer multiple of PI
+func reducePi(x float64) float64 {
+ const (
+ // extended precision value of PI:
+ DP1 = 3.14159265160560607910E0 // ?? 0x400921fb54000000
+ DP2 = 1.98418714791870343106E-9 // ?? 0x3e210b4610000000
+ DP3 = 1.14423774522196636802E-17 // ?? 0x3c6a62633145c06e
+ )
+ t := x / math.Pi
+ if t >= 0 {
+ t += 0.5
+ } else {
+ t -= 0.5
+ }
+ t = float64(int64(t)) // int64(t) = the multiple
+ return ((x - t*DP1) - t*DP2) - t*DP3
+}
+
+// Taylor series expansion for cosh(2y) - cos(2x)
+func tanSeries(z complex128) float64 {
+ const MACHEP = 1.0 / (1 << 53)
+ x := math.Abs(2 * real(z))
+ y := math.Abs(2 * imag(z))
+ x = reducePi(x)
+ x = x * x
+ y = y * y
+ x2 := 1.0
+ y2 := 1.0
+ f := 1.0
+ rn := 0.0
+ d := 0.0
+ for {
+ rn += 1
+ f *= rn
+ rn += 1
+ f *= rn
+ x2 *= x
+ y2 *= y
+ t := y2 + x2
+ t /= f
+ d += t
+
+ rn += 1
+ f *= rn
+ rn += 1
+ f *= rn
+ x2 *= x
+ y2 *= y
+ t = y2 - x2
+ t /= f
+ d += t
+ if math.Abs(t/d) <= MACHEP {
+ break
+ }
+ }
+ return d
+}
+
+// Complex circular cotangent
+//
+// DESCRIPTION:
+//
+// If
+// z = x + iy,
+//
+// then
+//
+// sin 2x - i sinh 2y
+// w = --------------------.
+// cosh 2y - cos 2x
+//
+// On the real axis, the denominator has zeros at even
+// multiples of PI/2. Near these points it is evaluated
+// by a Taylor series.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10,+10 3000 6.5e-17 1.6e-17
+// IEEE -10,+10 30000 9.2e-16 1.2e-16
+// Also tested by ctan * ccot = 1 + i0.
+
+// Cot returns the cotangent of x.
+func Cot(x complex128) complex128 {
+ d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
+ if math.Abs(d) < 0.25 {
+ d = tanSeries(x)
+ }
+ if d == 0 {
+ return Inf()
+ }
+ return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
+}
diff --git a/src/math/const.go b/src/math/const.go
new file mode 100644
index 0000000..f1247c3
--- /dev/null
+++ b/src/math/const.go
@@ -0,0 +1,51 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package math provides basic constants and mathematical functions.
+package math
+
+// Mathematical constants.
+// Reference: http://oeis.org/Axxxxxx
+const (
+ E = 2.71828182845904523536028747135266249775724709369995957496696763 // A001113
+ Pi = 3.14159265358979323846264338327950288419716939937510582097494459 // A000796
+ Phi = 1.61803398874989484820458683436563811772030917980576286213544862 // A001622
+
+ Sqrt2 = 1.41421356237309504880168872420969807856967187537694807317667974 // A002193
+ SqrtE = 1.64872127070012814684865078781416357165377610071014801157507931 // A019774
+ SqrtPi = 1.77245385090551602729816748334114518279754945612238712821380779 // A002161
+ SqrtPhi = 1.27201964951406896425242246173749149171560804184009624861664038 // A139339
+
+ Ln2 = 0.693147180559945309417232121458176568075500134360255254120680009 // A002162
+ Log2E = 1 / Ln2
+ Ln10 = 2.30258509299404568401799145468436420760110148862877297603332790 // A002392
+ Log10E = 1 / Ln10
+)
+
+// Floating-point limit values.
+// Max is the largest finite value representable by the type.
+// SmallestNonzero is the smallest positive, non-zero value representable by the type.
+const (
+ MaxFloat32 = 3.40282346638528859811704183484516925440e+38 // 2**127 * (2**24 - 1) / 2**23
+ SmallestNonzeroFloat32 = 1.401298464324817070923729583289916131280e-45 // 1 / 2**(127 - 1 + 23)
+
+ MaxFloat64 = 1.797693134862315708145274237317043567981e+308 // 2**1023 * (2**53 - 1) / 2**52
+ SmallestNonzeroFloat64 = 4.940656458412465441765687928682213723651e-324 // 1 / 2**(1023 - 1 + 52)
+)
+
+// Integer limit values.
+const (
+ MaxInt8 = 1<<7 - 1
+ MinInt8 = -1 << 7
+ MaxInt16 = 1<<15 - 1
+ MinInt16 = -1 << 15
+ MaxInt32 = 1<<31 - 1
+ MinInt32 = -1 << 31
+ MaxInt64 = 1<<63 - 1
+ MinInt64 = -1 << 63
+ MaxUint8 = 1<<8 - 1
+ MaxUint16 = 1<<16 - 1
+ MaxUint32 = 1<<32 - 1
+ MaxUint64 = 1<<64 - 1
+)
diff --git a/src/math/copysign.go b/src/math/copysign.go
new file mode 100644
index 0000000..719c64b
--- /dev/null
+++ b/src/math/copysign.go
@@ -0,0 +1,12 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Copysign returns a value with the magnitude
+// of x and the sign of y.
+func Copysign(x, y float64) float64 {
+ const sign = 1 << 63
+ return Float64frombits(Float64bits(x)&^sign | Float64bits(y)&sign)
+}
diff --git a/src/math/dim.go b/src/math/dim.go
new file mode 100644
index 0000000..1c634d4
--- /dev/null
+++ b/src/math/dim.go
@@ -0,0 +1,72 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Dim returns the maximum of x-y or 0.
+//
+// Special cases are:
+// Dim(+Inf, +Inf) = NaN
+// Dim(-Inf, -Inf) = NaN
+// Dim(x, NaN) = Dim(NaN, x) = NaN
+func Dim(x, y float64) float64
+
+func dim(x, y float64) float64 {
+ return max(x-y, 0)
+}
+
+// Max returns the larger of x or y.
+//
+// Special cases are:
+// Max(x, +Inf) = Max(+Inf, x) = +Inf
+// Max(x, NaN) = Max(NaN, x) = NaN
+// Max(+0, ±0) = Max(±0, +0) = +0
+// Max(-0, -0) = -0
+func Max(x, y float64) float64
+
+func max(x, y float64) float64 {
+ // special cases
+ switch {
+ case IsInf(x, 1) || IsInf(y, 1):
+ return Inf(1)
+ case IsNaN(x) || IsNaN(y):
+ return NaN()
+ case x == 0 && x == y:
+ if Signbit(x) {
+ return y
+ }
+ return x
+ }
+ if x > y {
+ return x
+ }
+ return y
+}
+
+// Min returns the smaller of x or y.
+//
+// Special cases are:
+// Min(x, -Inf) = Min(-Inf, x) = -Inf
+// Min(x, NaN) = Min(NaN, x) = NaN
+// Min(-0, ±0) = Min(±0, -0) = -0
+func Min(x, y float64) float64
+
+func min(x, y float64) float64 {
+ // special cases
+ switch {
+ case IsInf(x, -1) || IsInf(y, -1):
+ return Inf(-1)
+ case IsNaN(x) || IsNaN(y):
+ return NaN()
+ case x == 0 && x == y:
+ if Signbit(x) {
+ return x
+ }
+ return y
+ }
+ if x < y {
+ return x
+ }
+ return y
+}
diff --git a/src/math/dim_386.s b/src/math/dim_386.s
new file mode 100644
index 0000000..c8194fe
--- /dev/null
+++ b/src/math/dim_386.s
@@ -0,0 +1,14 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Dim(SB),NOSPLIT,$0
+ JMP ·dim(SB)
+
+TEXT ·Max(SB),NOSPLIT,$0
+ JMP ·max(SB)
+
+TEXT ·Min(SB),NOSPLIT,$0
+ JMP ·min(SB)
diff --git a/src/math/dim_amd64.s b/src/math/dim_amd64.s
new file mode 100644
index 0000000..622cc3f
--- /dev/null
+++ b/src/math/dim_amd64.s
@@ -0,0 +1,144 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+
+// func Dim(x, y float64) float64
+TEXT ·Dim(SB),NOSPLIT,$0
+ // (+Inf, +Inf) special case
+ MOVQ x+0(FP), BX
+ MOVQ y+8(FP), CX
+ MOVQ $PosInf, AX
+ CMPQ AX, BX
+ JNE dim2
+ CMPQ AX, CX
+ JEQ bothInf
+dim2: // (-Inf, -Inf) special case
+ MOVQ $NegInf, AX
+ CMPQ AX, BX
+ JNE dim3
+ CMPQ AX, CX
+ JEQ bothInf
+dim3: // (NaN, x) or (x, NaN)
+ MOVQ $~(1<<63), DX
+ MOVQ $NaN, AX
+ ANDQ DX, BX // x = |x|
+ CMPQ AX, BX
+ JLE isDimNaN
+ ANDQ DX, CX // y = |y|
+ CMPQ AX, CX
+ JLE isDimNaN
+
+ MOVSD x+0(FP), X0
+ SUBSD y+8(FP), X0
+ MOVSD $(0.0), X1
+ MAXSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+bothInf: // Dim(-Inf, -Inf) or Dim(+Inf, +Inf)
+ MOVQ $NaN, AX
+isDimNaN:
+ MOVQ AX, ret+16(FP)
+ RET
+
+// func ·Max(x, y float64) float64
+TEXT ·Max(SB),NOSPLIT,$0
+ // +Inf special cases
+ MOVQ $PosInf, AX
+ MOVQ x+0(FP), R8
+ CMPQ AX, R8
+ JEQ isPosInf
+ MOVQ y+8(FP), R9
+ CMPQ AX, R9
+ JEQ isPosInf
+ // NaN special cases
+ MOVQ $~(1<<63), DX // bit mask
+ MOVQ $NaN, AX
+ MOVQ R8, BX
+ ANDQ DX, BX // x = |x|
+ CMPQ AX, BX
+ JLE isMaxNaN
+ MOVQ R9, CX
+ ANDQ DX, CX // y = |y|
+ CMPQ AX, CX
+ JLE isMaxNaN
+ // ±0 special cases
+ ORQ CX, BX
+ JEQ isMaxZero
+
+ MOVQ R8, X0
+ MOVQ R9, X1
+ MAXSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+isMaxNaN: // return NaN
+isPosInf: // return +Inf
+ MOVQ AX, ret+16(FP)
+ RET
+isMaxZero:
+ MOVQ $(1<<63), AX // -0.0
+ CMPQ AX, R8
+ JEQ +3(PC)
+ MOVQ R8, ret+16(FP) // return 0
+ RET
+ MOVQ R9, ret+16(FP) // return other 0
+ RET
+
+/*
+ MOVQ $0, AX
+ CMPQ AX, R8
+ JNE +3(PC)
+ MOVQ R8, ret+16(FP) // return 0
+ RET
+ MOVQ R9, ret+16(FP) // return other 0
+ RET
+*/
+
+// func Min(x, y float64) float64
+TEXT ·Min(SB),NOSPLIT,$0
+ // -Inf special cases
+ MOVQ $NegInf, AX
+ MOVQ x+0(FP), R8
+ CMPQ AX, R8
+ JEQ isNegInf
+ MOVQ y+8(FP), R9
+ CMPQ AX, R9
+ JEQ isNegInf
+ // NaN special cases
+ MOVQ $~(1<<63), DX
+ MOVQ $NaN, AX
+ MOVQ R8, BX
+ ANDQ DX, BX // x = |x|
+ CMPQ AX, BX
+ JLE isMinNaN
+ MOVQ R9, CX
+ ANDQ DX, CX // y = |y|
+ CMPQ AX, CX
+ JLE isMinNaN
+ // ±0 special cases
+ ORQ CX, BX
+ JEQ isMinZero
+
+ MOVQ R8, X0
+ MOVQ R9, X1
+ MINSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+isMinNaN: // return NaN
+isNegInf: // return -Inf
+ MOVQ AX, ret+16(FP)
+ RET
+isMinZero:
+ MOVQ $(1<<63), AX // -0.0
+ CMPQ AX, R8
+ JEQ +3(PC)
+ MOVQ R9, ret+16(FP) // return other 0
+ RET
+ MOVQ R8, ret+16(FP) // return -0
+ RET
+
diff --git a/src/math/dim_amd64p32.s b/src/math/dim_amd64p32.s
new file mode 100644
index 0000000..e5e3447
--- /dev/null
+++ b/src/math/dim_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "dim_amd64.s"
diff --git a/src/math/dim_arm.s b/src/math/dim_arm.s
new file mode 100644
index 0000000..be66950
--- /dev/null
+++ b/src/math/dim_arm.s
@@ -0,0 +1,14 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Dim(SB),NOSPLIT,$0
+ B ·dim(SB)
+
+TEXT ·Min(SB),NOSPLIT,$0
+ B ·min(SB)
+
+TEXT ·Max(SB),NOSPLIT,$0
+ B ·max(SB)
diff --git a/src/math/erf.go b/src/math/erf.go
new file mode 100644
index 0000000..4cd80f8
--- /dev/null
+++ b/src/math/erf.go
@@ -0,0 +1,335 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point error function and complementary error function.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/s_erf.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// double erf(double x)
+// double erfc(double x)
+// x
+// 2 |\
+// erf(x) = --------- | exp(-t*t)dt
+// sqrt(pi) \|
+// 0
+//
+// erfc(x) = 1-erf(x)
+// Note that
+// erf(-x) = -erf(x)
+// erfc(-x) = 2 - erfc(x)
+//
+// Method:
+// 1. For |x| in [0, 0.84375]
+// erf(x) = x + x*R(x**2)
+// erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
+// = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
+// where R = P/Q where P is an odd poly of degree 8 and
+// Q is an odd poly of degree 10.
+// -57.90
+// | R - (erf(x)-x)/x | <= 2
+//
+//
+// Remark. The formula is derived by noting
+// erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
+// and that
+// 2/sqrt(pi) = 1.128379167095512573896158903121545171688
+// is close to one. The interval is chosen because the fix
+// point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
+// near 0.6174), and by some experiment, 0.84375 is chosen to
+// guarantee the error is less than one ulp for erf.
+//
+// 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
+// c = 0.84506291151 rounded to single (24 bits)
+// erf(x) = sign(x) * (c + P1(s)/Q1(s))
+// erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
+// 1+(c+P1(s)/Q1(s)) if x < 0
+// |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
+// Remark: here we use the taylor series expansion at x=1.
+// erf(1+s) = erf(1) + s*Poly(s)
+// = 0.845.. + P1(s)/Q1(s)
+// That is, we use rational approximation to approximate
+// erf(1+s) - (c = (single)0.84506291151)
+// Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
+// where
+// P1(s) = degree 6 poly in s
+// Q1(s) = degree 6 poly in s
+//
+// 3. For x in [1.25,1/0.35(~2.857143)],
+// erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
+// erf(x) = 1 - erfc(x)
+// where
+// R1(z) = degree 7 poly in z, (z=1/x**2)
+// S1(z) = degree 8 poly in z
+//
+// 4. For x in [1/0.35,28]
+// erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
+// = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
+// = 2.0 - tiny (if x <= -6)
+// erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
+// erf(x) = sign(x)*(1.0 - tiny)
+// where
+// R2(z) = degree 6 poly in z, (z=1/x**2)
+// S2(z) = degree 7 poly in z
+//
+// Note1:
+// To compute exp(-x*x-0.5625+R/S), let s be a single
+// precision number and s := x; then
+// -x*x = -s*s + (s-x)*(s+x)
+// exp(-x*x-0.5626+R/S) =
+// exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
+// Note2:
+// Here 4 and 5 make use of the asymptotic series
+// exp(-x*x)
+// erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
+// x*sqrt(pi)
+// We use rational approximation to approximate
+// g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625
+// Here is the error bound for R1/S1 and R2/S2
+// |R1/S1 - f(x)| < 2**(-62.57)
+// |R2/S2 - f(x)| < 2**(-61.52)
+//
+// 5. For inf > x >= 28
+// erf(x) = sign(x) *(1 - tiny) (raise inexact)
+// erfc(x) = tiny*tiny (raise underflow) if x > 0
+// = 2 - tiny if x<0
+//
+// 7. Special case:
+// erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
+// erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
+// erfc/erf(NaN) is NaN
+
+const (
+ erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000
+ // Coefficients for approximation to erf in [0, 0.84375]
+ efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69
+ efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69
+ pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68
+ pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
+ pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
+ pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4
+ pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
+ qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09
+ qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA
+ qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F
+ qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10
+ qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120
+ // Coefficients for approximation to erf in [0.84375, 1.25]
+ pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
+ pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D
+ pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
+ pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4
+ pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
+ pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB
+ pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
+ qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323
+ qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33
+ qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7
+ qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F
+ qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C
+ qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D
+ // Coefficients for approximation to erfc in [1.25, 1/0.35]
+ ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
+ ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
+ ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
+ ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
+ ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
+ ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
+ ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
+ ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
+ sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687
+ sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721
+ sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71
+ sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868
+ sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314
+ sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C
+ sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93
+ sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62
+ // Coefficients for approximation to erfc in [1/.35, 28]
+ rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
+ rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
+ rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
+ rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
+ rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
+ rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
+ rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
+ sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190
+ sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A
+ sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118
+ sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A
+ sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6
+ sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763
+ sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
+)
+
+// Erf returns the error function of x.
+//
+// Special cases are:
+// Erf(+Inf) = 1
+// Erf(-Inf) = -1
+// Erf(NaN) = NaN
+func Erf(x float64) float64 {
+ const (
+ VeryTiny = 2.848094538889218e-306 // 0x0080000000000000
+ Small = 1.0 / (1 << 28) // 2**-28
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 1
+ case IsInf(x, -1):
+ return -1
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x < 0.84375 { // |x| < 0.84375
+ var temp float64
+ if x < Small { // |x| < 2**-28
+ if x < VeryTiny {
+ temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
+ } else {
+ temp = x + efx*x
+ }
+ } else {
+ z := x * x
+ r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
+ s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
+ y := r / s
+ temp = x + x*y
+ }
+ if sign {
+ return -temp
+ }
+ return temp
+ }
+ if x < 1.25 { // 0.84375 <= |x| < 1.25
+ s := x - 1
+ P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
+ Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
+ if sign {
+ return -erx - P/Q
+ }
+ return erx + P/Q
+ }
+ if x >= 6 { // inf > |x| >= 6
+ if sign {
+ return -1
+ }
+ return 1
+ }
+ s := 1 / (x * x)
+ var R, S float64
+ if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
+ R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
+ S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
+ } else { // |x| >= 1 / 0.35 ~ 2.857143
+ R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
+ S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
+ }
+ z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
+ r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
+ if sign {
+ return r/x - 1
+ }
+ return 1 - r/x
+}
+
+// Erfc returns the complementary error function of x.
+//
+// Special cases are:
+// Erfc(+Inf) = 0
+// Erfc(-Inf) = 2
+// Erfc(NaN) = NaN
+func Erfc(x float64) float64 {
+ const Tiny = 1.0 / (1 << 56) // 2**-56
+ // special cases
+ switch {
+ case IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ case IsInf(x, -1):
+ return 2
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x < 0.84375 { // |x| < 0.84375
+ var temp float64
+ if x < Tiny { // |x| < 2**-56
+ temp = x
+ } else {
+ z := x * x
+ r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
+ s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
+ y := r / s
+ if x < 0.25 { // |x| < 1/4
+ temp = x + x*y
+ } else {
+ temp = 0.5 + (x*y + (x - 0.5))
+ }
+ }
+ if sign {
+ return 1 + temp
+ }
+ return 1 - temp
+ }
+ if x < 1.25 { // 0.84375 <= |x| < 1.25
+ s := x - 1
+ P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
+ Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
+ if sign {
+ return 1 + erx + P/Q
+ }
+ return 1 - erx - P/Q
+
+ }
+ if x < 28 { // |x| < 28
+ s := 1 / (x * x)
+ var R, S float64
+ if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
+ R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
+ S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
+ } else { // |x| >= 1 / 0.35 ~ 2.857143
+ if sign && x > 6 {
+ return 2 // x < -6
+ }
+ R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
+ S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
+ }
+ z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
+ r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
+ if sign {
+ return 2 - r/x
+ }
+ return r / x
+ }
+ if sign {
+ return 2
+ }
+ return 0
+}
diff --git a/src/math/exp.go b/src/math/exp.go
new file mode 100644
index 0000000..f31585f
--- /dev/null
+++ b/src/math/exp.go
@@ -0,0 +1,191 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Exp returns e**x, the base-e exponential of x.
+//
+// Special cases are:
+// Exp(+Inf) = +Inf
+// Exp(NaN) = NaN
+// Very large values overflow to 0 or +Inf.
+// Very small values underflow to 1.
+func Exp(x float64) float64
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+//
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// exp(x)
+// Returns the exponential of x.
+//
+// Method
+// 1. Argument reduction:
+// Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+// Given x, find r and integer k such that
+//
+// x = k*ln2 + r, |r| <= 0.5*ln2.
+//
+// Here r will be represented as r = hi-lo for better
+// accuracy.
+//
+// 2. Approximation of exp(r) by a special rational function on
+// the interval [0,0.34658]:
+// Write
+// R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
+// We use a special Remes algorithm on [0,0.34658] to generate
+// a polynomial of degree 5 to approximate R. The maximum error
+// of this polynomial approximation is bounded by 2**-59. In
+// other words,
+// R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+// (where z=r*r, and the values of P1 to P5 are listed below)
+// and
+// | 5 | -59
+// | 2.0+P1*z+...+P5*z - R(z) | <= 2
+// | |
+// The computation of exp(r) thus becomes
+// 2*r
+// exp(r) = 1 + -------
+// R - r
+// r*R1(r)
+// = 1 + r + ----------- (for better accuracy)
+// 2 - R1(r)
+// where
+// 2 4 10
+// R1(r) = r - (P1*r + P2*r + ... + P5*r ).
+//
+// 3. Scale back to obtain exp(x):
+// From step 1, we have
+// exp(x) = 2**k * exp(r)
+//
+// Special cases:
+// exp(INF) is INF, exp(NaN) is NaN;
+// exp(-INF) is 0, and
+// for finite argument, only exp(0)=1 is exact.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Misc. info.
+// For IEEE double
+// if x > 7.09782712893383973096e+02 then exp(x) overflow
+// if x < -7.45133219101941108420e+02 then exp(x) underflow
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+
+func exp(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01
+ Ln2Lo = 1.90821492927058770002e-10
+ Log2e = 1.44269504088896338700e+00
+
+ Overflow = 7.09782712893383973096e+02
+ Underflow = -7.45133219101941108420e+02
+ NearZero = 1.0 / (1 << 28) // 2**-28
+ )
+
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 1):
+ return x
+ case IsInf(x, -1):
+ return 0
+ case x > Overflow:
+ return Inf(1)
+ case x < Underflow:
+ return 0
+ case -NearZero < x && x < NearZero:
+ return 1 + x
+ }
+
+ // reduce; computed as r = hi - lo for extra precision.
+ var k int
+ switch {
+ case x < 0:
+ k = int(Log2e*x - 0.5)
+ case x > 0:
+ k = int(Log2e*x + 0.5)
+ }
+ hi := x - float64(k)*Ln2Hi
+ lo := float64(k) * Ln2Lo
+
+ // compute
+ return expmulti(hi, lo, k)
+}
+
+// Exp2 returns 2**x, the base-2 exponential of x.
+//
+// Special cases are the same as Exp.
+func Exp2(x float64) float64
+
+func exp2(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01
+ Ln2Lo = 1.90821492927058770002e-10
+
+ Overflow = 1.0239999999999999e+03
+ Underflow = -1.0740e+03
+ )
+
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 1):
+ return x
+ case IsInf(x, -1):
+ return 0
+ case x > Overflow:
+ return Inf(1)
+ case x < Underflow:
+ return 0
+ }
+
+ // argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
+ // computed as r = hi - lo for extra precision.
+ var k int
+ switch {
+ case x > 0:
+ k = int(x + 0.5)
+ case x < 0:
+ k = int(x - 0.5)
+ }
+ t := x - float64(k)
+ hi := t * Ln2Hi
+ lo := -t * Ln2Lo
+
+ // compute
+ return expmulti(hi, lo, k)
+}
+
+// exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
+func expmulti(hi, lo float64, k int) float64 {
+ const (
+ P1 = 1.66666666666666019037e-01 /* 0x3FC55555; 0x5555553E */
+ P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
+ P3 = 6.61375632143793436117e-05 /* 0x3F11566A; 0xAF25DE2C */
+ P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
+ P5 = 4.13813679705723846039e-08 /* 0x3E663769; 0x72BEA4D0 */
+ )
+
+ r := hi - lo
+ t := r * r
+ c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
+ y := 1 - ((lo - (r*c)/(2-c)) - hi)
+ // TODO(rsc): make sure Ldexp can handle boundary k
+ return Ldexp(y, k)
+}
diff --git a/src/math/exp2_386.s b/src/math/exp2_386.s
new file mode 100644
index 0000000..7d11920
--- /dev/null
+++ b/src/math/exp2_386.s
@@ -0,0 +1,40 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Exp2(x float64) float64
+TEXT ·Exp2(SB),NOSPLIT,$0
+// test bits for not-finite
+ MOVL x_hi+4(FP), AX
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FMOVD x+0(FP), F0 // F0=x
+ FMOVD F0, F1 // F0=x, F1=x
+ FRNDINT // F0=int(x), F1=x
+ FSUBD F0, F1 // F0=int(x), F1=x-int(x)
+ FXCHD F0, F1 // F0=x-int(x), F1=int(x)
+ F2XM1 // F0=2**(x-int(x))-1, F1=int(x)
+ FLD1 // F0=1, F1=2**(x-int(x))-1, F2=int(x)
+ FADDDP F0, F1 // F0=2**(x-int(x)), F1=int(x)
+ FSCALE // F0=2**x, F1=int(x)
+ FMOVDP F0, F1 // F0=2**x
+ FMOVDP F0, ret+8(FP)
+ RET
+not_finite:
+// test bits for -Inf
+ MOVL x_hi+4(FP), BX
+ MOVL x_lo+0(FP), CX
+ CMPL BX, $0xfff00000
+ JNE not_neginf
+ CMPL CX, $0
+ JNE not_neginf
+ MOVL $0, ret_lo+8(FP)
+ MOVL $0, ret_hi+12(FP)
+ RET
+not_neginf:
+ MOVL CX, ret_lo+8(FP)
+ MOVL BX, ret_hi+12(FP)
+ RET
diff --git a/src/math/exp2_amd64.s b/src/math/exp2_amd64.s
new file mode 100644
index 0000000..903c835
--- /dev/null
+++ b/src/math/exp2_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Exp2(SB),NOSPLIT,$0
+ JMP ·exp2(SB)
diff --git a/src/math/exp2_amd64p32.s b/src/math/exp2_amd64p32.s
new file mode 100644
index 0000000..4d38309
--- /dev/null
+++ b/src/math/exp2_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "exp2_amd64.s"
diff --git a/src/math/exp2_arm.s b/src/math/exp2_arm.s
new file mode 100644
index 0000000..58283cd
--- /dev/null
+++ b/src/math/exp2_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Exp2(SB),NOSPLIT,$0
+ B ·exp2(SB)
diff --git a/src/math/exp_386.s b/src/math/exp_386.s
new file mode 100644
index 0000000..6a478a5
--- /dev/null
+++ b/src/math/exp_386.s
@@ -0,0 +1,41 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Exp(x float64) float64
+TEXT ·Exp(SB),NOSPLIT,$0
+// test bits for not-finite
+ MOVL x_hi+4(FP), AX
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FLDL2E // F0=log2(e)
+ FMULD x+0(FP), F0 // F0=x*log2(e)
+ FMOVD F0, F1 // F0=x*log2(e), F1=x*log2(e)
+ FRNDINT // F0=int(x*log2(e)), F1=x*log2(e)
+ FSUBD F0, F1 // F0=int(x*log2(e)), F1=x*log2(e)-int(x*log2(e))
+ FXCHD F0, F1 // F0=x*log2(e)-int(x*log2(e)), F1=int(x*log2(e))
+ F2XM1 // F0=2**(x*log2(e)-int(x*log2(e)))-1, F1=int(x*log2(e))
+ FLD1 // F0=1, F1=2**(x*log2(e)-int(x*log2(e)))-1, F2=int(x*log2(e))
+ FADDDP F0, F1 // F0=2**(x*log2(e)-int(x*log2(e))), F1=int(x*log2(e))
+ FSCALE // F0=e**x, F1=int(x*log2(e))
+ FMOVDP F0, F1 // F0=e**x
+ FMOVDP F0, ret+8(FP)
+ RET
+not_finite:
+// test bits for -Inf
+ MOVL x_hi+4(FP), BX
+ MOVL x_lo+0(FP), CX
+ CMPL BX, $0xfff00000
+ JNE not_neginf
+ CMPL CX, $0
+ JNE not_neginf
+ FLDZ // F0=0
+ FMOVDP F0, ret+8(FP)
+ RET
+not_neginf:
+ MOVL CX, ret_lo+8(FP)
+ MOVL BX, ret_hi+12(FP)
+ RET
diff --git a/src/math/exp_amd64.s b/src/math/exp_amd64.s
new file mode 100644
index 0000000..d9cf8fd
--- /dev/null
+++ b/src/math/exp_amd64.s
@@ -0,0 +1,114 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// The method is based on a paper by Naoki Shibata: "Efficient evaluation
+// methods of elementary functions suitable for SIMD computation", Proc.
+// of International Supercomputing Conference 2010 (ISC'10), pp. 25 -- 32
+// (May 2010). The paper is available at
+// http://www.springerlink.com/content/340228x165742104/
+//
+// The original code and the constants below are from the author's
+// implementation available at http://freshmeat.net/projects/sleef.
+// The README file says, "The software is in public domain.
+// You can use the software without any obligation."
+//
+// This code is a simplified version of the original.
+
+#define LN2 0.6931471805599453094172321214581766 // log_e(2)
+#define LOG2E 1.4426950408889634073599246810018920 // 1/LN2
+#define LN2U 0.69314718055966295651160180568695068359375 // upper half LN2
+#define LN2L 0.28235290563031577122588448175013436025525412068e-12 // lower half LN2
+#define T0 1.0
+#define T1 0.5
+#define T2 1.6666666666666666667e-1
+#define T3 4.1666666666666666667e-2
+#define T4 8.3333333333333333333e-3
+#define T5 1.3888888888888888889e-3
+#define T6 1.9841269841269841270e-4
+#define T7 2.4801587301587301587e-5
+#define PosInf 0x7FF0000000000000
+#define NegInf 0xFFF0000000000000
+
+// func Exp(x float64) float64
+TEXT ·Exp(SB),NOSPLIT,$0
+// test bits for not-finite
+ MOVQ x+0(FP), BX
+ MOVQ $~(1<<63), AX // sign bit mask
+ MOVQ BX, DX
+ ANDQ AX, DX
+ MOVQ $PosInf, AX
+ CMPQ AX, DX
+ JLE notFinite
+ MOVQ BX, X0
+ MOVSD $LOG2E, X1
+ MULSD X0, X1
+ CVTSD2SL X1, BX // BX = exponent
+ CVTSL2SD BX, X1
+ MOVSD $LN2U, X2
+ MULSD X1, X2
+ SUBSD X2, X0
+ MOVSD $LN2L, X2
+ MULSD X1, X2
+ SUBSD X2, X0
+ // reduce argument
+ MULSD $0.0625, X0
+ // Taylor series evaluation
+ MOVSD $T7, X1
+ MULSD X0, X1
+ ADDSD $T6, X1
+ MULSD X0, X1
+ ADDSD $T5, X1
+ MULSD X0, X1
+ ADDSD $T4, X1
+ MULSD X0, X1
+ ADDSD $T3, X1
+ MULSD X0, X1
+ ADDSD $T2, X1
+ MULSD X0, X1
+ ADDSD $T1, X1
+ MULSD X0, X1
+ ADDSD $T0, X1
+ MULSD X1, X0
+ MOVSD $2.0, X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ MOVSD $2.0, X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ MOVSD $2.0, X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ MOVSD $2.0, X1
+ ADDSD X0, X1
+ MULSD X1, X0
+ ADDSD $1.0, X0
+ // return fr * 2**exponent
+ MOVL $0x3FF, AX // bias
+ ADDL AX, BX
+ JLE underflow
+ CMPL BX, $0x7FF
+ JGE overflow
+ MOVL $52, CX
+ SHLQ CX, BX
+ MOVQ BX, X1
+ MULSD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
+notFinite:
+ // test bits for -Inf
+ MOVQ $NegInf, AX
+ CMPQ AX, BX
+ JNE notNegInf
+ // -Inf, return 0
+underflow: // return 0
+ MOVQ $0, AX
+ MOVQ AX, ret+8(FP)
+ RET
+overflow: // return +Inf
+ MOVQ $PosInf, BX
+notNegInf: // NaN or +Inf, return x
+ MOVQ BX, ret+8(FP)
+ RET
diff --git a/src/math/exp_amd64p32.s b/src/math/exp_amd64p32.s
new file mode 100644
index 0000000..98ac2e9
--- /dev/null
+++ b/src/math/exp_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "exp_amd64.s"
diff --git a/src/math/exp_arm.s b/src/math/exp_arm.s
new file mode 100644
index 0000000..ce36d03
--- /dev/null
+++ b/src/math/exp_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Exp(SB),NOSPLIT,$0
+ B ·exp(SB)
diff --git a/src/math/expm1.go b/src/math/expm1.go
new file mode 100644
index 0000000..8f56e15
--- /dev/null
+++ b/src/math/expm1.go
@@ -0,0 +1,237 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/s_expm1.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// expm1(x)
+// Returns exp(x)-1, the exponential of x minus 1.
+//
+// Method
+// 1. Argument reduction:
+// Given x, find r and integer k such that
+//
+// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
+//
+// Here a correction term c will be computed to compensate
+// the error in r when rounded to a floating-point number.
+//
+// 2. Approximating expm1(r) by a special rational function on
+// the interval [0,0.34658]:
+// Since
+// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ...
+// we define R1(r*r) by
+// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r)
+// That is,
+// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+// = 1 - r**2/60 + r**4/2520 - r**6/100800 + ...
+// We use a special Reme algorithm on [0,0.347] to generate
+// a polynomial of degree 5 in r*r to approximate R1. The
+// maximum error of this polynomial approximation is bounded
+// by 2**-61. In other words,
+// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+// where Q1 = -1.6666666666666567384E-2,
+// Q2 = 3.9682539681370365873E-4,
+// Q3 = -9.9206344733435987357E-6,
+// Q4 = 2.5051361420808517002E-7,
+// Q5 = -6.2843505682382617102E-9;
+// (where z=r*r, and the values of Q1 to Q5 are listed below)
+// with error bounded by
+// | 5 | -61
+// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
+// | |
+//
+// expm1(r) = exp(r)-1 is then computed by the following
+// specific way which minimize the accumulation rounding error:
+// 2 3
+// r r [ 3 - (R1 + R1*r/2) ]
+// expm1(r) = r + --- + --- * [--------------------]
+// 2 2 [ 6 - r*(3 - R1*r/2) ]
+//
+// To compensate the error in the argument reduction, we use
+// expm1(r+c) = expm1(r) + c + expm1(r)*c
+// ~ expm1(r) + c + r*c
+// Thus c+r*c will be added in as the correction terms for
+// expm1(r+c). Now rearrange the term to avoid optimization
+// screw up:
+// ( 2 2 )
+// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
+// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
+// ( )
+//
+// = r - E
+// 3. Scale back to obtain expm1(x):
+// From step 1, we have
+// expm1(x) = either 2**k*[expm1(r)+1] - 1
+// = or 2**k*[expm1(r) + (1-2**-k)]
+// 4. Implementation notes:
+// (A). To save one multiplication, we scale the coefficient Qi
+// to Qi*2**i, and replace z by (x**2)/2.
+// (B). To achieve maximum accuracy, we compute expm1(x) by
+// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
+// (ii) if k=0, return r-E
+// (iii) if k=-1, return 0.5*(r-E)-0.5
+// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
+// else return 1.0+2.0*(r-E);
+// (v) if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1)
+// (vi) if k <= 20, return 2**k((1-2**-k)-(E-r)), else
+// (vii) return 2**k(1-((E+2**-k)-r))
+//
+// Special cases:
+// expm1(INF) is INF, expm1(NaN) is NaN;
+// expm1(-INF) is -1, and
+// for finite argument, only expm1(0)=0 is exact.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Misc. info.
+// For IEEE double
+// if x > 7.09782712893383973096e+02 then expm1(x) overflow
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+//
+
+// Expm1 returns e**x - 1, the base-e exponential of x minus 1.
+// It is more accurate than Exp(x) - 1 when x is near zero.
+//
+// Special cases are:
+// Expm1(+Inf) = +Inf
+// Expm1(-Inf) = -1
+// Expm1(NaN) = NaN
+// Very large values overflow to -1 or +Inf.
+func Expm1(x float64) float64
+
+func expm1(x float64) float64 {
+ const (
+ Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
+ Ln2X56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
+ Ln2HalfX3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
+ Ln2Half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
+ Ln2Hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
+ Ln2Lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
+ InvLn2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
+ Tiny = 1.0 / (1 << 54) // 2**-54 = 0x3c90000000000000
+ // scaled coefficients related to expm1
+ Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
+ Q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
+ Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
+ Q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
+ Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
+ )
+
+ // special cases
+ switch {
+ case IsInf(x, 1) || IsNaN(x):
+ return x
+ case IsInf(x, -1):
+ return -1
+ }
+
+ absx := x
+ sign := false
+ if x < 0 {
+ absx = -absx
+ sign = true
+ }
+
+ // filter out huge argument
+ if absx >= Ln2X56 { // if |x| >= 56 * ln2
+ if absx >= Othreshold { // if |x| >= 709.78...
+ return Inf(1) // overflow
+ }
+ if sign {
+ return -1 // x < -56*ln2, return -1.0
+ }
+ }
+
+ // argument reduction
+ var c float64
+ var k int
+ if absx > Ln2Half { // if |x| > 0.5 * ln2
+ var hi, lo float64
+ if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2
+ if !sign {
+ hi = x - Ln2Hi
+ lo = Ln2Lo
+ k = 1
+ } else {
+ hi = x + Ln2Hi
+ lo = -Ln2Lo
+ k = -1
+ }
+ } else {
+ if !sign {
+ k = int(InvLn2*x + 0.5)
+ } else {
+ k = int(InvLn2*x - 0.5)
+ }
+ t := float64(k)
+ hi = x - t*Ln2Hi // t * Ln2Hi is exact here
+ lo = t * Ln2Lo
+ }
+ x = hi - lo
+ c = (hi - x) - lo
+ } else if absx < Tiny { // when |x| < 2**-54, return x
+ return x
+ } else {
+ k = 0
+ }
+
+ // x is now in primary range
+ hfx := 0.5 * x
+ hxs := x * hfx
+ r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))))
+ t := 3 - r1*hfx
+ e := hxs * ((r1 - t) / (6.0 - x*t))
+ if k != 0 {
+ e = (x*(e-c) - c)
+ e -= hxs
+ switch {
+ case k == -1:
+ return 0.5*(x-e) - 0.5
+ case k == 1:
+ if x < -0.25 {
+ return -2 * (e - (x + 0.5))
+ }
+ return 1 + 2*(x-e)
+ case k <= -2 || k > 56: // suffice to return exp(x)-1
+ y := 1 - (e - x)
+ y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
+ return y - 1
+ }
+ if k < 20 {
+ t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k
+ y := t - (e - x)
+ y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
+ return y
+ }
+ t := Float64frombits(uint64((0x3ff - k) << 52)) // 2**-k
+ y := x - (e + t)
+ y += 1
+ y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
+ return y
+ }
+ return x - (x*e - hxs) // c is 0
+}
diff --git a/src/math/expm1_386.s b/src/math/expm1_386.s
new file mode 100644
index 0000000..a48ca8a
--- /dev/null
+++ b/src/math/expm1_386.s
@@ -0,0 +1,57 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Expm1(x float64) float64
+TEXT ·Expm1(SB),NOSPLIT,$0
+ FLDLN2 // F0=log(2) = 1/log2(e) ~ 0.693147
+ FMOVD x+0(FP), F0 // F0=x, F1=1/log2(e)
+ FABS // F0=|x|, F1=1/log2(e)
+ FUCOMPP F0, F1 // compare F0 to F1
+ FSTSW AX
+ SAHF
+ JCC use_exp // jump if F0 >= F1
+ FLDL2E // F0=log2(e)
+ FMULD x+0(FP), F0 // F0=x*log2(e) (-1<F0<1)
+ F2XM1 // F0=e**x-1 = 2**(x*log2(e))-1
+ FMOVDP F0, ret+8(FP)
+ RET
+use_exp:
+// test bits for not-finite
+ MOVL x_hi+4(FP), AX
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FLDL2E // F0=log2(e)
+ FMULD x+0(FP), F0 // F0=x*log2(e)
+ FMOVD F0, F1 // F0=x*log2(e), F1=x*log2(e)
+ FRNDINT // F0=int(x*log2(e)), F1=x*log2(e)
+ FSUBD F0, F1 // F0=int(x*log2(e)), F1=x*log2(e)-int(x*log2(e))
+ FXCHD F0, F1 // F0=x*log2(e)-int(x*log2(e)), F1=int(x*log2(e))
+ F2XM1 // F0=2**(x*log2(e)-int(x*log2(e)))-1, F1=int(x*log2(e))
+ FLD1 // F0=1, F1=2**(x*log2(e)-int(x*log2(e)))-1, F2=int(x*log2(e))
+ FADDDP F0, F1 // F0=2**(x*log2(e)-int(x*log2(e))), F1=int(x*log2(e))
+ FSCALE // F0=e**x, F1=int(x*log2(e))
+ FMOVDP F0, F1 // F0=e**x
+ FLD1 // F0=1, F1=e**x
+ FSUBDP F0, F1 // F0=e**x-1
+ FMOVDP F0, ret+8(FP)
+ RET
+not_finite:
+// test bits for -Inf
+ MOVL x_hi+4(FP), BX
+ MOVL x_lo+0(FP), CX
+ CMPL BX, $0xfff00000
+ JNE not_neginf
+ CMPL CX, $0
+ JNE not_neginf
+ FLD1 // F0=1
+ FCHS // F0=-1
+ FMOVDP F0, ret+8(FP)
+ RET
+not_neginf:
+ MOVL CX, ret_lo+8(FP)
+ MOVL BX, ret_hi+12(FP)
+ RET
diff --git a/src/math/expm1_amd64.s b/src/math/expm1_amd64.s
new file mode 100644
index 0000000..b7d5a3b
--- /dev/null
+++ b/src/math/expm1_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Expm1(SB),NOSPLIT,$0
+ JMP ·expm1(SB)
diff --git a/src/math/expm1_amd64p32.s b/src/math/expm1_amd64p32.s
new file mode 100644
index 0000000..709ebef
--- /dev/null
+++ b/src/math/expm1_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "expm1_amd64.s"
diff --git a/src/math/expm1_arm.s b/src/math/expm1_arm.s
new file mode 100644
index 0000000..5f80d87
--- /dev/null
+++ b/src/math/expm1_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Expm1(SB),NOSPLIT,$0
+ B ·expm1(SB)
diff --git a/src/math/export_test.go b/src/math/export_test.go
new file mode 100644
index 0000000..02992d7
--- /dev/null
+++ b/src/math/export_test.go
@@ -0,0 +1,11 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Export internal functions for testing.
+var ExpGo = exp
+var Exp2Go = exp2
+var HypotGo = hypot
+var SqrtGo = sqrt
diff --git a/src/math/floor.go b/src/math/floor.go
new file mode 100644
index 0000000..9d30629
--- /dev/null
+++ b/src/math/floor.go
@@ -0,0 +1,56 @@
+// Copyright 2009-2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Floor returns the greatest integer value less than or equal to x.
+//
+// Special cases are:
+// Floor(±0) = ±0
+// Floor(±Inf) = ±Inf
+// Floor(NaN) = NaN
+func Floor(x float64) float64
+
+func floor(x float64) float64 {
+ if x == 0 || IsNaN(x) || IsInf(x, 0) {
+ return x
+ }
+ if x < 0 {
+ d, fract := Modf(-x)
+ if fract != 0.0 {
+ d = d + 1
+ }
+ return -d
+ }
+ d, _ := Modf(x)
+ return d
+}
+
+// Ceil returns the least integer value greater than or equal to x.
+//
+// Special cases are:
+// Ceil(±0) = ±0
+// Ceil(±Inf) = ±Inf
+// Ceil(NaN) = NaN
+func Ceil(x float64) float64
+
+func ceil(x float64) float64 {
+ return -Floor(-x)
+}
+
+// Trunc returns the integer value of x.
+//
+// Special cases are:
+// Trunc(±0) = ±0
+// Trunc(±Inf) = ±Inf
+// Trunc(NaN) = NaN
+func Trunc(x float64) float64
+
+func trunc(x float64) float64 {
+ if x == 0 || IsNaN(x) || IsInf(x, 0) {
+ return x
+ }
+ d, _ := Modf(x)
+ return d
+}
diff --git a/src/math/floor_386.s b/src/math/floor_386.s
new file mode 100644
index 0000000..31c9b17
--- /dev/null
+++ b/src/math/floor_386.s
@@ -0,0 +1,46 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Ceil(x float64) float64
+TEXT ·Ceil(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ANDW $0xf3ff, AX
+ ORW $0x0800, AX // Rounding Control set to +Inf
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=Ceil(x)
+ FLDCW -2(SP) // load old Control Word
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Floor(x float64) float64
+TEXT ·Floor(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ANDW $0xf3ff, AX
+ ORW $0x0400, AX // Rounding Control set to -Inf
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=Floor(x)
+ FLDCW -2(SP) // load old Control Word
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Trunc(x float64) float64
+TEXT ·Trunc(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ORW $0x0c00, AX // Rounding Control set to truncate
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=Trunc(x)
+ FLDCW -2(SP) // load old Control Word
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/floor_amd64.s b/src/math/floor_amd64.s
new file mode 100644
index 0000000..67b7cde
--- /dev/null
+++ b/src/math/floor_amd64.s
@@ -0,0 +1,76 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define Big 0x4330000000000000 // 2**52
+
+// func Floor(x float64) float64
+TEXT ·Floor(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MOVQ $~(1<<63), DX // sign bit mask
+ ANDQ AX,DX // DX = |x|
+ SUBQ $1,DX
+ MOVQ $(Big - 1), CX // if |x| >= 2**52-1 or IsNaN(x) or |x| == 0, return x
+ CMPQ DX,CX
+ JAE isBig_floor
+ MOVQ AX, X0 // X0 = x
+ CVTTSD2SQ X0, AX
+ CVTSQ2SD AX, X1 // X1 = float(int(x))
+ CMPSD X1, X0, 1 // compare LT; X0 = 0xffffffffffffffff or 0
+ MOVSD $(-1.0), X2
+ ANDPD X2, X0 // if x < float(int(x)) {X0 = -1} else {X0 = 0}
+ ADDSD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
+isBig_floor:
+ MOVQ AX, ret+8(FP) // return x
+ RET
+
+// func Ceil(x float64) float64
+TEXT ·Ceil(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MOVQ $~(1<<63), DX // sign bit mask
+ MOVQ AX, BX // BX = copy of x
+ ANDQ DX, BX // BX = |x|
+ MOVQ $Big, CX // if |x| >= 2**52 or IsNaN(x), return x
+ CMPQ BX, CX
+ JAE isBig_ceil
+ MOVQ AX, X0 // X0 = x
+ MOVQ DX, X2 // X2 = sign bit mask
+ CVTTSD2SQ X0, AX
+ ANDNPD X0, X2 // X2 = sign
+ CVTSQ2SD AX, X1 // X1 = float(int(x))
+ CMPSD X1, X0, 2 // compare LE; X0 = 0xffffffffffffffff or 0
+ ORPD X2, X1 // if X1 = 0.0, incorporate sign
+ MOVSD $1.0, X3
+ ANDNPD X3, X0
+ ORPD X2, X0 // if float(int(x)) <= x {X0 = 1} else {X0 = -0}
+ ADDSD X1, X0
+ MOVSD X0, ret+8(FP)
+ RET
+isBig_ceil:
+ MOVQ AX, ret+8(FP)
+ RET
+
+// func Trunc(x float64) float64
+TEXT ·Trunc(SB),NOSPLIT,$0
+ MOVQ x+0(FP), AX
+ MOVQ $~(1<<63), DX // sign bit mask
+ MOVQ AX, BX // BX = copy of x
+ ANDQ DX, BX // BX = |x|
+ MOVQ $Big, CX // if |x| >= 2**52 or IsNaN(x), return x
+ CMPQ BX, CX
+ JAE isBig_trunc
+ MOVQ AX, X0
+ MOVQ DX, X2 // X2 = sign bit mask
+ CVTTSD2SQ X0, AX
+ ANDNPD X0, X2 // X2 = sign
+ CVTSQ2SD AX, X0 // X0 = float(int(x))
+ ORPD X2, X0 // if X0 = 0.0, incorporate sign
+ MOVSD X0, ret+8(FP)
+ RET
+isBig_trunc:
+ MOVQ AX, ret+8(FP) // return x
+ RET
diff --git a/src/math/floor_amd64p32.s b/src/math/floor_amd64p32.s
new file mode 100644
index 0000000..5b87d7a
--- /dev/null
+++ b/src/math/floor_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "floor_amd64.s"
diff --git a/src/math/floor_arm.s b/src/math/floor_arm.s
new file mode 100644
index 0000000..5909176
--- /dev/null
+++ b/src/math/floor_arm.s
@@ -0,0 +1,14 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Floor(SB),NOSPLIT,$0
+ B ·floor(SB)
+
+TEXT ·Ceil(SB),NOSPLIT,$0
+ B ·ceil(SB)
+
+TEXT ·Trunc(SB),NOSPLIT,$0
+ B ·trunc(SB)
diff --git a/src/math/frexp.go b/src/math/frexp.go
new file mode 100644
index 0000000..0e26feb
--- /dev/null
+++ b/src/math/frexp.go
@@ -0,0 +1,33 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Frexp breaks f into a normalized fraction
+// and an integral power of two.
+// It returns frac and exp satisfying f == frac × 2**exp,
+// with the absolute value of frac in the interval [½, 1).
+//
+// Special cases are:
+// Frexp(±0) = ±0, 0
+// Frexp(±Inf) = ±Inf, 0
+// Frexp(NaN) = NaN, 0
+func Frexp(f float64) (frac float64, exp int)
+
+func frexp(f float64) (frac float64, exp int) {
+ // special cases
+ switch {
+ case f == 0:
+ return f, 0 // correctly return -0
+ case IsInf(f, 0) || IsNaN(f):
+ return f, 0
+ }
+ f, exp = normalize(f)
+ x := Float64bits(f)
+ exp += int((x>>shift)&mask) - bias + 1
+ x &^= mask << shift
+ x |= (-1 + bias) << shift
+ frac = Float64frombits(x)
+ return
+}
diff --git a/src/math/frexp_386.s b/src/math/frexp_386.s
new file mode 100644
index 0000000..5bff7e2
--- /dev/null
+++ b/src/math/frexp_386.s
@@ -0,0 +1,25 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Frexp(f float64) (frac float64, exp int)
+TEXT ·Frexp(SB),NOSPLIT,$0
+ FMOVD f+0(FP), F0 // F0=f
+ FXAM
+ FSTSW AX
+ SAHF
+ JNP nan_zero_inf
+ JCS nan_zero_inf
+ FXTRACT // F0=f (0<=f<1), F1=e
+ FMULD $(0.5), F0 // F0=f (0.5<=f<1), F1=e
+ FMOVDP F0, frac+8(FP) // F0=e
+ FLD1 // F0=1, F1=e
+ FADDDP F0, F1 // F0=e+1
+ FMOVLP F0, exp+16(FP) // (int=int32)
+ RET
+nan_zero_inf:
+ FMOVDP F0, frac+8(FP) // F0=e
+ MOVL $0, exp+16(FP) // exp=0
+ RET
diff --git a/src/math/frexp_amd64.s b/src/math/frexp_amd64.s
new file mode 100644
index 0000000..93a3210
--- /dev/null
+++ b/src/math/frexp_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Frexp(SB),NOSPLIT,$0
+ JMP ·frexp(SB)
diff --git a/src/math/frexp_amd64p32.s b/src/math/frexp_amd64p32.s
new file mode 100644
index 0000000..fbb5645
--- /dev/null
+++ b/src/math/frexp_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "frexp_amd64.s"
diff --git a/src/math/frexp_arm.s b/src/math/frexp_arm.s
new file mode 100644
index 0000000..7842eca
--- /dev/null
+++ b/src/math/frexp_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Frexp(SB),NOSPLIT,$0
+ B ·frexp(SB)
diff --git a/src/math/gamma.go b/src/math/gamma.go
new file mode 100644
index 0000000..164f54f
--- /dev/null
+++ b/src/math/gamma.go
@@ -0,0 +1,202 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/cprob/gamma.c.
+// The go code is a simplified version of the original C.
+//
+// tgamma.c
+//
+// Gamma function
+//
+// SYNOPSIS:
+//
+// double x, y, tgamma();
+// extern int signgam;
+//
+// y = tgamma( x );
+//
+// DESCRIPTION:
+//
+// Returns gamma function of the argument. The result is
+// correctly signed, and the sign (+1 or -1) is also
+// returned in a global (extern) variable named signgam.
+// This variable is also filled in by the logarithmic gamma
+// function lgamma().
+//
+// Arguments |x| <= 34 are reduced by recurrence and the function
+// approximated by a rational function of degree 6/7 in the
+// interval (2,3). Large arguments are handled by Stirling's
+// formula. Large negative arguments are made positive using
+// a reflection formula.
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -34, 34 10000 1.3e-16 2.5e-17
+// IEEE -170,-33 20000 2.3e-15 3.3e-16
+// IEEE -33, 33 20000 9.4e-16 2.2e-16
+// IEEE 33, 171.6 20000 2.3e-15 3.2e-16
+//
+// Error for arguments outside the test range will be larger
+// owing to error amplification by the exponential function.
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+var _gamP = [...]float64{
+ 1.60119522476751861407e-04,
+ 1.19135147006586384913e-03,
+ 1.04213797561761569935e-02,
+ 4.76367800457137231464e-02,
+ 2.07448227648435975150e-01,
+ 4.94214826801497100753e-01,
+ 9.99999999999999996796e-01,
+}
+var _gamQ = [...]float64{
+ -2.31581873324120129819e-05,
+ 5.39605580493303397842e-04,
+ -4.45641913851797240494e-03,
+ 1.18139785222060435552e-02,
+ 3.58236398605498653373e-02,
+ -2.34591795718243348568e-01,
+ 7.14304917030273074085e-02,
+ 1.00000000000000000320e+00,
+}
+var _gamS = [...]float64{
+ 7.87311395793093628397e-04,
+ -2.29549961613378126380e-04,
+ -2.68132617805781232825e-03,
+ 3.47222221605458667310e-03,
+ 8.33333333333482257126e-02,
+}
+
+// Gamma function computed by Stirling's formula.
+// The polynomial is valid for 33 <= x <= 172.
+func stirling(x float64) float64 {
+ const (
+ SqrtTwoPi = 2.506628274631000502417
+ MaxStirling = 143.01608
+ )
+ w := 1 / x
+ w = 1 + w*((((_gamS[0]*w+_gamS[1])*w+_gamS[2])*w+_gamS[3])*w+_gamS[4])
+ y := Exp(x)
+ if x > MaxStirling { // avoid Pow() overflow
+ v := Pow(x, 0.5*x-0.25)
+ y = v * (v / y)
+ } else {
+ y = Pow(x, x-0.5) / y
+ }
+ y = SqrtTwoPi * y * w
+ return y
+}
+
+// Gamma returns the Gamma function of x.
+//
+// Special cases are:
+// Gamma(+Inf) = +Inf
+// Gamma(+0) = +Inf
+// Gamma(-0) = -Inf
+// Gamma(x) = NaN for integer x < 0
+// Gamma(-Inf) = NaN
+// Gamma(NaN) = NaN
+func Gamma(x float64) float64 {
+ const Euler = 0.57721566490153286060651209008240243104215933593992 // A001620
+ // special cases
+ switch {
+ case isNegInt(x) || IsInf(x, -1) || IsNaN(x):
+ return NaN()
+ case x == 0:
+ if Signbit(x) {
+ return Inf(-1)
+ }
+ return Inf(1)
+ case x < -170.5674972726612 || x > 171.61447887182298:
+ return Inf(1)
+ }
+ q := Abs(x)
+ p := Floor(q)
+ if q > 33 {
+ if x >= 0 {
+ return stirling(x)
+ }
+ signgam := 1
+ if ip := int(p); ip&1 == 0 {
+ signgam = -1
+ }
+ z := q - p
+ if z > 0.5 {
+ p = p + 1
+ z = q - p
+ }
+ z = q * Sin(Pi*z)
+ if z == 0 {
+ return Inf(signgam)
+ }
+ z = Pi / (Abs(z) * stirling(q))
+ return float64(signgam) * z
+ }
+
+ // Reduce argument
+ z := 1.0
+ for x >= 3 {
+ x = x - 1
+ z = z * x
+ }
+ for x < 0 {
+ if x > -1e-09 {
+ goto small
+ }
+ z = z / x
+ x = x + 1
+ }
+ for x < 2 {
+ if x < 1e-09 {
+ goto small
+ }
+ z = z / x
+ x = x + 1
+ }
+
+ if x == 2 {
+ return z
+ }
+
+ x = x - 2
+ p = (((((x*_gamP[0]+_gamP[1])*x+_gamP[2])*x+_gamP[3])*x+_gamP[4])*x+_gamP[5])*x + _gamP[6]
+ q = ((((((x*_gamQ[0]+_gamQ[1])*x+_gamQ[2])*x+_gamQ[3])*x+_gamQ[4])*x+_gamQ[5])*x+_gamQ[6])*x + _gamQ[7]
+ return z * p / q
+
+small:
+ if x == 0 {
+ return Inf(1)
+ }
+ return z / ((1 + Euler*x) * x)
+}
+
+func isNegInt(x float64) bool {
+ if x < 0 {
+ _, xf := Modf(x)
+ return xf == 0
+ }
+ return false
+}
diff --git a/src/math/hypot.go b/src/math/hypot.go
new file mode 100644
index 0000000..2087cb0
--- /dev/null
+++ b/src/math/hypot.go
@@ -0,0 +1,43 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Hypot -- sqrt(p*p + q*q), but overflows only if the result does.
+*/
+
+// Hypot returns Sqrt(p*p + q*q), taking care to avoid
+// unnecessary overflow and underflow.
+//
+// Special cases are:
+// Hypot(±Inf, q) = +Inf
+// Hypot(p, ±Inf) = +Inf
+// Hypot(NaN, q) = NaN
+// Hypot(p, NaN) = NaN
+func Hypot(p, q float64) float64
+
+func hypot(p, q float64) float64 {
+ // special cases
+ switch {
+ case IsInf(p, 0) || IsInf(q, 0):
+ return Inf(1)
+ case IsNaN(p) || IsNaN(q):
+ return NaN()
+ }
+ if p < 0 {
+ p = -p
+ }
+ if q < 0 {
+ q = -q
+ }
+ if p < q {
+ p, q = q, p
+ }
+ if p == 0 {
+ return 0
+ }
+ q = q / p
+ return p * Sqrt(1+q*q)
+}
diff --git a/src/math/hypot_386.s b/src/math/hypot_386.s
new file mode 100644
index 0000000..d321f46
--- /dev/null
+++ b/src/math/hypot_386.s
@@ -0,0 +1,59 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Hypot(p, q float64) float64
+TEXT ·Hypot(SB),NOSPLIT,$0
+// test bits for not-finite
+ MOVL p_hi+4(FP), AX // high word p
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ MOVL q_hi+12(FP), AX // high word q
+ ANDL $0x7ff00000, AX
+ CMPL AX, $0x7ff00000
+ JEQ not_finite
+ FMOVD p+0(FP), F0 // F0=p
+ FABS // F0=|p|
+ FMOVD q+8(FP), F0 // F0=q, F1=|p|
+ FABS // F0=|q|, F1=|p|
+ FUCOMI F0, F1 // compare F0 to F1
+ JCC 2(PC) // jump if F0 >= F1
+ FXCHD F0, F1 // F0=|p| (larger), F1=|q| (smaller)
+ FTST // compare F0 to 0
+ FSTSW AX
+ ANDW $0x4000, AX
+ JNE 10(PC) // jump if F0 = 0
+ FXCHD F0, F1 // F0=q (smaller), F1=p (larger)
+ FDIVD F1, F0 // F0=q(=q/p), F1=p
+ FMULD F0, F0 // F0=q*q, F1=p
+ FLD1 // F0=1, F1=q*q, F2=p
+ FADDDP F0, F1 // F0=1+q*q, F1=p
+ FSQRT // F0=sqrt(1+q*q), F1=p
+ FMULDP F0, F1 // F0=p*sqrt(1+q*q)
+ FMOVDP F0, ret+16(FP)
+ RET
+ FMOVDP F0, F1 // F0=0
+ FMOVDP F0, ret+16(FP)
+ RET
+not_finite:
+// test bits for -Inf or +Inf
+ MOVL p_hi+4(FP), AX // high word p
+ ORL p_lo+0(FP), AX // low word p
+ ANDL $0x7fffffff, AX
+ CMPL AX, $0x7ff00000
+ JEQ is_inf
+ MOVL q_hi+12(FP), AX // high word q
+ ORL q_lo+8(FP), AX // low word q
+ ANDL $0x7fffffff, AX
+ CMPL AX, $0x7ff00000
+ JEQ is_inf
+ MOVL $0x7ff80000, ret_hi+20(FP) // return NaN = 0x7FF8000000000001
+ MOVL $0x00000001, ret_lo+16(FP)
+ RET
+is_inf:
+ MOVL AX, ret_hi+20(FP) // return +Inf = 0x7FF0000000000000
+ MOVL $0x00000000, ret_lo+16(FP)
+ RET
diff --git a/src/math/hypot_amd64.s b/src/math/hypot_amd64.s
new file mode 100644
index 0000000..a68eebc
--- /dev/null
+++ b/src/math/hypot_amd64.s
@@ -0,0 +1,52 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+
+// func Hypot(p, q float64) float64
+TEXT ·Hypot(SB),NOSPLIT,$0
+ // test bits for special cases
+ MOVQ p+0(FP), BX
+ MOVQ $~(1<<63), AX
+ ANDQ AX, BX // p = |p|
+ MOVQ q+8(FP), CX
+ ANDQ AX, CX // q = |q|
+ MOVQ $PosInf, AX
+ CMPQ AX, BX
+ JLE isInfOrNaN
+ CMPQ AX, CX
+ JLE isInfOrNaN
+ // hypot = max * sqrt(1 + (min/max)**2)
+ MOVQ BX, X0
+ MOVQ CX, X1
+ ORQ CX, BX
+ JEQ isZero
+ MOVAPD X0, X2
+ MAXSD X1, X0
+ MINSD X2, X1
+ DIVSD X0, X1
+ MULSD X1, X1
+ ADDSD $1.0, X1
+ SQRTSD X1, X1
+ MULSD X1, X0
+ MOVSD X0, ret+16(FP)
+ RET
+isInfOrNaN:
+ CMPQ AX, BX
+ JEQ isInf
+ CMPQ AX, CX
+ JEQ isInf
+ MOVQ $NaN, AX
+ MOVQ AX, ret+16(FP) // return NaN
+ RET
+isInf:
+ MOVQ AX, ret+16(FP) // return +Inf
+ RET
+isZero:
+ MOVQ $0, AX
+ MOVQ AX, ret+16(FP) // return 0
+ RET
diff --git a/src/math/hypot_amd64p32.s b/src/math/hypot_amd64p32.s
new file mode 100644
index 0000000..b84542a
--- /dev/null
+++ b/src/math/hypot_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "hypot_amd64.s"
diff --git a/src/math/hypot_arm.s b/src/math/hypot_arm.s
new file mode 100644
index 0000000..9c8abca
--- /dev/null
+++ b/src/math/hypot_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Hypot(SB),NOSPLIT,$0
+ B ·hypot(SB)
diff --git a/src/math/j0.go b/src/math/j0.go
new file mode 100644
index 0000000..c20a9b2
--- /dev/null
+++ b/src/math/j0.go
@@ -0,0 +1,429 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order zero.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_j0.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_j0(x), __ieee754_y0(x)
+// Bessel function of the first and second kinds of order zero.
+// Method -- j0(x):
+// 1. For tiny x, we use j0(x) = 1 - x**2/4 + x**4/64 - ...
+// 2. Reduce x to |x| since j0(x)=j0(-x), and
+// for x in (0,2)
+// j0(x) = 1-z/4+ z**2*R0/S0, where z = x*x;
+// (precision: |j0-1+z/4-z**2R0/S0 |<2**-63.67 )
+// for x in (2,inf)
+// j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
+// where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
+// as follow:
+// cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
+// = 1/sqrt(2) * (cos(x) + sin(x))
+// sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
+// = 1/sqrt(2) * (sin(x) - cos(x))
+// (To avoid cancellation, use
+// sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+// to compute the worse one.)
+//
+// 3 Special cases
+// j0(nan)= nan
+// j0(0) = 1
+// j0(inf) = 0
+//
+// Method -- y0(x):
+// 1. For x<2.
+// Since
+// y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x**2/4 - ...)
+// therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
+// We use the following function to approximate y0,
+// y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x**2
+// where
+// U(z) = u00 + u01*z + ... + u06*z**6
+// V(z) = 1 + v01*z + ... + v04*z**4
+// with absolute approximation error bounded by 2**-72.
+// Note: For tiny x, U/V = u0 and j0(x)~1, hence
+// y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
+// 2. For x>=2.
+// y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
+// where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
+// by the method mentioned above.
+// 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
+//
+
+// J0 returns the order-zero Bessel function of the first kind.
+//
+// Special cases are:
+// J0(±Inf) = 0
+// J0(0) = 1
+// J0(NaN) = NaN
+func J0(x float64) float64 {
+ const (
+ Huge = 1e300
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ // R0/S0 on [0, 2]
+ R02 = 1.56249999999999947958e-02 // 0x3F8FFFFFFFFFFFFD
+ R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9
+ R04 = 1.82954049532700665670e-06 // 0x3EBEB1D10C503919
+ R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE
+ S01 = 1.56191029464890010492e-02 // 0x3F8FFCE882C8C2A4
+ S02 = 1.16926784663337450260e-04 // 0x3F1EA6D2DD57DBF4
+ S03 = 5.13546550207318111446e-07 // 0x3EA13B54CE84D5A9
+ S04 = 1.16614003333790000205e-09 // 0x3E1408BCF4745D8F
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return x
+ case IsInf(x, 0):
+ return 0
+ case x == 0:
+ return 1
+ }
+
+ if x < 0 {
+ x = -x
+ }
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := s - c
+ cc := s + c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := -Cos(x + x)
+ if s*c < 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+
+ // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
+ // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
+
+ var z float64
+ if x > Two129 { // |x| > ~6.8056e+38
+ z = (1 / SqrtPi) * cc / Sqrt(x)
+ } else {
+ u := pzero(x)
+ v := qzero(x)
+ z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
+ }
+ return z // |x| >= 2.0
+ }
+ if x < TwoM13 { // |x| < ~1.2207e-4
+ if x < TwoM27 {
+ return 1 // |x| < ~7.4506e-9
+ }
+ return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4
+ }
+ z := x * x
+ r := z * (R02 + z*(R03+z*(R04+z*R05)))
+ s := 1 + z*(S01+z*(S02+z*(S03+z*S04)))
+ if x < 1 {
+ return 1 + z*(-0.25+(r/s)) // |x| < 1.00
+ }
+ u := 0.5 * x
+ return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0
+}
+
+// Y0 returns the order-zero Bessel function of the second kind.
+//
+// Special cases are:
+// Y0(+Inf) = 0
+// Y0(0) = -Inf
+// Y0(x < 0) = NaN
+// Y0(NaN) = NaN
+func Y0(x float64) float64 {
+ const (
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ U00 = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F
+ U01 = 1.76666452509181115538e-01 // 0x3FC69D019DE9E3FC
+ U02 = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97
+ U03 = 3.47453432093683650238e-04 // 0x3F36C54D20B29B6B
+ U04 = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD
+ U05 = 1.95590137035022920206e-08 // 0x3E5500573B4EABD4
+ U06 = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8
+ V01 = 1.27304834834123699328e-02 // 0x3F8A127091C9C71A
+ V02 = 7.60068627350353253702e-05 // 0x3F13ECBBF578C6C1
+ V03 = 2.59150851840457805467e-07 // 0x3E91642D7FF202FD
+ V04 = 4.41110311332675467403e-10 // 0x3DFE50183BD6D9EF
+ )
+ // special cases
+ switch {
+ case x < 0 || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ case x == 0:
+ return Inf(-1)
+ }
+
+ if x >= 2 { // |x| >= 2.0
+
+ // y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
+ // where x0 = x-pi/4
+ // Better formula:
+ // cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
+ // = 1/sqrt(2) * (sin(x) + cos(x))
+ // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+ // = 1/sqrt(2) * (sin(x) - cos(x))
+ // To avoid cancellation, use
+ // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+ // to compute the worse one.
+
+ s, c := Sincos(x)
+ ss := s - c
+ cc := s + c
+
+ // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
+ // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := -Cos(x + x)
+ if s*c < 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+ var z float64
+ if x > Two129 { // |x| > ~6.8056e+38
+ z = (1 / SqrtPi) * ss / Sqrt(x)
+ } else {
+ u := pzero(x)
+ v := qzero(x)
+ z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
+ }
+ return z // |x| >= 2.0
+ }
+ if x <= TwoM27 {
+ return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9
+ }
+ z := x * x
+ u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06)))))
+ v := 1 + z*(V01+z*(V02+z*(V03+z*V04)))
+ return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0
+}
+
+// The asymptotic expansions of pzero is
+// 1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x.
+// For x >= 2, We approximate pzero by
+// pzero(x) = 1 + (R/S)
+// where R = pR0 + pR1*s**2 + pR2*s**4 + ... + pR5*s**10
+// S = 1 + pS0*s**2 + ... + pS4*s**10
+// and
+// | pzero(x)-1-R/S | <= 2 ** ( -60.26)
+
+// for x in [inf, 8]=1/[0,0.125]
+var p0R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ -7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
+ -8.08167041275349795626e+00, // 0xC02029D0B44FA779
+ -2.57063105679704847262e+02, // 0xC07011027B19E863
+ -2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
+ -5.25304380490729545272e+03, // 0xC0B4850B36CC643D
+}
+var p0S8 = [5]float64{
+ 1.16534364619668181717e+02, // 0x405D223307A96751
+ 3.83374475364121826715e+03, // 0x40ADF37D50596938
+ 4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
+ 1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD
+ 4.76277284146730962675e+04, // 0x40E741774F2C49DC
+}
+
+// for x in [8,4.5454]=1/[0.125,0.22001]
+var p0R5 = [6]float64{
+ -1.14125464691894502584e-11, // 0xBDA918B147E495CC
+ -7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
+ -4.15961064470587782438e+00, // 0xC010A370F90C6BBF
+ -6.76747652265167261021e+01, // 0xC050EB2F5A7D1783
+ -3.31231299649172967747e+02, // 0xC074B3B36742CC63
+ -3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
+}
+var p0S5 = [5]float64{
+ 6.07539382692300335975e+01, // 0x404E60810C98C5DE
+ 1.05125230595704579173e+03, // 0x40906D025C7E2864
+ 5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
+ 9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38
+ 2.40605815922939109441e+03, // 0x40A2CC1DC70BE864
+}
+
+// for x in [4.547,2.8571]=1/[0.2199,0.35001]
+var p0R3 = [6]float64{
+ -2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
+ -7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
+ -2.40903221549529611423e+00, // 0xC00345B2AEA48074
+ -2.19659774734883086467e+01, // 0xC035F74A4CB94E14
+ -5.80791704701737572236e+01, // 0xC04D0A22420A1A45
+ -3.14479470594888503854e+01, // 0xC03F72ACA892D80F
+}
+var p0S3 = [5]float64{
+ 3.58560338055209726349e+01, // 0x4041ED9284077DD3
+ 3.61513983050303863820e+02, // 0x40769839464A7C0E
+ 1.19360783792111533330e+03, // 0x4092A66E6D1061D6
+ 1.12799679856907414432e+03, // 0x40919FFCB8C39B7E
+ 1.73580930813335754692e+02, // 0x4065B296FC379081
+}
+
+// for x in [2.8570,2]=1/[0.3499,0.5]
+var p0R2 = [6]float64{
+ -8.87534333032526411254e-08, // 0xBE77D316E927026D
+ -7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
+ -1.45073846780952986357e+00, // 0xBFF736398A24A843
+ -7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3
+ -1.11931668860356747786e+01, // 0xC02662E6C5246303
+ -3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
+}
+var p0S2 = [5]float64{
+ 2.22202997532088808441e+01, // 0x40363865908B5959
+ 1.36206794218215208048e+02, // 0x4061069E0EE8878F
+ 2.70470278658083486789e+02, // 0x4070E78642EA079B
+ 1.53875394208320329881e+02, // 0x40633C033AB6FAFF
+ 1.46576176948256193810e+01, // 0x402D50B344391809
+}
+
+func pzero(x float64) float64 {
+ var p [6]float64
+ var q [5]float64
+ if x >= 8 {
+ p = p0R8
+ q = p0S8
+ } else if x >= 4.5454 {
+ p = p0R5
+ q = p0S5
+ } else if x >= 2.8571 {
+ p = p0R3
+ q = p0S3
+ } else if x >= 2 {
+ p = p0R2
+ q = p0S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
+ return 1 + r/s
+}
+
+// For x >= 8, the asymptotic expansions of qzero is
+// -1/8 s + 75/1024 s**3 - ..., where s = 1/x.
+// We approximate pzero by
+// qzero(x) = s*(-1.25 + (R/S))
+// where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10
+// S = 1 + qS0*s**2 + ... + qS5*s**12
+// and
+// | qzero(x)/s +1.25-R/S | <= 2**(-61.22)
+
+// for x in [inf, 8]=1/[0,0.125]
+var q0R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ 7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
+ 1.17682064682252693899e+01, // 0x402789525BB334D6
+ 5.57673380256401856059e+02, // 0x40816D6315301825
+ 8.85919720756468632317e+03, // 0x40C14D993E18F46D
+ 3.70146267776887834771e+04, // 0x40E212D40E901566
+}
+var q0S8 = [6]float64{
+ 1.63776026895689824414e+02, // 0x406478D5365B39BC
+ 8.09834494656449805916e+03, // 0x40BFA2584E6B0563
+ 1.42538291419120476348e+05, // 0x4101665254D38C3F
+ 8.03309257119514397345e+05, // 0x412883DA83A52B43
+ 8.40501579819060512818e+05, // 0x4129A66B28DE0B3D
+ -3.43899293537866615225e+05, // 0xC114FD6D2C9530C5
+}
+
+// for x in [8,4.5454]=1/[0.125,0.22001]
+var q0R5 = [6]float64{
+ 1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
+ 7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
+ 5.83563508962056953777e+00, // 0x401757B0B9953DD3
+ 1.35111577286449829671e+02, // 0x4060E3920A8788E9
+ 1.02724376596164097464e+03, // 0x40900CF99DC8C481
+ 1.98997785864605384631e+03, // 0x409F17E953C6E3A6
+}
+var q0S5 = [6]float64{
+ 8.27766102236537761883e+01, // 0x4054B1B3FB5E1543
+ 2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE
+ 1.88472887785718085070e+04, // 0x40D267D27B591E6D
+ 5.67511122894947329769e+04, // 0x40EBB5E397E02372
+ 3.59767538425114471465e+04, // 0x40E191181F7A54A0
+ -5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609
+}
+
+// for x in [4.547,2.8571]=1/[0.2199,0.35001]
+var q0R3 = [6]float64{
+ 4.37741014089738620906e-09, // 0x3E32CD036ADECB82
+ 7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
+ 3.34423137516170720929e+00, // 0x400AC0FC61149CF5
+ 4.26218440745412650017e+01, // 0x40454F98962DAEDD
+ 1.70808091340565596283e+02, // 0x406559DBE25EFD1F
+ 1.66733948696651168575e+02, // 0x4064D77C81FA21E0
+}
+var q0S3 = [6]float64{
+ 4.87588729724587182091e+01, // 0x40486122BFE343A6
+ 7.09689221056606015736e+02, // 0x40862D8386544EB3
+ 3.70414822620111362994e+03, // 0x40ACF04BE44DFC63
+ 6.46042516752568917582e+03, // 0x40B93C6CD7C76A28
+ 2.51633368920368957333e+03, // 0x40A3A8AAD94FB1C0
+ -1.49247451836156386662e+02, // 0xC062A7EB201CF40F
+}
+
+// for x in [2.8570,2]=1/[0.3499,0.5]
+var q0R2 = [6]float64{
+ 1.50444444886983272379e-07, // 0x3E84313B54F76BDB
+ 7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
+ 1.99819174093815998816e+00, // 0x3FFFF897E727779C
+ 1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5
+ 3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
+ 1.62527075710929267416e+01, // 0x403040B171814BB4
+}
+var q0S2 = [6]float64{
+ 3.03655848355219184498e+01, // 0x403E5D96F7C07AED
+ 2.69348118608049844624e+02, // 0x4070D591E4D14B40
+ 8.44783757595320139444e+02, // 0x408A664522B3BF22
+ 8.82935845112488550512e+02, // 0x408B977C9C5CC214
+ 2.12666388511798828631e+02, // 0x406A95530E001365
+ -5.31095493882666946917e+00, // 0xC0153E6AF8B32931
+}
+
+func qzero(x float64) float64 {
+ var p, q [6]float64
+ if x >= 8 {
+ p = q0R8
+ q = q0S8
+ } else if x >= 4.5454 {
+ p = q0R5
+ q = q0S5
+ } else if x >= 2.8571 {
+ p = q0R3
+ q = q0S3
+ } else if x >= 2 {
+ p = q0R2
+ q = q0S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
+ return (-0.125 + r/s) / x
+}
diff --git a/src/math/j1.go b/src/math/j1.go
new file mode 100644
index 0000000..7ac186b
--- /dev/null
+++ b/src/math/j1.go
@@ -0,0 +1,422 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order one.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_j1.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_j1(x), __ieee754_y1(x)
+// Bessel function of the first and second kinds of order one.
+// Method -- j1(x):
+// 1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ...
+// 2. Reduce x to |x| since j1(x)=-j1(-x), and
+// for x in (0,2)
+// j1(x) = x/2 + x*z*R0/S0, where z = x*x;
+// (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
+// for x in (2,inf)
+// j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
+// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+// where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+// as follow:
+// cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+// = 1/sqrt(2) * (sin(x) - cos(x))
+// sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+// = -1/sqrt(2) * (sin(x) + cos(x))
+// (To avoid cancellation, use
+// sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+// to compute the worse one.)
+//
+// 3 Special cases
+// j1(nan)= nan
+// j1(0) = 0
+// j1(inf) = 0
+//
+// Method -- y1(x):
+// 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
+// 2. For x<2.
+// Since
+// y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...)
+// therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
+// We use the following function to approximate y1,
+// y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2
+// where for x in [0,2] (abs err less than 2**-65.89)
+// U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4
+// V(z) = 1 + v0[0]*z + ... + v0[4]*z**5
+// Note: For tiny x, 1/x dominate y1 and hence
+// y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
+// 3. For x>=2.
+// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+// where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+// by method mentioned above.
+
+// J1 returns the order-one Bessel function of the first kind.
+//
+// Special cases are:
+// J1(±Inf) = 0
+// J1(NaN) = NaN
+func J1(x float64) float64 {
+ const (
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ // R0/S0 on [0, 2]
+ R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000
+ R01 = 1.40705666955189706048e-03 // 0x3F570D9F98472C61
+ R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668
+ R03 = 4.96727999609584448412e-08 // 0x3E6AAAFA46CA0BD9
+ S01 = 1.91537599538363460805e-02 // 0x3F939D0B12637E53
+ S02 = 1.85946785588630915560e-04 // 0x3F285F56B9CDF664
+ S03 = 1.17718464042623683263e-06 // 0x3EB3BFF8333F8498
+ S04 = 5.04636257076217042715e-09 // 0x3E35AC88C97DFF2C
+ S05 = 1.23542274426137913908e-11 // 0x3DAB2ACFCFB97ED8
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return x
+ case IsInf(x, 0) || x == 0:
+ return 0
+ }
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := -s - c
+ cc := s - c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := Cos(x + x)
+ if s*c > 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+
+ // j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
+ // y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
+
+ var z float64
+ if x > Two129 {
+ z = (1 / SqrtPi) * cc / Sqrt(x)
+ } else {
+ u := pone(x)
+ v := qone(x)
+ z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
+ }
+ if sign {
+ return -z
+ }
+ return z
+ }
+ if x < TwoM27 { // |x|<2**-27
+ return 0.5 * x // inexact if x!=0 necessary
+ }
+ z := x * x
+ r := z * (R00 + z*(R01+z*(R02+z*R03)))
+ s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
+ r *= x
+ z = 0.5*x + r/s
+ if sign {
+ return -z
+ }
+ return z
+}
+
+// Y1 returns the order-one Bessel function of the second kind.
+//
+// Special cases are:
+// Y1(+Inf) = 0
+// Y1(0) = -Inf
+// Y1(x < 0) = NaN
+// Y1(NaN) = NaN
+func Y1(x float64) float64 {
+ const (
+ TwoM54 = 1.0 / (1 << 54) // 2**-54 0x3c90000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ U00 = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
+ U01 = 5.04438716639811282616e-02 // 0x3FA9D3C776292CD1
+ U02 = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
+ U03 = 2.35252600561610495928e-05 // 0x3EF8AB038FA6B88E
+ U04 = -9.19099158039878874504e-08 // 0xBE78AC00569105B8
+ V00 = 1.99167318236649903973e-02 // 0x3F94650D3F4DA9F0
+ V01 = 2.02552581025135171496e-04 // 0x3F2A8C896C257764
+ V02 = 1.35608801097516229404e-06 // 0x3EB6C05A894E8CA6
+ V03 = 6.22741452364621501295e-09 // 0x3E3ABF1D5BA69A86
+ V04 = 1.66559246207992079114e-11 // 0x3DB25039DACA772A
+ )
+ // special cases
+ switch {
+ case x < 0 || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ case x == 0:
+ return Inf(-1)
+ }
+
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := -s - c
+ cc := s - c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := Cos(x + x)
+ if s*c > 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+ // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
+ // where x0 = x-3pi/4
+ // Better formula:
+ // cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+ // = 1/sqrt(2) * (sin(x) - cos(x))
+ // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
+ // = -1/sqrt(2) * (cos(x) + sin(x))
+ // To avoid cancellation, use
+ // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+ // to compute the worse one.
+
+ var z float64
+ if x > Two129 {
+ z = (1 / SqrtPi) * ss / Sqrt(x)
+ } else {
+ u := pone(x)
+ v := qone(x)
+ z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
+ }
+ return z
+ }
+ if x <= TwoM54 { // x < 2**-54
+ return -(2 / Pi) / x
+ }
+ z := x * x
+ u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
+ v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
+ return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
+}
+
+// For x >= 8, the asymptotic expansions of pone is
+// 1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x.
+// We approximate pone by
+// pone(x) = 1 + (R/S)
+// where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10
+// S = 1 + ps0*s**2 + ... + ps4*s**10
+// and
+// | pone(x)-1-R/S | <= 2**(-60.06)
+
+// for x in [inf, 8]=1/[0,0.125]
+var p1R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ 1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
+ 1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
+ 4.12051854307378562225e+02, // 0x4079C0D4652EA590
+ 3.87474538913960532227e+03, // 0x40AE457DA3A532CC
+ 7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
+}
+var p1S8 = [5]float64{
+ 1.14207370375678408436e+02, // 0x405C8D458E656CAC
+ 3.65093083420853463394e+03, // 0x40AC85DC964D274F
+ 3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
+ 9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
+ 3.08042720627888811578e+04, // 0x40DE1511697A0B2D
+}
+
+// for x in [8,4.5454] = 1/[0.125,0.22001]
+var p1R5 = [6]float64{
+ 1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
+ 1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
+ 6.80275127868432871736e+00, // 0x401B36046E6315E3
+ 1.08308182990189109773e+02, // 0x405B13B9452602ED
+ 5.17636139533199752805e+02, // 0x40802D16D052D649
+ 5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
+}
+var p1S5 = [5]float64{
+ 5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
+ 9.91401418733614377743e+02, // 0x408EFB361B066701
+ 5.35326695291487976647e+03, // 0x40B4E9445706B6FB
+ 7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
+ 1.50404688810361062679e+03, // 0x40978030036F5E51
+}
+
+// for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
+var p1R3 = [6]float64{
+ 3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
+ 1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
+ 3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
+ 3.51194035591636932736e+01, // 0x40418F489DA6D129
+ 9.10550110750781271918e+01, // 0x4056C3854D2C1837
+ 4.85590685197364919645e+01, // 0x4048478F8EA83EE5
+}
+var p1S3 = [5]float64{
+ 3.47913095001251519989e+01, // 0x40416549A134069C
+ 3.36762458747825746741e+02, // 0x40750C3307F1A75F
+ 1.04687139975775130551e+03, // 0x40905B7C5037D523
+ 8.90811346398256432622e+02, // 0x408BD67DA32E31E9
+ 1.03787932439639277504e+02, // 0x4059F26D7C2EED53
+}
+
+// for x in [2.8570,2] = 1/[0.3499,0.5]
+var p1R2 = [6]float64{
+ 1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
+ 1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
+ 2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
+ 1.22426109148261232917e+01, // 0x40287C377F71A964
+ 1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
+ 5.07352312588818499250e+00, // 0x40144B49A574C1FE
+}
+var p1S2 = [5]float64{
+ 2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
+ 1.25290227168402751090e+02, // 0x405F529314F92CD5
+ 2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
+ 1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
+ 8.36463893371618283368e+00, // 0x4020BAB1F44E5192
+}
+
+func pone(x float64) float64 {
+ var p [6]float64
+ var q [5]float64
+ if x >= 8 {
+ p = p1R8
+ q = p1S8
+ } else if x >= 4.5454 {
+ p = p1R5
+ q = p1S5
+ } else if x >= 2.8571 {
+ p = p1R3
+ q = p1S3
+ } else if x >= 2 {
+ p = p1R2
+ q = p1S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
+ return 1 + r/s
+}
+
+// For x >= 8, the asymptotic expansions of qone is
+// 3/8 s - 105/1024 s**3 - ..., where s = 1/x.
+// We approximate qone by
+// qone(x) = s*(0.375 + (R/S))
+// where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10
+// S = 1 + qs1*s**2 + ... + qs6*s**12
+// and
+// | qone(x)/s -0.375-R/S | <= 2**(-61.13)
+
+// for x in [inf, 8] = 1/[0,0.125]
+var q1R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ -1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
+ -1.62717534544589987888e+01, // 0xC0304591A26779F7
+ -7.59601722513950107896e+02, // 0xC087BCD053E4B576
+ -1.18498066702429587167e+04, // 0xC0C724E740F87415
+ -4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
+}
+var q1S8 = [6]float64{
+ 1.61395369700722909556e+02, // 0x40642CA6DE5BCDE5
+ 7.82538599923348465381e+03, // 0x40BE9162D0D88419
+ 1.33875336287249578163e+05, // 0x4100579AB0B75E98
+ 7.19657723683240939863e+05, // 0x4125F65372869C19
+ 6.66601232617776375264e+05, // 0x412457D27719AD5C
+ -2.94490264303834643215e+05, // 0xC111F9690EA5AA18
+}
+
+// for x in [8,4.5454] = 1/[0.125,0.22001]
+var q1R5 = [6]float64{
+ -2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
+ -1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
+ -8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
+ -1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
+ -1.37319376065508163265e+03, // 0xC09574C66931734F
+ -2.61244440453215656817e+03, // 0xC0A468E388FDA79D
+}
+var q1S5 = [6]float64{
+ 8.12765501384335777857e+01, // 0x405451B2FF5A11B2
+ 1.99179873460485964642e+03, // 0x409F1F31E77BF839
+ 1.74684851924908907677e+04, // 0x40D10F1F0D64CE29
+ 4.98514270910352279316e+04, // 0x40E8576DAABAD197
+ 2.79480751638918118260e+04, // 0x40DB4B04CF7C364B
+ -4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
+}
+
+// for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
+var q1R3 = [6]float64{
+ -5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
+ -1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
+ -4.61011581139473403113e+00, // 0xC01270C23302D9FF
+ -5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
+ -2.28244540737631695038e+02, // 0xC06C87D34718D55F
+ -2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
+}
+var q1S3 = [6]float64{
+ 4.76651550323729509273e+01, // 0x4047D523CCD367E4
+ 6.73865112676699709482e+02, // 0x40850EEBC031EE3E
+ 3.38015286679526343505e+03, // 0x40AA684E448E7C9A
+ 5.54772909720722782367e+03, // 0x40B5ABBAA61D54A6
+ 1.90311919338810798763e+03, // 0x409DBC7A0DD4DF4B
+ -1.35201191444307340817e+02, // 0xC060E670290A311F
+}
+
+// for x in [2.8570,2] = 1/[0.3499,0.5]
+var q1R2 = [6]float64{
+ -1.78381727510958865572e-07, // 0xBE87F12644C626D2
+ -1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
+ -2.75220568278187460720e+00, // 0xC006048469BB4EDA
+ -1.96636162643703720221e+01, // 0xC033A9E2C168907F
+ -4.23253133372830490089e+01, // 0xC04529A3DE104AAA
+ -2.13719211703704061733e+01, // 0xC0355F3639CF6E52
+}
+var q1S2 = [6]float64{
+ 2.95333629060523854548e+01, // 0x403D888A78AE64FF
+ 2.52981549982190529136e+02, // 0x406F9F68DB821CBA
+ 7.57502834868645436472e+02, // 0x4087AC05CE49A0F7
+ 7.39393205320467245656e+02, // 0x40871B2548D4C029
+ 1.55949003336666123687e+02, // 0x40637E5E3C3ED8D4
+ -4.95949898822628210127e+00, // 0xC013D686E71BE86B
+}
+
+func qone(x float64) float64 {
+ var p, q [6]float64
+ if x >= 8 {
+ p = q1R8
+ q = q1S8
+ } else if x >= 4.5454 {
+ p = q1R5
+ q = q1S5
+ } else if x >= 2.8571 {
+ p = q1R3
+ q = q1S3
+ } else if x >= 2 {
+ p = q1R2
+ q = q1S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
+ return (0.375 + r/s) / x
+}
diff --git a/src/math/jn.go b/src/math/jn.go
new file mode 100644
index 0000000..a7909eb
--- /dev/null
+++ b/src/math/jn.go
@@ -0,0 +1,306 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order n.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_jn.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_jn(n, x), __ieee754_yn(n, x)
+// floating point Bessel's function of the 1st and 2nd kind
+// of order n
+//
+// Special cases:
+// y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
+// y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
+// Note 2. About jn(n,x), yn(n,x)
+// For n=0, j0(x) is called,
+// for n=1, j1(x) is called,
+// for n<x, forward recursion is used starting
+// from values of j0(x) and j1(x).
+// for n>x, a continued fraction approximation to
+// j(n,x)/j(n-1,x) is evaluated and then backward
+// recursion is used starting from a supposed value
+// for j(n,x). The resulting value of j(0,x) is
+// compared with the actual value to correct the
+// supposed value of j(n,x).
+//
+// yn(n,x) is similar in all respects, except
+// that forward recursion is used for all
+// values of n>1.
+
+// Jn returns the order-n Bessel function of the first kind.
+//
+// Special cases are:
+// Jn(n, ±Inf) = 0
+// Jn(n, NaN) = NaN
+func Jn(n int, x float64) float64 {
+ const (
+ TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
+ Two302 = 1 << 302 // 2**302 0x52D0000000000000
+ )
+ // special cases
+ switch {
+ case IsNaN(x):
+ return x
+ case IsInf(x, 0):
+ return 0
+ }
+ // J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
+ // Thus, J(-n, x) = J(n, -x)
+
+ if n == 0 {
+ return J0(x)
+ }
+ if x == 0 {
+ return 0
+ }
+ if n < 0 {
+ n, x = -n, -x
+ }
+ if n == 1 {
+ return J1(x)
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ if n&1 == 1 {
+ sign = true // odd n and negative x
+ }
+ }
+ var b float64
+ if float64(n) <= x {
+ // Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
+ if x >= Two302 { // x > 2**302
+
+ // (x >> n**2)
+ // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Let s=sin(x), c=cos(x),
+ // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ //
+ // n sin(xn)*sqt2 cos(xn)*sqt2
+ // ----------------------------------
+ // 0 s-c c+s
+ // 1 -s-c -c+s
+ // 2 -s+c -c-s
+ // 3 s+c c-s
+
+ var temp float64
+ switch n & 3 {
+ case 0:
+ temp = Cos(x) + Sin(x)
+ case 1:
+ temp = -Cos(x) + Sin(x)
+ case 2:
+ temp = -Cos(x) - Sin(x)
+ case 3:
+ temp = Cos(x) - Sin(x)
+ }
+ b = (1 / SqrtPi) * temp / Sqrt(x)
+ } else {
+ b = J1(x)
+ for i, a := 1, J0(x); i < n; i++ {
+ a, b = b, b*(float64(i+i)/x)-a // avoid underflow
+ }
+ }
+ } else {
+ if x < TwoM29 { // x < 2**-29
+ // x is tiny, return the first Taylor expansion of J(n,x)
+ // J(n,x) = 1/n!*(x/2)**n - ...
+
+ if n > 33 { // underflow
+ b = 0
+ } else {
+ temp := x * 0.5
+ b = temp
+ a := 1.0
+ for i := 2; i <= n; i++ {
+ a *= float64(i) // a = n!
+ b *= temp // b = (x/2)**n
+ }
+ b /= a
+ }
+ } else {
+ // use backward recurrence
+ // x x**2 x**2
+ // J(n,x)/J(n-1,x) = ---- ------ ------ .....
+ // 2n - 2(n+1) - 2(n+2)
+ //
+ // 1 1 1
+ // (for large x) = ---- ------ ------ .....
+ // 2n 2(n+1) 2(n+2)
+ // -- - ------ - ------ -
+ // x x x
+ //
+ // Let w = 2n/x and h=2/x, then the above quotient
+ // is equal to the continued fraction:
+ // 1
+ // = -----------------------
+ // 1
+ // w - -----------------
+ // 1
+ // w+h - ---------
+ // w+2h - ...
+ //
+ // To determine how many terms needed, let
+ // Q(0) = w, Q(1) = w(w+h) - 1,
+ // Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
+ // When Q(k) > 1e4 good for single
+ // When Q(k) > 1e9 good for double
+ // When Q(k) > 1e17 good for quadruple
+
+ // determine k
+ w := float64(n+n) / x
+ h := 2 / x
+ q0 := w
+ z := w + h
+ q1 := w*z - 1
+ k := 1
+ for q1 < 1e9 {
+ k += 1
+ z += h
+ q0, q1 = q1, z*q1-q0
+ }
+ m := n + n
+ t := 0.0
+ for i := 2 * (n + k); i >= m; i -= 2 {
+ t = 1 / (float64(i)/x - t)
+ }
+ a := t
+ b = 1
+ // estimate log((2/x)**n*n!) = n*log(2/x)+n*ln(n)
+ // Hence, if n*(log(2n/x)) > ...
+ // single 8.8722839355e+01
+ // double 7.09782712893383973096e+02
+ // long double 1.1356523406294143949491931077970765006170e+04
+ // then recurrent value may overflow and the result is
+ // likely underflow to zero
+
+ tmp := float64(n)
+ v := 2 / x
+ tmp = tmp * Log(Abs(v*tmp))
+ if tmp < 7.09782712893383973096e+02 {
+ for i := n - 1; i > 0; i-- {
+ di := float64(i + i)
+ a, b = b, b*di/x-a
+ di -= 2
+ }
+ } else {
+ for i := n - 1; i > 0; i-- {
+ di := float64(i + i)
+ a, b = b, b*di/x-a
+ di -= 2
+ // scale b to avoid spurious overflow
+ if b > 1e100 {
+ a /= b
+ t /= b
+ b = 1
+ }
+ }
+ }
+ b = t * J0(x) / b
+ }
+ }
+ if sign {
+ return -b
+ }
+ return b
+}
+
+// Yn returns the order-n Bessel function of the second kind.
+//
+// Special cases are:
+// Yn(n, +Inf) = 0
+// Yn(n > 0, 0) = -Inf
+// Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
+// Y1(n, x < 0) = NaN
+// Y1(n, NaN) = NaN
+func Yn(n int, x float64) float64 {
+ const Two302 = 1 << 302 // 2**302 0x52D0000000000000
+ // special cases
+ switch {
+ case x < 0 || IsNaN(x):
+ return NaN()
+ case IsInf(x, 1):
+ return 0
+ }
+
+ if n == 0 {
+ return Y0(x)
+ }
+ if x == 0 {
+ if n < 0 && n&1 == 1 {
+ return Inf(1)
+ }
+ return Inf(-1)
+ }
+ sign := false
+ if n < 0 {
+ n = -n
+ if n&1 == 1 {
+ sign = true // sign true if n < 0 && |n| odd
+ }
+ }
+ if n == 1 {
+ if sign {
+ return -Y1(x)
+ }
+ return Y1(x)
+ }
+ var b float64
+ if x >= Two302 { // x > 2**302
+ // (x >> n**2)
+ // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Let s=sin(x), c=cos(x),
+ // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ //
+ // n sin(xn)*sqt2 cos(xn)*sqt2
+ // ----------------------------------
+ // 0 s-c c+s
+ // 1 -s-c -c+s
+ // 2 -s+c -c-s
+ // 3 s+c c-s
+
+ var temp float64
+ switch n & 3 {
+ case 0:
+ temp = Sin(x) - Cos(x)
+ case 1:
+ temp = -Sin(x) - Cos(x)
+ case 2:
+ temp = -Sin(x) + Cos(x)
+ case 3:
+ temp = Sin(x) + Cos(x)
+ }
+ b = (1 / SqrtPi) * temp / Sqrt(x)
+ } else {
+ a := Y0(x)
+ b = Y1(x)
+ // quit if b is -inf
+ for i := 1; i < n && !IsInf(b, -1); i++ {
+ a, b = b, (float64(i+i)/x)*b-a
+ }
+ }
+ if sign {
+ return -b
+ }
+ return b
+}
diff --git a/src/math/ldexp.go b/src/math/ldexp.go
new file mode 100644
index 0000000..b5d2a5e
--- /dev/null
+++ b/src/math/ldexp.go
@@ -0,0 +1,45 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Ldexp is the inverse of Frexp.
+// It returns frac × 2**exp.
+//
+// Special cases are:
+// Ldexp(±0, exp) = ±0
+// Ldexp(±Inf, exp) = ±Inf
+// Ldexp(NaN, exp) = NaN
+func Ldexp(frac float64, exp int) float64
+
+func ldexp(frac float64, exp int) float64 {
+ // special cases
+ switch {
+ case frac == 0:
+ return frac // correctly return -0
+ case IsInf(frac, 0) || IsNaN(frac):
+ return frac
+ }
+ frac, e := normalize(frac)
+ exp += e
+ x := Float64bits(frac)
+ exp += int(x>>shift)&mask - bias
+ if exp < -1074 {
+ return Copysign(0, frac) // underflow
+ }
+ if exp > 1023 { // overflow
+ if frac < 0 {
+ return Inf(-1)
+ }
+ return Inf(1)
+ }
+ var m float64 = 1
+ if exp < -1022 { // denormal
+ exp += 52
+ m = 1.0 / (1 << 52) // 2**-52
+ }
+ x &^= mask << shift
+ x |= uint64(exp+bias) << shift
+ return m * Float64frombits(x)
+}
diff --git a/src/math/ldexp_386.s b/src/math/ldexp_386.s
new file mode 100644
index 0000000..ac8e8ba
--- /dev/null
+++ b/src/math/ldexp_386.s
@@ -0,0 +1,14 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Ldexp(frac float64, exp int) float64
+TEXT ·Ldexp(SB),NOSPLIT,$0
+ FMOVL exp+8(FP), F0 // F0=exp
+ FMOVD frac+0(FP), F0 // F0=frac, F1=e
+ FSCALE // F0=x*2**e, F1=e
+ FMOVDP F0, F1 // F0=x*2**e
+ FMOVDP F0, ret+12(FP)
+ RET
diff --git a/src/math/ldexp_amd64.s b/src/math/ldexp_amd64.s
new file mode 100644
index 0000000..6063a64
--- /dev/null
+++ b/src/math/ldexp_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Ldexp(SB),NOSPLIT,$0
+ JMP ·ldexp(SB)
diff --git a/src/math/ldexp_amd64p32.s b/src/math/ldexp_amd64p32.s
new file mode 100644
index 0000000..9aa9d9d
--- /dev/null
+++ b/src/math/ldexp_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "ldexp_amd64.s"
diff --git a/src/math/ldexp_arm.s b/src/math/ldexp_arm.s
new file mode 100644
index 0000000..fcffa2e
--- /dev/null
+++ b/src/math/ldexp_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Ldexp(SB),NOSPLIT,$0
+ B ·ldexp(SB)
diff --git a/src/math/lgamma.go b/src/math/lgamma.go
new file mode 100644
index 0000000..6a02c41
--- /dev/null
+++ b/src/math/lgamma.go
@@ -0,0 +1,365 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point logarithm of the Gamma function.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_lgamma_r(x, signgamp)
+// Reentrant version of the logarithm of the Gamma function
+// with user provided pointer for the sign of Gamma(x).
+//
+// Method:
+// 1. Argument Reduction for 0 < x <= 8
+// Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
+// reduce x to a number in [1.5,2.5] by
+// lgamma(1+s) = log(s) + lgamma(s)
+// for example,
+// lgamma(7.3) = log(6.3) + lgamma(6.3)
+// = log(6.3*5.3) + lgamma(5.3)
+// = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
+// 2. Polynomial approximation of lgamma around its
+// minimum (ymin=1.461632144968362245) to maintain monotonicity.
+// On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
+// Let z = x-ymin;
+// lgamma(x) = -1.214862905358496078218 + z**2*poly(z)
+// poly(z) is a 14 degree polynomial.
+// 2. Rational approximation in the primary interval [2,3]
+// We use the following approximation:
+// s = x-2.0;
+// lgamma(x) = 0.5*s + s*P(s)/Q(s)
+// with accuracy
+// |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
+// Our algorithms are based on the following observation
+//
+// zeta(2)-1 2 zeta(3)-1 3
+// lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
+// 2 3
+//
+// where Euler = 0.5772156649... is the Euler constant, which
+// is very close to 0.5.
+//
+// 3. For x>=8, we have
+// lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
+// (better formula:
+// lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
+// Let z = 1/x, then we approximation
+// f(z) = lgamma(x) - (x-0.5)(log(x)-1)
+// by
+// 3 5 11
+// w = w0 + w1*z + w2*z + w3*z + ... + w6*z
+// where
+// |w - f(z)| < 2**-58.74
+//
+// 4. For negative x, since (G is gamma function)
+// -x*G(-x)*G(x) = pi/sin(pi*x),
+// we have
+// G(x) = pi/(sin(pi*x)*(-x)*G(-x))
+// since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
+// Hence, for x<0, signgam = sign(sin(pi*x)) and
+// lgamma(x) = log(|Gamma(x)|)
+// = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
+// Note: one should avoid computing pi*(-x) directly in the
+// computation of sin(pi*(-x)).
+//
+// 5. Special Cases
+// lgamma(2+s) ~ s*(1-Euler) for tiny s
+// lgamma(1)=lgamma(2)=0
+// lgamma(x) ~ -log(x) for tiny x
+// lgamma(0) = lgamma(inf) = inf
+// lgamma(-integer) = +-inf
+//
+//
+
+var _lgamA = [...]float64{
+ 7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
+ 3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
+ 6.73523010531292681824e-02, // 0x3FB13E001A5562A7
+ 2.05808084325167332806e-02, // 0x3F951322AC92547B
+ 7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
+ 2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
+ 1.19270763183362067845e-03, // 0x3F538A94116F3F5D
+ 5.10069792153511336608e-04, // 0x3F40B6C689B99C00
+ 2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
+ 1.08011567247583939954e-04, // 0x3F1C5088987DFB07
+ 2.52144565451257326939e-05, // 0x3EFA7074428CFA52
+ 4.48640949618915160150e-05, // 0x3F07858E90A45837
+}
+var _lgamR = [...]float64{
+ 1.0, // placeholder
+ 1.39200533467621045958e+00, // 0x3FF645A762C4AB74
+ 7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
+ 1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
+ 1.86459191715652901344e-02, // 0x3F9317EA742ED475
+ 7.77942496381893596434e-04, // 0x3F497DDACA41A95B
+ 7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
+}
+var _lgamS = [...]float64{
+ -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
+ 2.14982415960608852501e-01, // 0x3FCB848B36E20878
+ 3.25778796408930981787e-01, // 0x3FD4D98F4F139F59
+ 1.46350472652464452805e-01, // 0x3FC2BB9CBEE5F2F7
+ 2.66422703033638609560e-02, // 0x3F9B481C7E939961
+ 1.84028451407337715652e-03, // 0x3F5E26B67368F239
+ 3.19475326584100867617e-05, // 0x3F00BFECDD17E945
+}
+var _lgamT = [...]float64{
+ 4.83836122723810047042e-01, // 0x3FDEF72BC8EE38A2
+ -1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
+ 6.46249402391333854778e-02, // 0x3FB08B4294D5419B
+ -3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
+ 1.79706750811820387126e-02, // 0x3F9266E7970AF9EC
+ -1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
+ 6.10053870246291332635e-03, // 0x3F78FCE0E370E344
+ -3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
+ 2.25964780900612472250e-03, // 0x3F6282D32E15C915
+ -1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
+ 8.81081882437654011382e-04, // 0x3F4CDF0CEF61A8E9
+ -5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
+ 3.15632070903625950361e-04, // 0x3F34AF6D6C0EBBF7
+ -3.12754168375120860518e-04, // 0xBF347F24ECC38C38
+ 3.35529192635519073543e-04, // 0x3F35FD3EE8C2D3F4
+}
+var _lgamU = [...]float64{
+ -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
+ 6.32827064025093366517e-01, // 0x3FE4401E8B005DFF
+ 1.45492250137234768737e+00, // 0x3FF7475CD119BD6F
+ 9.77717527963372745603e-01, // 0x3FEF497644EA8450
+ 2.28963728064692451092e-01, // 0x3FCD4EAEF6010924
+ 1.33810918536787660377e-02, // 0x3F8B678BBF2BAB09
+}
+var _lgamV = [...]float64{
+ 1.0,
+ 2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
+ 2.12848976379893395361e+00, // 0x40010725A42B18F5
+ 7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
+ 1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
+ 3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
+}
+var _lgamW = [...]float64{
+ 4.18938533204672725052e-01, // 0x3FDACFE390C97D69
+ 8.33333333333329678849e-02, // 0x3FB555555555553B
+ -2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
+ 7.93650558643019558500e-04, // 0x3F4A019F98CF38B6
+ -5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
+ 8.36339918996282139126e-04, // 0x3F4B67BA4CDAD5D1
+ -1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
+}
+
+// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
+//
+// Special cases are:
+// Lgamma(+Inf) = +Inf
+// Lgamma(0) = +Inf
+// Lgamma(-integer) = +Inf
+// Lgamma(-Inf) = -Inf
+// Lgamma(NaN) = NaN
+func Lgamma(x float64) (lgamma float64, sign int) {
+ const (
+ Ymin = 1.461632144968362245
+ Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+ Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+ Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17
+ Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22
+ Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F
+ Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
+ // Tt = -(tail of Tf)
+ Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
+ )
+ // special cases
+ sign = 1
+ switch {
+ case IsNaN(x):
+ lgamma = x
+ return
+ case IsInf(x, 0):
+ lgamma = x
+ return
+ case x == 0:
+ lgamma = Inf(1)
+ return
+ }
+
+ neg := false
+ if x < 0 {
+ x = -x
+ neg = true
+ }
+
+ if x < Tiny { // if |x| < 2**-70, return -log(|x|)
+ if neg {
+ sign = -1
+ }
+ lgamma = -Log(x)
+ return
+ }
+ var nadj float64
+ if neg {
+ if x >= Two52 { // |x| >= 2**52, must be -integer
+ lgamma = Inf(1)
+ return
+ }
+ t := sinPi(x)
+ if t == 0 {
+ lgamma = Inf(1) // -integer
+ return
+ }
+ nadj = Log(Pi / Abs(t*x))
+ if t < 0 {
+ sign = -1
+ }
+ }
+
+ switch {
+ case x == 1 || x == 2: // purge off 1 and 2
+ lgamma = 0
+ return
+ case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
+ var y float64
+ var i int
+ if x <= 0.9 {
+ lgamma = -Log(x)
+ switch {
+ case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <= 0.9
+ y = 1 - x
+ i = 0
+ case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
+ y = x - (Tc - 1)
+ i = 1
+ default: // 0 < x < 0.2316
+ y = x
+ i = 2
+ }
+ } else {
+ lgamma = 0
+ switch {
+ case x >= (Ymin + 0.27): // 1.7316 <= x < 2
+ y = 2 - x
+ i = 0
+ case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
+ y = x - Tc
+ i = 1
+ default: // 0.9 < x < 1.2316
+ y = x - 1
+ i = 2
+ }
+ }
+ switch i {
+ case 0:
+ z := y * y
+ p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
+ p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
+ p := y*p1 + p2
+ lgamma += (p - 0.5*y)
+ case 1:
+ z := y * y
+ w := z * y
+ p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
+ p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
+ p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
+ p := z*p1 - (Tt - w*(p2+y*p3))
+ lgamma += (Tf + p)
+ case 2:
+ p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
+ p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
+ lgamma += (-0.5*y + p1/p2)
+ }
+ case x < 8: // 2 <= x < 8
+ i := int(x)
+ y := x - float64(i)
+ p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
+ q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
+ lgamma = 0.5*y + p/q
+ z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
+ switch i {
+ case 7:
+ z *= (y + 6)
+ fallthrough
+ case 6:
+ z *= (y + 5)
+ fallthrough
+ case 5:
+ z *= (y + 4)
+ fallthrough
+ case 4:
+ z *= (y + 3)
+ fallthrough
+ case 3:
+ z *= (y + 2)
+ lgamma += Log(z)
+ }
+ case x < Two58: // 8 <= x < 2**58
+ t := Log(x)
+ z := 1 / x
+ y := z * z
+ w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
+ lgamma = (x-0.5)*(t-1) + w
+ default: // 2**58 <= x <= Inf
+ lgamma = x * (Log(x) - 1)
+ }
+ if neg {
+ lgamma = nadj - lgamma
+ }
+ return
+}
+
+// sinPi(x) is a helper function for negative x
+func sinPi(x float64) float64 {
+ const (
+ Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+ Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+ )
+ if x < 0.25 {
+ return -Sin(Pi * x)
+ }
+
+ // argument reduction
+ z := Floor(x)
+ var n int
+ if z != x { // inexact
+ x = Mod(x, 2)
+ n = int(x * 4)
+ } else {
+ if x >= Two53 { // x must be even
+ x = 0
+ n = 0
+ } else {
+ if x < Two52 {
+ z = x + Two52 // exact
+ }
+ n = int(1 & Float64bits(z))
+ x = float64(n)
+ n <<= 2
+ }
+ }
+ switch n {
+ case 0:
+ x = Sin(Pi * x)
+ case 1, 2:
+ x = Cos(Pi * (0.5 - x))
+ case 3, 4:
+ x = Sin(Pi * (1 - x))
+ case 5, 6:
+ x = -Cos(Pi * (x - 1.5))
+ default:
+ x = Sin(Pi * (x - 2))
+ }
+ return -x
+}
diff --git a/src/math/log.go b/src/math/log.go
new file mode 100644
index 0000000..818f00a
--- /dev/null
+++ b/src/math/log.go
@@ -0,0 +1,123 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point logarithm.
+*/
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
+// and came with this notice. The go code is a simpler
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_log(x)
+// Return the logarithm of x
+//
+// Method :
+// 1. Argument Reduction: find k and f such that
+// x = 2**k * (1+f),
+// where sqrt(2)/2 < 1+f < sqrt(2) .
+//
+// 2. Approximation of log(1+f).
+// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+// = 2s + s*R
+// We use a special Reme algorithm on [0,0.1716] to generate
+// a polynomial of degree 14 to approximate R. The maximum error
+// of this polynomial approximation is bounded by 2**-58.45. In
+// other words,
+// 2 4 6 8 10 12 14
+// R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s +L6*s +L7*s
+// (the values of L1 to L7 are listed in the program) and
+// | 2 14 | -58.45
+// | L1*s +...+L7*s - R(z) | <= 2
+// | |
+// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+// In order to guarantee error in log below 1ulp, we compute log by
+// log(1+f) = f - s*(f - R) (if f is not too large)
+// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
+//
+// 3. Finally, log(x) = k*Ln2 + log(1+f).
+// = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
+// Here Ln2 is split into two floating point number:
+// Ln2_hi + Ln2_lo,
+// where n*Ln2_hi is always exact for |n| < 2000.
+//
+// Special cases:
+// log(x) is NaN with signal if x < 0 (including -INF) ;
+// log(+INF) is +INF; log(0) is -INF with signal;
+// log(NaN) is that NaN with no signal.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+
+// Log returns the natural logarithm of x.
+//
+// Special cases are:
+// Log(+Inf) = +Inf
+// Log(0) = -Inf
+// Log(x < 0) = NaN
+// Log(NaN) = NaN
+func Log(x float64) float64
+
+func log(x float64) float64 {
+ const (
+ Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
+ Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
+ L1 = 6.666666666666735130e-01 /* 3FE55555 55555593 */
+ L2 = 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
+ L3 = 2.857142874366239149e-01 /* 3FD24924 94229359 */
+ L4 = 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
+ L5 = 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
+ L6 = 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
+ L7 = 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
+ )
+
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 1):
+ return x
+ case x < 0:
+ return NaN()
+ case x == 0:
+ return Inf(-1)
+ }
+
+ // reduce
+ f1, ki := Frexp(x)
+ if f1 < Sqrt2/2 {
+ f1 *= 2
+ ki--
+ }
+ f := f1 - 1
+ k := float64(ki)
+
+ // compute
+ s := f / (2 + f)
+ s2 := s * s
+ s4 := s2 * s2
+ t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
+ t2 := s4 * (L2 + s4*(L4+s4*L6))
+ R := t1 + t2
+ hfsq := 0.5 * f * f
+ return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
+}
diff --git a/src/math/log10.go b/src/math/log10.go
new file mode 100644
index 0000000..95cfbf4
--- /dev/null
+++ b/src/math/log10.go
@@ -0,0 +1,22 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Log10 returns the decimal logarithm of x.
+// The special cases are the same as for Log.
+func Log10(x float64) float64
+
+func log10(x float64) float64 {
+ return Log(x) * (1 / Ln10)
+}
+
+// Log2 returns the binary logarithm of x.
+// The special cases are the same as for Log.
+func Log2(x float64) float64
+
+func log2(x float64) float64 {
+ frac, exp := Frexp(x)
+ return Log(frac)*(1/Ln2) + float64(exp)
+}
diff --git a/src/math/log10_386.s b/src/math/log10_386.s
new file mode 100644
index 0000000..2897f3c
--- /dev/null
+++ b/src/math/log10_386.s
@@ -0,0 +1,21 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Log10(x float64) float64
+TEXT ·Log10(SB),NOSPLIT,$0
+ FLDLG2 // F0=log10(2)
+ FMOVD x+0(FP), F0 // F0=x, F1=log10(2)
+ FYL2X // F0=log10(x)=log2(x)*log10(2)
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Log2(x float64) float64
+TEXT ·Log2(SB),NOSPLIT,$0
+ FLD1 // F0=1
+ FMOVD x+0(FP), F0 // F0=x, F1=1
+ FYL2X // F0=log2(x)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/log10_amd64.s b/src/math/log10_amd64.s
new file mode 100644
index 0000000..8382ba7
--- /dev/null
+++ b/src/math/log10_amd64.s
@@ -0,0 +1,11 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Log10(SB),NOSPLIT,$0
+ JMP ·log10(SB)
+
+TEXT ·Log2(SB),NOSPLIT,$0
+ JMP ·log2(SB)
diff --git a/src/math/log10_amd64p32.s b/src/math/log10_amd64p32.s
new file mode 100644
index 0000000..bf43841
--- /dev/null
+++ b/src/math/log10_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "log10_amd64.s"
diff --git a/src/math/log10_arm.s b/src/math/log10_arm.s
new file mode 100644
index 0000000..dbcb835
--- /dev/null
+++ b/src/math/log10_arm.s
@@ -0,0 +1,11 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Log10(SB),NOSPLIT,$0
+ B ·log10(SB)
+
+TEXT ·Log2(SB),NOSPLIT,$0
+ B ·log2(SB)
diff --git a/src/math/log1p.go b/src/math/log1p.go
new file mode 100644
index 0000000..12b9868
--- /dev/null
+++ b/src/math/log1p.go
@@ -0,0 +1,200 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
+// and came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// double log1p(double x)
+//
+// Method :
+// 1. Argument Reduction: find k and f such that
+// 1+x = 2**k * (1+f),
+// where sqrt(2)/2 < 1+f < sqrt(2) .
+//
+// Note. If k=0, then f=x is exact. However, if k!=0, then f
+// may not be representable exactly. In that case, a correction
+// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
+// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
+// and add back the correction term c/u.
+// (Note: when x > 2**53, one can simply return log(x))
+//
+// 2. Approximation of log1p(f).
+// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+// = 2s + s*R
+// We use a special Reme algorithm on [0,0.1716] to generate
+// a polynomial of degree 14 to approximate R The maximum error
+// of this polynomial approximation is bounded by 2**-58.45. In
+// other words,
+// 2 4 6 8 10 12 14
+// R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
+// (the values of Lp1 to Lp7 are listed in the program)
+// and
+// | 2 14 | -58.45
+// | Lp1*s +...+Lp7*s - R(z) | <= 2
+// | |
+// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+// In order to guarantee error in log below 1ulp, we compute log
+// by
+// log1p(f) = f - (hfsq - s*(hfsq+R)).
+//
+// 3. Finally, log1p(x) = k*ln2 + log1p(f).
+// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+// Here ln2 is split into two floating point number:
+// ln2_hi + ln2_lo,
+// where n*ln2_hi is always exact for |n| < 2000.
+//
+// Special cases:
+// log1p(x) is NaN with signal if x < -1 (including -INF) ;
+// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
+// log1p(NaN) is that NaN with no signal.
+//
+// Accuracy:
+// according to an error analysis, the error is always less than
+// 1 ulp (unit in the last place).
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+//
+// Note: Assuming log() return accurate answer, the following
+// algorithm can be used to compute log1p(x) to within a few ULP:
+//
+// u = 1+x;
+// if(u==1.0) return x ; else
+// return log(u)*(x/(u-1.0));
+//
+// See HP-15C Advanced Functions Handbook, p.193.
+
+// Log1p returns the natural logarithm of 1 plus its argument x.
+// It is more accurate than Log(1 + x) when x is near zero.
+//
+// Special cases are:
+// Log1p(+Inf) = +Inf
+// Log1p(±0) = ±0
+// Log1p(-1) = -Inf
+// Log1p(x < -1) = NaN
+// Log1p(NaN) = NaN
+func Log1p(x float64) float64
+
+func log1p(x float64) float64 {
+ const (
+ Sqrt2M1 = 4.142135623730950488017e-01 // Sqrt(2)-1 = 0x3fda827999fcef34
+ Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
+ Small = 1.0 / (1 << 29) // 2**-29 = 0x3e20000000000000
+ Tiny = 1.0 / (1 << 54) // 2**-54
+ Two53 = 1 << 53 // 2**53
+ Ln2Hi = 6.93147180369123816490e-01 // 3fe62e42fee00000
+ Ln2Lo = 1.90821492927058770002e-10 // 3dea39ef35793c76
+ Lp1 = 6.666666666666735130e-01 // 3FE5555555555593
+ Lp2 = 3.999999999940941908e-01 // 3FD999999997FA04
+ Lp3 = 2.857142874366239149e-01 // 3FD2492494229359
+ Lp4 = 2.222219843214978396e-01 // 3FCC71C51D8E78AF
+ Lp5 = 1.818357216161805012e-01 // 3FC7466496CB03DE
+ Lp6 = 1.531383769920937332e-01 // 3FC39A09D078C69F
+ Lp7 = 1.479819860511658591e-01 // 3FC2F112DF3E5244
+ )
+
+ // special cases
+ switch {
+ case x < -1 || IsNaN(x): // includes -Inf
+ return NaN()
+ case x == -1:
+ return Inf(-1)
+ case IsInf(x, 1):
+ return Inf(1)
+ }
+
+ absx := x
+ if absx < 0 {
+ absx = -absx
+ }
+
+ var f float64
+ var iu uint64
+ k := 1
+ if absx < Sqrt2M1 { // |x| < Sqrt(2)-1
+ if absx < Small { // |x| < 2**-29
+ if absx < Tiny { // |x| < 2**-54
+ return x
+ }
+ return x - x*x*0.5
+ }
+ if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
+ // (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
+ k = 0
+ f = x
+ iu = 1
+ }
+ }
+ var c float64
+ if k != 0 {
+ var u float64
+ if absx < Two53 { // 1<<53
+ u = 1.0 + x
+ iu = Float64bits(u)
+ k = int((iu >> 52) - 1023)
+ if k > 0 {
+ c = 1.0 - (u - x)
+ } else {
+ c = x - (u - 1.0) // correction term
+ c /= u
+ }
+ } else {
+ u = x
+ iu = Float64bits(u)
+ k = int((iu >> 52) - 1023)
+ c = 0
+ }
+ iu &= 0x000fffffffffffff
+ if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
+ u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
+ } else {
+ k += 1
+ u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
+ iu = (0x0010000000000000 - iu) >> 2
+ }
+ f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
+ }
+ hfsq := 0.5 * f * f
+ var s, R, z float64
+ if iu == 0 { // |f| < 2**-20
+ if f == 0 {
+ if k == 0 {
+ return 0
+ } else {
+ c += float64(k) * Ln2Lo
+ return float64(k)*Ln2Hi + c
+ }
+ }
+ R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
+ if k == 0 {
+ return f - R
+ }
+ return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
+ }
+ s = f / (2.0 + f)
+ z = s * s
+ R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
+ if k == 0 {
+ return f - (hfsq - s*(hfsq+R))
+ }
+ return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
+}
diff --git a/src/math/log1p_386.s b/src/math/log1p_386.s
new file mode 100644
index 0000000..1c2d683
--- /dev/null
+++ b/src/math/log1p_386.s
@@ -0,0 +1,27 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Log1p(x float64) float64
+TEXT ·Log1p(SB),NOSPLIT,$0
+ FMOVD $(2.928932188134524e-01), F0
+ FMOVD x+0(FP), F0 // F0=x, F1=1-sqrt(2)/2 = 0.29289321881345247559915564
+ FABS // F0=|x|, F1=1-sqrt(2)/2
+ FUCOMPP F0, F1 // compare F0 to F1
+ FSTSW AX
+ FLDLN2 // F0=log(2)
+ ANDW $0x0100, AX
+ JEQ use_fyl2x // jump if F0 >= F1
+ FMOVD x+0(FP), F0 // F0=x, F1=log(2)
+ FYL2XP1 // F0=log(1+x)=log2(1+x)*log(2)
+ FMOVDP F0, ret+8(FP)
+ RET
+use_fyl2x:
+ FLD1 // F0=1, F2=log(2)
+ FADDD x+0(FP), F0 // F0=1+x, F1=log(2)
+ FYL2X // F0=log(1+x)=log2(1+x)*log(2)
+ FMOVDP F0, ret+8(FP)
+ RET
+
diff --git a/src/math/log1p_amd64.s b/src/math/log1p_amd64.s
new file mode 100644
index 0000000..1e58fb1
--- /dev/null
+++ b/src/math/log1p_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Log1p(SB),NOSPLIT,$0
+ JMP ·log1p(SB)
diff --git a/src/math/log1p_amd64p32.s b/src/math/log1p_amd64p32.s
new file mode 100644
index 0000000..a14b5e3
--- /dev/null
+++ b/src/math/log1p_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "log1p_amd64.s"
diff --git a/src/math/log1p_arm.s b/src/math/log1p_arm.s
new file mode 100644
index 0000000..95d5496
--- /dev/null
+++ b/src/math/log1p_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Log1p(SB),NOSPLIT,$0
+ B ·log1p(SB)
diff --git a/src/math/log_386.s b/src/math/log_386.s
new file mode 100644
index 0000000..ff998af
--- /dev/null
+++ b/src/math/log_386.s
@@ -0,0 +1,13 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Log(x float64) float64
+TEXT ·Log(SB),NOSPLIT,$0
+ FLDLN2 // F0=log(2)
+ FMOVD x+0(FP), F0 // F0=x, F1=log(2)
+ FYL2X // F0=log(x)=log2(x)*log(2)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/log_amd64.s b/src/math/log_amd64.s
new file mode 100644
index 0000000..84c60ab
--- /dev/null
+++ b/src/math/log_amd64.s
@@ -0,0 +1,111 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+#define HSqrt2 7.07106781186547524401e-01 // sqrt(2)/2
+#define Ln2Hi 6.93147180369123816490e-01 // 0x3fe62e42fee00000
+#define Ln2Lo 1.90821492927058770002e-10 // 0x3dea39ef35793c76
+#define L1 6.666666666666735130e-01 // 0x3FE5555555555593
+#define L2 3.999999999940941908e-01 // 0x3FD999999997FA04
+#define L3 2.857142874366239149e-01 // 0x3FD2492494229359
+#define L4 2.222219843214978396e-01 // 0x3FCC71C51D8E78AF
+#define L5 1.818357216161805012e-01 // 0x3FC7466496CB03DE
+#define L6 1.531383769920937332e-01 // 0x3FC39A09D078C69F
+#define L7 1.479819860511658591e-01 // 0x3FC2F112DF3E5244
+#define NaN 0x7FF8000000000001
+#define NegInf 0xFFF0000000000000
+#define PosInf 0x7FF0000000000000
+
+// func Log(x float64) float64
+TEXT ·Log(SB),NOSPLIT,$0
+ // test bits for special cases
+ MOVQ x+0(FP), BX
+ MOVQ $~(1<<63), AX // sign bit mask
+ ANDQ BX, AX
+ JEQ isZero
+ MOVQ $0, AX
+ CMPQ AX, BX
+ JGT isNegative
+ MOVQ $PosInf, AX
+ CMPQ AX, BX
+ JLE isInfOrNaN
+ // f1, ki := math.Frexp(x); k := float64(ki)
+ MOVQ BX, X0
+ MOVQ $0x000FFFFFFFFFFFFF, AX
+ MOVQ AX, X2
+ ANDPD X0, X2
+ MOVSD $0.5, X0 // 0x3FE0000000000000
+ ORPD X0, X2 // X2= f1
+ SHRQ $52, BX
+ ANDL $0x7FF, BX
+ SUBL $0x3FE, BX
+ CVTSL2SD BX, X1 // x1= k, x2= f1
+ // if f1 < math.Sqrt2/2 { k -= 1; f1 *= 2 }
+ MOVSD $HSqrt2, X0 // x0= 0.7071, x1= k, x2= f1
+ CMPSD X2, X0, 5 // cmpnlt; x0= 0 or ^0, x1= k, x2 = f1
+ MOVSD $1.0, X3 // x0= 0 or ^0, x1= k, x2 = f1, x3= 1
+ ANDPD X0, X3 // x0= 0 or ^0, x1= k, x2 = f1, x3= 0 or 1
+ SUBSD X3, X1 // x0= 0 or ^0, x1= k, x2 = f1, x3= 0 or 1
+ MOVSD $1.0, X0 // x0= 1, x1= k, x2= f1, x3= 0 or 1
+ ADDSD X0, X3 // x0= 1, x1= k, x2= f1, x3= 1 or 2
+ MULSD X3, X2 // x0= 1, x1= k, x2= f1
+ // f := f1 - 1
+ SUBSD X0, X2 // x1= k, x2= f
+ // s := f / (2 + f)
+ MOVSD $2.0, X0
+ ADDSD X2, X0
+ MOVAPD X2, X3
+ DIVSD X0, X3 // x1=k, x2= f, x3= s
+ // s2 := s * s
+ MOVAPD X3, X4 // x1= k, x2= f, x3= s
+ MULSD X4, X4 // x1= k, x2= f, x3= s, x4= s2
+ // s4 := s2 * s2
+ MOVAPD X4, X5 // x1= k, x2= f, x3= s, x4= s2
+ MULSD X5, X5 // x1= k, x2= f, x3= s, x4= s2, x5= s4
+ // t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
+ MOVSD $L7, X6
+ MULSD X5, X6
+ ADDSD $L5, X6
+ MULSD X5, X6
+ ADDSD $L3, X6
+ MULSD X5, X6
+ ADDSD $L1, X6
+ MULSD X6, X4 // x1= k, x2= f, x3= s, x4= t1, x5= s4
+ // t2 := s4 * (L2 + s4*(L4+s4*L6))
+ MOVSD $L6, X6
+ MULSD X5, X6
+ ADDSD $L4, X6
+ MULSD X5, X6
+ ADDSD $L2, X6
+ MULSD X6, X5 // x1= k, x2= f, x3= s, x4= t1, x5= t2
+ // R := t1 + t2
+ ADDSD X5, X4 // x1= k, x2= f, x3= s, x4= R
+ // hfsq := 0.5 * f * f
+ MOVSD $0.5, X0
+ MULSD X2, X0
+ MULSD X2, X0 // x0= hfsq, x1= k, x2= f, x3= s, x4= R
+ // return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
+ ADDSD X0, X4 // x0= hfsq, x1= k, x2= f, x3= s, x4= hfsq+R
+ MULSD X4, X3 // x0= hfsq, x1= k, x2= f, x3= s*(hfsq+R)
+ MOVSD $Ln2Lo, X4
+ MULSD X1, X4 // x4= k*Ln2Lo
+ ADDSD X4, X3 // x0= hfsq, x1= k, x2= f, x3= s*(hfsq+R)+k*Ln2Lo
+ SUBSD X3, X0 // x0= hfsq-(s*(hfsq+R)+k*Ln2Lo), x1= k, x2= f
+ SUBSD X2, X0 // x0= (hfsq-(s*(hfsq+R)+k*Ln2Lo))-f, x1= k
+ MULSD $Ln2Hi, X1 // x0= (hfsq-(s*(hfsq+R)+k*Ln2Lo))-f, x1= k*Ln2Hi
+ SUBSD X0, X1 // x1= k*Ln2Hi-((hfsq-(s*(hfsq+R)+k*Ln2Lo))-f)
+ MOVSD X1, ret+8(FP)
+ RET
+isInfOrNaN:
+ MOVQ BX, ret+8(FP) // +Inf or NaN, return x
+ RET
+isNegative:
+ MOVQ $NaN, AX
+ MOVQ AX, ret+8(FP) // return NaN
+ RET
+isZero:
+ MOVQ $NegInf, AX
+ MOVQ AX, ret+8(FP) // return -Inf
+ RET
diff --git a/src/math/log_amd64p32.s b/src/math/log_amd64p32.s
new file mode 100644
index 0000000..5058d60
--- /dev/null
+++ b/src/math/log_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "log_amd64.s"
diff --git a/src/math/log_arm.s b/src/math/log_arm.s
new file mode 100644
index 0000000..e21d036
--- /dev/null
+++ b/src/math/log_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Log(SB),NOSPLIT,$0
+ B ·log(SB)
diff --git a/src/math/logb.go b/src/math/logb.go
new file mode 100644
index 0000000..f2769d4
--- /dev/null
+++ b/src/math/logb.go
@@ -0,0 +1,50 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Logb returns the binary exponent of x.
+//
+// Special cases are:
+// Logb(±Inf) = +Inf
+// Logb(0) = -Inf
+// Logb(NaN) = NaN
+func Logb(x float64) float64 {
+ // special cases
+ switch {
+ case x == 0:
+ return Inf(-1)
+ case IsInf(x, 0):
+ return Inf(1)
+ case IsNaN(x):
+ return x
+ }
+ return float64(ilogb(x))
+}
+
+// Ilogb returns the binary exponent of x as an integer.
+//
+// Special cases are:
+// Ilogb(±Inf) = MaxInt32
+// Ilogb(0) = MinInt32
+// Ilogb(NaN) = MaxInt32
+func Ilogb(x float64) int {
+ // special cases
+ switch {
+ case x == 0:
+ return MinInt32
+ case IsNaN(x):
+ return MaxInt32
+ case IsInf(x, 0):
+ return MaxInt32
+ }
+ return ilogb(x)
+}
+
+// logb returns the binary exponent of x. It assumes x is finite and
+// non-zero.
+func ilogb(x float64) int {
+ x, exp := normalize(x)
+ return int((Float64bits(x)>>shift)&mask) - bias + exp
+}
diff --git a/src/math/mod.go b/src/math/mod.go
new file mode 100644
index 0000000..e1a414e
--- /dev/null
+++ b/src/math/mod.go
@@ -0,0 +1,50 @@
+// Copyright 2009-2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point mod function.
+*/
+
+// Mod returns the floating-point remainder of x/y.
+// The magnitude of the result is less than y and its
+// sign agrees with that of x.
+//
+// Special cases are:
+// Mod(±Inf, y) = NaN
+// Mod(NaN, y) = NaN
+// Mod(x, 0) = NaN
+// Mod(x, ±Inf) = x
+// Mod(x, NaN) = NaN
+func Mod(x, y float64) float64
+
+func mod(x, y float64) float64 {
+ if y == 0 || IsInf(x, 0) || IsNaN(x) || IsNaN(y) {
+ return NaN()
+ }
+ if y < 0 {
+ y = -y
+ }
+
+ yfr, yexp := Frexp(y)
+ sign := false
+ r := x
+ if x < 0 {
+ r = -x
+ sign = true
+ }
+
+ for r >= y {
+ rfr, rexp := Frexp(r)
+ if rfr < yfr {
+ rexp = rexp - 1
+ }
+ r = r - Ldexp(y, rexp-yexp)
+ }
+ if sign {
+ r = -r
+ }
+ return r
+}
diff --git a/src/math/mod_386.s b/src/math/mod_386.s
new file mode 100644
index 0000000..10ad98b
--- /dev/null
+++ b/src/math/mod_386.s
@@ -0,0 +1,17 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Mod(x, y float64) float64
+TEXT ·Mod(SB),NOSPLIT,$0
+ FMOVD y+8(FP), F0 // F0=y
+ FMOVD x+0(FP), F0 // F0=x, F1=y
+ FPREM // F0=reduced_x, F1=y
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=x-q*y
+ FMOVDP F0, ret+16(FP)
+ RET
diff --git a/src/math/mod_amd64.s b/src/math/mod_amd64.s
new file mode 100644
index 0000000..f99dbe2
--- /dev/null
+++ b/src/math/mod_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Mod(SB),NOSPLIT,$0
+ JMP ·mod(SB)
diff --git a/src/math/mod_amd64p32.s b/src/math/mod_amd64p32.s
new file mode 100644
index 0000000..c1b2311
--- /dev/null
+++ b/src/math/mod_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "mod_amd64.s"
diff --git a/src/math/mod_arm.s b/src/math/mod_arm.s
new file mode 100644
index 0000000..5afb359
--- /dev/null
+++ b/src/math/mod_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Mod(SB),NOSPLIT,$0
+ B ·mod(SB)
diff --git a/src/math/modf.go b/src/math/modf.go
new file mode 100644
index 0000000..1e8376a
--- /dev/null
+++ b/src/math/modf.go
@@ -0,0 +1,34 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Modf returns integer and fractional floating-point numbers
+// that sum to f. Both values have the same sign as f.
+//
+// Special cases are:
+// Modf(±Inf) = ±Inf, NaN
+// Modf(NaN) = NaN, NaN
+func Modf(f float64) (int float64, frac float64)
+
+func modf(f float64) (int float64, frac float64) {
+ if f < 1 {
+ if f < 0 {
+ int, frac = Modf(-f)
+ return -int, -frac
+ }
+ return 0, f
+ }
+
+ x := Float64bits(f)
+ e := uint(x>>shift)&mask - bias
+
+ // Keep the top 12+e bits, the integer part; clear the rest.
+ if e < 64-12 {
+ x &^= 1<<(64-12-e) - 1
+ }
+ int = Float64frombits(x)
+ frac = f - int
+ return
+}
diff --git a/src/math/modf_386.s b/src/math/modf_386.s
new file mode 100644
index 0000000..3debd3b
--- /dev/null
+++ b/src/math/modf_386.s
@@ -0,0 +1,21 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Modf(f float64) (int float64, frac float64)
+TEXT ·Modf(SB),NOSPLIT,$0
+ FMOVD f+0(FP), F0 // F0=f
+ FMOVD F0, F1 // F0=f, F1=f
+ FSTCW -2(SP) // save old Control Word
+ MOVW -2(SP), AX
+ ORW $0x0c00, AX // Rounding Control set to truncate
+ MOVW AX, -4(SP) // store new Control Word
+ FLDCW -4(SP) // load new Control Word
+ FRNDINT // F0=trunc(f), F1=f
+ FLDCW -2(SP) // load old Control Word
+ FSUBD F0, F1 // F0=trunc(f), F1=f-trunc(f)
+ FMOVDP F0, int+8(FP) // F0=f-trunc(f)
+ FMOVDP F0, frac+16(FP)
+ RET
diff --git a/src/math/modf_amd64.s b/src/math/modf_amd64.s
new file mode 100644
index 0000000..701cf72
--- /dev/null
+++ b/src/math/modf_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Modf(SB),NOSPLIT,$0
+ JMP ·modf(SB)
diff --git a/src/math/modf_amd64p32.s b/src/math/modf_amd64p32.s
new file mode 100644
index 0000000..5508c25
--- /dev/null
+++ b/src/math/modf_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "modf_amd64.s"
diff --git a/src/math/modf_arm.s b/src/math/modf_arm.s
new file mode 100644
index 0000000..ea3c8dc
--- /dev/null
+++ b/src/math/modf_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Modf(SB),NOSPLIT,$0
+ B ·modf(SB)
diff --git a/src/math/nextafter.go b/src/math/nextafter.go
new file mode 100644
index 0000000..bbb1399
--- /dev/null
+++ b/src/math/nextafter.go
@@ -0,0 +1,47 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Nextafter32 returns the next representable float32 value after x towards y.
+// Special cases:
+// Nextafter32(x, x) = x
+// Nextafter32(NaN, y) = NaN
+// Nextafter32(x, NaN) = NaN
+func Nextafter32(x, y float32) (r float32) {
+ switch {
+ case IsNaN(float64(x)) || IsNaN(float64(y)): // special case
+ r = float32(NaN())
+ case x == y:
+ r = x
+ case x == 0:
+ r = float32(Copysign(float64(Float32frombits(1)), float64(y)))
+ case (y > x) == (x > 0):
+ r = Float32frombits(Float32bits(x) + 1)
+ default:
+ r = Float32frombits(Float32bits(x) - 1)
+ }
+ return
+}
+
+// Nextafter returns the next representable float64 value after x towards y.
+// Special cases:
+// Nextafter64(x, x) = x
+// Nextafter64(NaN, y) = NaN
+// Nextafter64(x, NaN) = NaN
+func Nextafter(x, y float64) (r float64) {
+ switch {
+ case IsNaN(x) || IsNaN(y): // special case
+ r = NaN()
+ case x == y:
+ r = x
+ case x == 0:
+ r = Copysign(Float64frombits(1), y)
+ case (y > x) == (x > 0):
+ r = Float64frombits(Float64bits(x) + 1)
+ default:
+ r = Float64frombits(Float64bits(x) - 1)
+ }
+ return
+}
diff --git a/src/math/pow.go b/src/math/pow.go
new file mode 100644
index 0000000..77af256
--- /dev/null
+++ b/src/math/pow.go
@@ -0,0 +1,137 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+func isOddInt(x float64) bool {
+ xi, xf := Modf(x)
+ return xf == 0 && int64(xi)&1 == 1
+}
+
+// Special cases taken from FreeBSD's /usr/src/lib/msun/src/e_pow.c
+// updated by IEEE Std. 754-2008 "Section 9.2.1 Special values".
+
+// Pow returns x**y, the base-x exponential of y.
+//
+// Special cases are (in order):
+// Pow(x, ±0) = 1 for any x
+// Pow(1, y) = 1 for any y
+// Pow(x, 1) = x for any x
+// Pow(NaN, y) = NaN
+// Pow(x, NaN) = NaN
+// Pow(±0, y) = ±Inf for y an odd integer < 0
+// Pow(±0, -Inf) = +Inf
+// Pow(±0, +Inf) = +0
+// Pow(±0, y) = +Inf for finite y < 0 and not an odd integer
+// Pow(±0, y) = ±0 for y an odd integer > 0
+// Pow(±0, y) = +0 for finite y > 0 and not an odd integer
+// Pow(-1, ±Inf) = 1
+// Pow(x, +Inf) = +Inf for |x| > 1
+// Pow(x, -Inf) = +0 for |x| > 1
+// Pow(x, +Inf) = +0 for |x| < 1
+// Pow(x, -Inf) = +Inf for |x| < 1
+// Pow(+Inf, y) = +Inf for y > 0
+// Pow(+Inf, y) = +0 for y < 0
+// Pow(-Inf, y) = Pow(-0, -y)
+// Pow(x, y) = NaN for finite x < 0 and finite non-integer y
+func Pow(x, y float64) float64 {
+ switch {
+ case y == 0 || x == 1:
+ return 1
+ case y == 1:
+ return x
+ case y == 0.5:
+ return Sqrt(x)
+ case y == -0.5:
+ return 1 / Sqrt(x)
+ case IsNaN(x) || IsNaN(y):
+ return NaN()
+ case x == 0:
+ switch {
+ case y < 0:
+ if isOddInt(y) {
+ return Copysign(Inf(1), x)
+ }
+ return Inf(1)
+ case y > 0:
+ if isOddInt(y) {
+ return x
+ }
+ return 0
+ }
+ case IsInf(y, 0):
+ switch {
+ case x == -1:
+ return 1
+ case (Abs(x) < 1) == IsInf(y, 1):
+ return 0
+ default:
+ return Inf(1)
+ }
+ case IsInf(x, 0):
+ if IsInf(x, -1) {
+ return Pow(1/x, -y) // Pow(-0, -y)
+ }
+ switch {
+ case y < 0:
+ return 0
+ case y > 0:
+ return Inf(1)
+ }
+ }
+
+ absy := y
+ flip := false
+ if absy < 0 {
+ absy = -absy
+ flip = true
+ }
+ yi, yf := Modf(absy)
+ if yf != 0 && x < 0 {
+ return NaN()
+ }
+ if yi >= 1<<63 {
+ return Exp(y * Log(x))
+ }
+
+ // ans = a1 * 2**ae (= 1 for now).
+ a1 := 1.0
+ ae := 0
+
+ // ans *= x**yf
+ if yf != 0 {
+ if yf > 0.5 {
+ yf--
+ yi++
+ }
+ a1 = Exp(yf * Log(x))
+ }
+
+ // ans *= x**yi
+ // by multiplying in successive squarings
+ // of x according to bits of yi.
+ // accumulate powers of two into exp.
+ x1, xe := Frexp(x)
+ for i := int64(yi); i != 0; i >>= 1 {
+ if i&1 == 1 {
+ a1 *= x1
+ ae += xe
+ }
+ x1 *= x1
+ xe <<= 1
+ if x1 < .5 {
+ x1 += x1
+ xe--
+ }
+ }
+
+ // ans = a1*2**ae
+ // if flip { ans = 1 / ans }
+ // but in the opposite order
+ if flip {
+ a1 = 1 / a1
+ ae = -ae
+ }
+ return Ldexp(a1, ae)
+}
diff --git a/src/math/pow10.go b/src/math/pow10.go
new file mode 100644
index 0000000..f5ad28b
--- /dev/null
+++ b/src/math/pow10.go
@@ -0,0 +1,40 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// This table might overflow 127-bit exponent representations.
+// In that case, truncate it after 1.0e38.
+var pow10tab [70]float64
+
+// Pow10 returns 10**e, the base-10 exponential of e.
+//
+// Special cases are:
+// Pow10(e) = +Inf for e > 309
+// Pow10(e) = 0 for e < -324
+func Pow10(e int) float64 {
+ if e <= -325 {
+ return 0
+ } else if e > 309 {
+ return Inf(1)
+ }
+
+ if e < 0 {
+ return 1 / Pow10(-e)
+ }
+ if e < len(pow10tab) {
+ return pow10tab[e]
+ }
+ m := e / 2
+ return Pow10(m) * Pow10(e-m)
+}
+
+func init() {
+ pow10tab[0] = 1.0e0
+ pow10tab[1] = 1.0e1
+ for i := 2; i < len(pow10tab); i++ {
+ m := i / 2
+ pow10tab[i] = pow10tab[m] * pow10tab[i-m]
+ }
+}
diff --git a/src/math/rand/example_test.go b/src/math/rand/example_test.go
new file mode 100644
index 0000000..f429914
--- /dev/null
+++ b/src/math/rand/example_test.go
@@ -0,0 +1,97 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand_test
+
+import (
+ "fmt"
+ "math/rand"
+ "os"
+ "text/tabwriter"
+)
+
+// These tests serve as an example but also make sure we don't change
+// the output of the random number generator when given a fixed seed.
+
+func Example() {
+ rand.Seed(42) // Try changing this number!
+ answers := []string{
+ "It is certain",
+ "It is decidedly so",
+ "Without a doubt",
+ "Yes definitely",
+ "You may rely on it",
+ "As I see it yes",
+ "Most likely",
+ "Outlook good",
+ "Yes",
+ "Signs point to yes",
+ "Reply hazy try again",
+ "Ask again later",
+ "Better not tell you now",
+ "Cannot predict now",
+ "Concentrate and ask again",
+ "Don't count on it",
+ "My reply is no",
+ "My sources say no",
+ "Outlook not so good",
+ "Very doubtful",
+ }
+ fmt.Println("Magic 8-Ball says:", answers[rand.Intn(len(answers))])
+ // Output: Magic 8-Ball says: As I see it yes
+}
+
+// This example shows the use of each of the methods on a *Rand.
+// The use of the global functions is the same, without the receiver.
+func Example_rand() {
+ // Create and seed the generator.
+ // Typically a non-fixed seed should be used, such as time.Now().UnixNano().
+ // Using a fixed seed will produce the same output on every run.
+ r := rand.New(rand.NewSource(99))
+
+ // The tabwriter here helps us generate aligned output.
+ w := tabwriter.NewWriter(os.Stdout, 1, 1, 1, ' ', 0)
+ defer w.Flush()
+ show := func(name string, v1, v2, v3 interface{}) {
+ fmt.Fprintf(w, "%s\t%v\t%v\t%v\n", name, v1, v2, v3)
+ }
+
+ // Float32 and Float64 values are in [0, 1).
+ show("Float32", r.Float32(), r.Float32(), r.Float32())
+ show("Float64", r.Float64(), r.Float64(), r.Float64())
+
+ // ExpFloat64 values have an average of 1 but decay exponentially.
+ show("ExpFloat64", r.ExpFloat64(), r.ExpFloat64(), r.ExpFloat64())
+
+ // NormFloat64 values have an average of 0 and a standard deviation of 1.
+ show("NormFloat64", r.NormFloat64(), r.NormFloat64(), r.NormFloat64())
+
+ // Int31, Int63, and Uint32 generate values of the given width.
+ // The Int method (not shown) is like either Int31 or Int63
+ // depending on the size of 'int'.
+ show("Int31", r.Int31(), r.Int31(), r.Int31())
+ show("Int63", r.Int63(), r.Int63(), r.Int63())
+ show("Uint32", r.Int63(), r.Int63(), r.Int63())
+
+ // Intn, Int31n, and Int63n limit their output to be < n.
+ // They do so more carefully than using r.Int()%n.
+ show("Intn(10)", r.Intn(10), r.Intn(10), r.Intn(10))
+ show("Int31n(10)", r.Int31n(10), r.Int31n(10), r.Int31n(10))
+ show("Int63n(10)", r.Int63n(10), r.Int63n(10), r.Int63n(10))
+
+ // Perm generates a random permutation of the numbers [0, n).
+ show("Perm", r.Perm(5), r.Perm(5), r.Perm(5))
+ // Output:
+ // Float32 0.2635776 0.6358173 0.6718283
+ // Float64 0.628605430454327 0.4504798828572669 0.9562755949377957
+ // ExpFloat64 0.3362240648200941 1.4256072328483647 0.24354758816173044
+ // NormFloat64 0.17233959114940064 1.577014951434847 0.04259129641113857
+ // Int31 1501292890 1486668269 182840835
+ // Int63 3546343826724305832 5724354148158589552 5239846799706671610
+ // Uint32 5927547564735367388 637072299495207830 4128311955958246186
+ // Intn(10) 1 2 5
+ // Int31n(10) 4 7 8
+ // Int63n(10) 7 6 3
+ // Perm [1 4 2 3 0] [4 2 1 3 0] [1 2 4 0 3]
+}
diff --git a/src/math/rand/exp.go b/src/math/rand/exp.go
new file mode 100644
index 0000000..4bc110f
--- /dev/null
+++ b/src/math/rand/exp.go
@@ -0,0 +1,222 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "math"
+)
+
+/*
+ * Exponential distribution
+ *
+ * See "The Ziggurat Method for Generating Random Variables"
+ * (Marsaglia & Tsang, 2000)
+ * http://www.jstatsoft.org/v05/i08/paper [pdf]
+ */
+
+const (
+ re = 7.69711747013104972
+)
+
+// ExpFloat64 returns an exponentially distributed float64 in the range
+// (0, +math.MaxFloat64] with an exponential distribution whose rate parameter
+// (lambda) is 1 and whose mean is 1/lambda (1).
+// To produce a distribution with a different rate parameter,
+// callers can adjust the output using:
+//
+// sample = ExpFloat64() / desiredRateParameter
+//
+func (r *Rand) ExpFloat64() float64 {
+ for {
+ j := r.Uint32()
+ i := j & 0xFF
+ x := float64(j) * float64(we[i])
+ if j < ke[i] {
+ return x
+ }
+ if i == 0 {
+ return re - math.Log(r.Float64())
+ }
+ if fe[i]+float32(r.Float64())*(fe[i-1]-fe[i]) < float32(math.Exp(-x)) {
+ return x
+ }
+ }
+}
+
+var ke = [256]uint32{
+ 0xe290a139, 0x0, 0x9beadebc, 0xc377ac71, 0xd4ddb990,
+ 0xde893fb8, 0xe4a8e87c, 0xe8dff16a, 0xebf2deab, 0xee49a6e8,
+ 0xf0204efd, 0xf19bdb8e, 0xf2d458bb, 0xf3da104b, 0xf4b86d78,
+ 0xf577ad8a, 0xf61de83d, 0xf6afb784, 0xf730a573, 0xf7a37651,
+ 0xf80a5bb6, 0xf867189d, 0xf8bb1b4f, 0xf9079062, 0xf94d70ca,
+ 0xf98d8c7d, 0xf9c8928a, 0xf9ff175b, 0xfa319996, 0xfa6085f8,
+ 0xfa8c3a62, 0xfab5084e, 0xfadb36c8, 0xfaff0410, 0xfb20a6ea,
+ 0xfb404fb4, 0xfb5e2951, 0xfb7a59e9, 0xfb95038c, 0xfbae44ba,
+ 0xfbc638d8, 0xfbdcf892, 0xfbf29a30, 0xfc0731df, 0xfc1ad1ed,
+ 0xfc2d8b02, 0xfc3f6c4d, 0xfc5083ac, 0xfc60ddd1, 0xfc708662,
+ 0xfc7f8810, 0xfc8decb4, 0xfc9bbd62, 0xfca9027c, 0xfcb5c3c3,
+ 0xfcc20864, 0xfccdd70a, 0xfcd935e3, 0xfce42ab0, 0xfceebace,
+ 0xfcf8eb3b, 0xfd02c0a0, 0xfd0c3f59, 0xfd156b7b, 0xfd1e48d6,
+ 0xfd26daff, 0xfd2f2552, 0xfd372af7, 0xfd3eeee5, 0xfd4673e7,
+ 0xfd4dbc9e, 0xfd54cb85, 0xfd5ba2f2, 0xfd62451b, 0xfd68b415,
+ 0xfd6ef1da, 0xfd750047, 0xfd7ae120, 0xfd809612, 0xfd8620b4,
+ 0xfd8b8285, 0xfd90bcf5, 0xfd95d15e, 0xfd9ac10b, 0xfd9f8d36,
+ 0xfda43708, 0xfda8bf9e, 0xfdad2806, 0xfdb17141, 0xfdb59c46,
+ 0xfdb9a9fd, 0xfdbd9b46, 0xfdc170f6, 0xfdc52bd8, 0xfdc8ccac,
+ 0xfdcc542d, 0xfdcfc30b, 0xfdd319ef, 0xfdd6597a, 0xfdd98245,
+ 0xfddc94e5, 0xfddf91e6, 0xfde279ce, 0xfde54d1f, 0xfde80c52,
+ 0xfdeab7de, 0xfded5034, 0xfdefd5be, 0xfdf248e3, 0xfdf4aa06,
+ 0xfdf6f984, 0xfdf937b6, 0xfdfb64f4, 0xfdfd818d, 0xfdff8dd0,
+ 0xfe018a08, 0xfe03767a, 0xfe05536c, 0xfe07211c, 0xfe08dfc9,
+ 0xfe0a8fab, 0xfe0c30fb, 0xfe0dc3ec, 0xfe0f48b1, 0xfe10bf76,
+ 0xfe122869, 0xfe1383b4, 0xfe14d17c, 0xfe1611e7, 0xfe174516,
+ 0xfe186b2a, 0xfe19843e, 0xfe1a9070, 0xfe1b8fd6, 0xfe1c8289,
+ 0xfe1d689b, 0xfe1e4220, 0xfe1f0f26, 0xfe1fcfbc, 0xfe2083ed,
+ 0xfe212bc3, 0xfe21c745, 0xfe225678, 0xfe22d95f, 0xfe234ffb,
+ 0xfe23ba4a, 0xfe241849, 0xfe2469f2, 0xfe24af3c, 0xfe24e81e,
+ 0xfe25148b, 0xfe253474, 0xfe2547c7, 0xfe254e70, 0xfe25485a,
+ 0xfe25356a, 0xfe251586, 0xfe24e88f, 0xfe24ae64, 0xfe2466e1,
+ 0xfe2411df, 0xfe23af34, 0xfe233eb4, 0xfe22c02c, 0xfe22336b,
+ 0xfe219838, 0xfe20ee58, 0xfe20358c, 0xfe1f6d92, 0xfe1e9621,
+ 0xfe1daef0, 0xfe1cb7ac, 0xfe1bb002, 0xfe1a9798, 0xfe196e0d,
+ 0xfe1832fd, 0xfe16e5fe, 0xfe15869d, 0xfe141464, 0xfe128ed3,
+ 0xfe10f565, 0xfe0f478c, 0xfe0d84b1, 0xfe0bac36, 0xfe09bd73,
+ 0xfe07b7b5, 0xfe059a40, 0xfe03644c, 0xfe011504, 0xfdfeab88,
+ 0xfdfc26e9, 0xfdf98629, 0xfdf6c83b, 0xfdf3ec01, 0xfdf0f04a,
+ 0xfdedd3d1, 0xfdea953d, 0xfde7331e, 0xfde3abe9, 0xfddffdfb,
+ 0xfddc2791, 0xfdd826cd, 0xfdd3f9a8, 0xfdcf9dfc, 0xfdcb1176,
+ 0xfdc65198, 0xfdc15bb3, 0xfdbc2ce2, 0xfdb6c206, 0xfdb117be,
+ 0xfdab2a63, 0xfda4f5fd, 0xfd9e7640, 0xfd97a67a, 0xfd908192,
+ 0xfd8901f2, 0xfd812182, 0xfd78d98e, 0xfd7022bb, 0xfd66f4ed,
+ 0xfd5d4732, 0xfd530f9c, 0xfd48432b, 0xfd3cd59a, 0xfd30b936,
+ 0xfd23dea4, 0xfd16349e, 0xfd07a7a3, 0xfcf8219b, 0xfce7895b,
+ 0xfcd5c220, 0xfcc2aadb, 0xfcae1d5e, 0xfc97ed4e, 0xfc7fe6d4,
+ 0xfc65ccf3, 0xfc495762, 0xfc2a2fc8, 0xfc07ee19, 0xfbe213c1,
+ 0xfbb8051a, 0xfb890078, 0xfb5411a5, 0xfb180005, 0xfad33482,
+ 0xfa839276, 0xfa263b32, 0xf9b72d1c, 0xf930a1a2, 0xf889f023,
+ 0xf7b577d2, 0xf69c650c, 0xf51530f0, 0xf2cb0e3c, 0xeeefb15d,
+ 0xe6da6ecf,
+}
+var we = [256]float32{
+ 2.0249555e-09, 1.486674e-11, 2.4409617e-11, 3.1968806e-11,
+ 3.844677e-11, 4.4228204e-11, 4.9516443e-11, 5.443359e-11,
+ 5.905944e-11, 6.344942e-11, 6.7643814e-11, 7.1672945e-11,
+ 7.556032e-11, 7.932458e-11, 8.298079e-11, 8.654132e-11,
+ 9.0016515e-11, 9.3415074e-11, 9.674443e-11, 1.0001099e-10,
+ 1.03220314e-10, 1.06377254e-10, 1.09486115e-10, 1.1255068e-10,
+ 1.1557435e-10, 1.1856015e-10, 1.2151083e-10, 1.2442886e-10,
+ 1.2731648e-10, 1.3017575e-10, 1.3300853e-10, 1.3581657e-10,
+ 1.3860142e-10, 1.4136457e-10, 1.4410738e-10, 1.4683108e-10,
+ 1.4953687e-10, 1.5222583e-10, 1.54899e-10, 1.5755733e-10,
+ 1.6020171e-10, 1.6283301e-10, 1.6545203e-10, 1.6805951e-10,
+ 1.7065617e-10, 1.732427e-10, 1.7581973e-10, 1.7838787e-10,
+ 1.8094774e-10, 1.8349985e-10, 1.8604476e-10, 1.8858298e-10,
+ 1.9111498e-10, 1.9364126e-10, 1.9616223e-10, 1.9867835e-10,
+ 2.0119004e-10, 2.0369768e-10, 2.0620168e-10, 2.087024e-10,
+ 2.1120022e-10, 2.136955e-10, 2.1618855e-10, 2.1867974e-10,
+ 2.2116936e-10, 2.2365775e-10, 2.261452e-10, 2.2863202e-10,
+ 2.311185e-10, 2.3360494e-10, 2.360916e-10, 2.3857874e-10,
+ 2.4106667e-10, 2.4355562e-10, 2.4604588e-10, 2.485377e-10,
+ 2.5103128e-10, 2.5352695e-10, 2.560249e-10, 2.585254e-10,
+ 2.6102867e-10, 2.6353494e-10, 2.6604446e-10, 2.6855745e-10,
+ 2.7107416e-10, 2.7359479e-10, 2.761196e-10, 2.7864877e-10,
+ 2.8118255e-10, 2.8372119e-10, 2.8626485e-10, 2.888138e-10,
+ 2.9136826e-10, 2.939284e-10, 2.9649452e-10, 2.9906677e-10,
+ 3.016454e-10, 3.0423064e-10, 3.0682268e-10, 3.0942177e-10,
+ 3.1202813e-10, 3.1464195e-10, 3.1726352e-10, 3.19893e-10,
+ 3.2253064e-10, 3.251767e-10, 3.2783135e-10, 3.3049485e-10,
+ 3.3316744e-10, 3.3584938e-10, 3.3854083e-10, 3.4124212e-10,
+ 3.4395342e-10, 3.46675e-10, 3.4940711e-10, 3.5215003e-10,
+ 3.5490397e-10, 3.5766917e-10, 3.6044595e-10, 3.6323455e-10,
+ 3.660352e-10, 3.6884823e-10, 3.7167386e-10, 3.745124e-10,
+ 3.773641e-10, 3.802293e-10, 3.8310827e-10, 3.860013e-10,
+ 3.8890866e-10, 3.918307e-10, 3.9476775e-10, 3.9772008e-10,
+ 4.0068804e-10, 4.0367196e-10, 4.0667217e-10, 4.09689e-10,
+ 4.1272286e-10, 4.1577405e-10, 4.1884296e-10, 4.2192994e-10,
+ 4.250354e-10, 4.281597e-10, 4.313033e-10, 4.3446652e-10,
+ 4.3764986e-10, 4.408537e-10, 4.4407847e-10, 4.4732465e-10,
+ 4.5059267e-10, 4.5388301e-10, 4.571962e-10, 4.6053267e-10,
+ 4.6389292e-10, 4.6727755e-10, 4.70687e-10, 4.741219e-10,
+ 4.7758275e-10, 4.810702e-10, 4.845848e-10, 4.8812715e-10,
+ 4.9169796e-10, 4.9529775e-10, 4.989273e-10, 5.0258725e-10,
+ 5.0627835e-10, 5.100013e-10, 5.1375687e-10, 5.1754584e-10,
+ 5.21369e-10, 5.2522725e-10, 5.2912136e-10, 5.330522e-10,
+ 5.370208e-10, 5.4102806e-10, 5.45075e-10, 5.491625e-10,
+ 5.532918e-10, 5.5746385e-10, 5.616799e-10, 5.6594107e-10,
+ 5.7024857e-10, 5.746037e-10, 5.7900773e-10, 5.834621e-10,
+ 5.8796823e-10, 5.925276e-10, 5.971417e-10, 6.018122e-10,
+ 6.065408e-10, 6.113292e-10, 6.1617933e-10, 6.2109295e-10,
+ 6.260722e-10, 6.3111916e-10, 6.3623595e-10, 6.4142497e-10,
+ 6.4668854e-10, 6.5202926e-10, 6.5744976e-10, 6.6295286e-10,
+ 6.6854156e-10, 6.742188e-10, 6.79988e-10, 6.858526e-10,
+ 6.9181616e-10, 6.978826e-10, 7.04056e-10, 7.103407e-10,
+ 7.167412e-10, 7.2326256e-10, 7.2990985e-10, 7.366886e-10,
+ 7.4360473e-10, 7.5066453e-10, 7.5787476e-10, 7.6524265e-10,
+ 7.7277595e-10, 7.80483e-10, 7.883728e-10, 7.9645507e-10,
+ 8.047402e-10, 8.1323964e-10, 8.219657e-10, 8.309319e-10,
+ 8.401528e-10, 8.496445e-10, 8.594247e-10, 8.6951274e-10,
+ 8.799301e-10, 8.9070046e-10, 9.018503e-10, 9.134092e-10,
+ 9.254101e-10, 9.378904e-10, 9.508923e-10, 9.644638e-10,
+ 9.786603e-10, 9.935448e-10, 1.0091913e-09, 1.025686e-09,
+ 1.0431306e-09, 1.0616465e-09, 1.08138e-09, 1.1025096e-09,
+ 1.1252564e-09, 1.1498986e-09, 1.1767932e-09, 1.206409e-09,
+ 1.2393786e-09, 1.276585e-09, 1.3193139e-09, 1.3695435e-09,
+ 1.4305498e-09, 1.508365e-09, 1.6160854e-09, 1.7921248e-09,
+}
+var fe = [256]float32{
+ 1, 0.9381437, 0.90046996, 0.87170434, 0.8477855, 0.8269933,
+ 0.8084217, 0.7915276, 0.77595687, 0.7614634, 0.7478686,
+ 0.7350381, 0.72286767, 0.71127474, 0.70019263, 0.6895665,
+ 0.67935055, 0.6695063, 0.66000086, 0.65080583, 0.6418967,
+ 0.63325197, 0.6248527, 0.6166822, 0.60872537, 0.60096896,
+ 0.5934009, 0.58601034, 0.5787874, 0.57172304, 0.5648092,
+ 0.5580383, 0.5514034, 0.5448982, 0.5385169, 0.53225386,
+ 0.5261042, 0.52006316, 0.5141264, 0.50828975, 0.5025495,
+ 0.496902, 0.49134386, 0.485872, 0.48048335, 0.4751752,
+ 0.46994483, 0.46478975, 0.45970762, 0.45469615, 0.44975325,
+ 0.44487688, 0.44006512, 0.43531612, 0.43062815, 0.42599955,
+ 0.42142874, 0.4169142, 0.41245446, 0.40804818, 0.403694,
+ 0.3993907, 0.39513698, 0.39093173, 0.38677382, 0.38266218,
+ 0.37859577, 0.37457356, 0.37059465, 0.3666581, 0.362763,
+ 0.35890847, 0.35509375, 0.351318, 0.3475805, 0.34388044,
+ 0.34021714, 0.3365899, 0.33299807, 0.32944095, 0.32591796,
+ 0.3224285, 0.3189719, 0.31554767, 0.31215525, 0.30879408,
+ 0.3054636, 0.3021634, 0.29889292, 0.2956517, 0.29243928,
+ 0.28925523, 0.28609908, 0.28297043, 0.27986884, 0.27679393,
+ 0.2737453, 0.2707226, 0.2677254, 0.26475343, 0.26180625,
+ 0.25888354, 0.25598502, 0.2531103, 0.25025907, 0.24743107,
+ 0.24462597, 0.24184346, 0.23908329, 0.23634516, 0.23362878,
+ 0.23093392, 0.2282603, 0.22560766, 0.22297576, 0.22036438,
+ 0.21777324, 0.21520215, 0.21265087, 0.21011916, 0.20760682,
+ 0.20511365, 0.20263945, 0.20018397, 0.19774707, 0.19532852,
+ 0.19292815, 0.19054577, 0.1881812, 0.18583426, 0.18350479,
+ 0.1811926, 0.17889754, 0.17661946, 0.17435817, 0.17211354,
+ 0.1698854, 0.16767362, 0.16547804, 0.16329853, 0.16113494,
+ 0.15898713, 0.15685499, 0.15473837, 0.15263714, 0.15055119,
+ 0.14848037, 0.14642459, 0.14438373, 0.14235765, 0.14034624,
+ 0.13834943, 0.13636707, 0.13439907, 0.13244532, 0.13050574,
+ 0.1285802, 0.12666863, 0.12477092, 0.12288698, 0.12101672,
+ 0.119160056, 0.1173169, 0.115487166, 0.11367077, 0.11186763,
+ 0.11007768, 0.10830083, 0.10653701, 0.10478614, 0.10304816,
+ 0.101323, 0.09961058, 0.09791085, 0.09622374, 0.09454919,
+ 0.09288713, 0.091237515, 0.08960028, 0.087975375, 0.08636274,
+ 0.08476233, 0.083174095, 0.081597984, 0.08003395, 0.07848195,
+ 0.076941945, 0.07541389, 0.07389775, 0.072393484, 0.07090106,
+ 0.069420435, 0.06795159, 0.066494495, 0.06504912, 0.063615434,
+ 0.062193416, 0.060783047, 0.059384305, 0.057997175,
+ 0.05662164, 0.05525769, 0.053905312, 0.052564494, 0.051235236,
+ 0.049917534, 0.048611384, 0.047316793, 0.046033762, 0.0447623,
+ 0.043502413, 0.042254124, 0.041017443, 0.039792392,
+ 0.038578995, 0.037377283, 0.036187284, 0.035009038,
+ 0.033842582, 0.032687962, 0.031545233, 0.030414443, 0.02929566,
+ 0.02818895, 0.027094385, 0.026012046, 0.024942026, 0.023884421,
+ 0.022839336, 0.021806888, 0.020787204, 0.019780423, 0.0187867,
+ 0.0178062, 0.016839107, 0.015885621, 0.014945968, 0.014020392,
+ 0.013109165, 0.012212592, 0.011331013, 0.01046481, 0.009614414,
+ 0.008780315, 0.007963077, 0.0071633533, 0.006381906,
+ 0.0056196423, 0.0048776558, 0.004157295, 0.0034602648,
+ 0.0027887989, 0.0021459677, 0.0015362998, 0.0009672693,
+ 0.00045413437,
+}
diff --git a/src/math/rand/normal.go b/src/math/rand/normal.go
new file mode 100644
index 0000000..ba4ea54
--- /dev/null
+++ b/src/math/rand/normal.go
@@ -0,0 +1,157 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "math"
+)
+
+/*
+ * Normal distribution
+ *
+ * See "The Ziggurat Method for Generating Random Variables"
+ * (Marsaglia & Tsang, 2000)
+ * http://www.jstatsoft.org/v05/i08/paper [pdf]
+ */
+
+const (
+ rn = 3.442619855899
+)
+
+func absInt32(i int32) uint32 {
+ if i < 0 {
+ return uint32(-i)
+ }
+ return uint32(i)
+}
+
+// NormFloat64 returns a normally distributed float64 in the range
+// [-math.MaxFloat64, +math.MaxFloat64] with
+// standard normal distribution (mean = 0, stddev = 1).
+// To produce a different normal distribution, callers can
+// adjust the output using:
+//
+// sample = NormFloat64() * desiredStdDev + desiredMean
+//
+func (r *Rand) NormFloat64() float64 {
+ for {
+ j := int32(r.Uint32()) // Possibly negative
+ i := j & 0x7F
+ x := float64(j) * float64(wn[i])
+ if absInt32(j) < kn[i] {
+ // This case should be hit better than 99% of the time.
+ return x
+ }
+
+ if i == 0 {
+ // This extra work is only required for the base strip.
+ for {
+ x = -math.Log(r.Float64()) * (1.0 / rn)
+ y := -math.Log(r.Float64())
+ if y+y >= x*x {
+ break
+ }
+ }
+ if j > 0 {
+ return rn + x
+ }
+ return -rn - x
+ }
+ if fn[i]+float32(r.Float64())*(fn[i-1]-fn[i]) < float32(math.Exp(-.5*x*x)) {
+ return x
+ }
+ }
+}
+
+var kn = [128]uint32{
+ 0x76ad2212, 0x0, 0x600f1b53, 0x6ce447a6, 0x725b46a2,
+ 0x7560051d, 0x774921eb, 0x789a25bd, 0x799045c3, 0x7a4bce5d,
+ 0x7adf629f, 0x7b5682a6, 0x7bb8a8c6, 0x7c0ae722, 0x7c50cce7,
+ 0x7c8cec5b, 0x7cc12cd6, 0x7ceefed2, 0x7d177e0b, 0x7d3b8883,
+ 0x7d5bce6c, 0x7d78dd64, 0x7d932886, 0x7dab0e57, 0x7dc0dd30,
+ 0x7dd4d688, 0x7de73185, 0x7df81cea, 0x7e07c0a3, 0x7e163efa,
+ 0x7e23b587, 0x7e303dfd, 0x7e3beec2, 0x7e46db77, 0x7e51155d,
+ 0x7e5aabb3, 0x7e63abf7, 0x7e6c222c, 0x7e741906, 0x7e7b9a18,
+ 0x7e82adfa, 0x7e895c63, 0x7e8fac4b, 0x7e95a3fb, 0x7e9b4924,
+ 0x7ea0a0ef, 0x7ea5b00d, 0x7eaa7ac3, 0x7eaf04f3, 0x7eb3522a,
+ 0x7eb765a5, 0x7ebb4259, 0x7ebeeafd, 0x7ec2620a, 0x7ec5a9c4,
+ 0x7ec8c441, 0x7ecbb365, 0x7ece78ed, 0x7ed11671, 0x7ed38d62,
+ 0x7ed5df12, 0x7ed80cb4, 0x7eda175c, 0x7edc0005, 0x7eddc78e,
+ 0x7edf6ebf, 0x7ee0f647, 0x7ee25ebe, 0x7ee3a8a9, 0x7ee4d473,
+ 0x7ee5e276, 0x7ee6d2f5, 0x7ee7a620, 0x7ee85c10, 0x7ee8f4cd,
+ 0x7ee97047, 0x7ee9ce59, 0x7eea0eca, 0x7eea3147, 0x7eea3568,
+ 0x7eea1aab, 0x7ee9e071, 0x7ee98602, 0x7ee90a88, 0x7ee86d08,
+ 0x7ee7ac6a, 0x7ee6c769, 0x7ee5bc9c, 0x7ee48a67, 0x7ee32efc,
+ 0x7ee1a857, 0x7edff42f, 0x7ede0ffa, 0x7edbf8d9, 0x7ed9ab94,
+ 0x7ed7248d, 0x7ed45fae, 0x7ed1585c, 0x7ece095f, 0x7eca6ccb,
+ 0x7ec67be2, 0x7ec22eee, 0x7ebd7d1a, 0x7eb85c35, 0x7eb2c075,
+ 0x7eac9c20, 0x7ea5df27, 0x7e9e769f, 0x7e964c16, 0x7e8d44ba,
+ 0x7e834033, 0x7e781728, 0x7e6b9933, 0x7e5d8a1a, 0x7e4d9ded,
+ 0x7e3b737a, 0x7e268c2f, 0x7e0e3ff5, 0x7df1aa5d, 0x7dcf8c72,
+ 0x7da61a1e, 0x7d72a0fb, 0x7d30e097, 0x7cd9b4ab, 0x7c600f1a,
+ 0x7ba90bdc, 0x7a722176, 0x77d664e5,
+}
+var wn = [128]float32{
+ 1.7290405e-09, 1.2680929e-10, 1.6897518e-10, 1.9862688e-10,
+ 2.2232431e-10, 2.4244937e-10, 2.601613e-10, 2.7611988e-10,
+ 2.9073963e-10, 3.042997e-10, 3.1699796e-10, 3.289802e-10,
+ 3.4035738e-10, 3.5121603e-10, 3.616251e-10, 3.7164058e-10,
+ 3.8130857e-10, 3.9066758e-10, 3.9975012e-10, 4.08584e-10,
+ 4.1719309e-10, 4.2559822e-10, 4.338176e-10, 4.418672e-10,
+ 4.497613e-10, 4.5751258e-10, 4.651324e-10, 4.7263105e-10,
+ 4.8001775e-10, 4.87301e-10, 4.944885e-10, 5.015873e-10,
+ 5.0860405e-10, 5.155446e-10, 5.2241467e-10, 5.2921934e-10,
+ 5.359635e-10, 5.426517e-10, 5.4928817e-10, 5.5587696e-10,
+ 5.624219e-10, 5.6892646e-10, 5.753941e-10, 5.818282e-10,
+ 5.882317e-10, 5.946077e-10, 6.00959e-10, 6.072884e-10,
+ 6.135985e-10, 6.19892e-10, 6.2617134e-10, 6.3243905e-10,
+ 6.386974e-10, 6.449488e-10, 6.511956e-10, 6.5744005e-10,
+ 6.6368433e-10, 6.699307e-10, 6.7618144e-10, 6.824387e-10,
+ 6.8870465e-10, 6.949815e-10, 7.012715e-10, 7.075768e-10,
+ 7.1389966e-10, 7.202424e-10, 7.266073e-10, 7.329966e-10,
+ 7.394128e-10, 7.4585826e-10, 7.5233547e-10, 7.58847e-10,
+ 7.653954e-10, 7.719835e-10, 7.7861395e-10, 7.852897e-10,
+ 7.920138e-10, 7.987892e-10, 8.0561924e-10, 8.125073e-10,
+ 8.194569e-10, 8.2647167e-10, 8.3355556e-10, 8.407127e-10,
+ 8.479473e-10, 8.55264e-10, 8.6266755e-10, 8.7016316e-10,
+ 8.777562e-10, 8.8545243e-10, 8.932582e-10, 9.0117996e-10,
+ 9.09225e-10, 9.174008e-10, 9.2571584e-10, 9.341788e-10,
+ 9.427997e-10, 9.515889e-10, 9.605579e-10, 9.697193e-10,
+ 9.790869e-10, 9.88676e-10, 9.985036e-10, 1.0085882e-09,
+ 1.0189509e-09, 1.0296151e-09, 1.0406069e-09, 1.0519566e-09,
+ 1.063698e-09, 1.0758702e-09, 1.0885183e-09, 1.1016947e-09,
+ 1.1154611e-09, 1.1298902e-09, 1.1450696e-09, 1.1611052e-09,
+ 1.1781276e-09, 1.1962995e-09, 1.2158287e-09, 1.2369856e-09,
+ 1.2601323e-09, 1.2857697e-09, 1.3146202e-09, 1.347784e-09,
+ 1.3870636e-09, 1.4357403e-09, 1.5008659e-09, 1.6030948e-09,
+}
+var fn = [128]float32{
+ 1, 0.9635997, 0.9362827, 0.9130436, 0.89228165, 0.87324303,
+ 0.8555006, 0.8387836, 0.8229072, 0.8077383, 0.793177,
+ 0.7791461, 0.7655842, 0.7524416, 0.73967725, 0.7272569,
+ 0.7151515, 0.7033361, 0.69178915, 0.68049186, 0.6694277,
+ 0.658582, 0.6479418, 0.63749546, 0.6272325, 0.6171434,
+ 0.6072195, 0.5974532, 0.58783704, 0.5783647, 0.56903,
+ 0.5598274, 0.5507518, 0.54179835, 0.5329627, 0.52424055,
+ 0.5156282, 0.50712204, 0.49871865, 0.49041483, 0.48220766,
+ 0.4740943, 0.46607214, 0.4581387, 0.45029163, 0.44252872,
+ 0.43484783, 0.427247, 0.41972435, 0.41227803, 0.40490642,
+ 0.39760786, 0.3903808, 0.3832238, 0.37613547, 0.36911446,
+ 0.3621595, 0.35526937, 0.34844297, 0.34167916, 0.33497685,
+ 0.3283351, 0.3217529, 0.3152294, 0.30876362, 0.30235484,
+ 0.29600215, 0.28970486, 0.2834622, 0.2772735, 0.27113807,
+ 0.2650553, 0.25902456, 0.2530453, 0.24711695, 0.241239,
+ 0.23541094, 0.22963232, 0.2239027, 0.21822165, 0.21258877,
+ 0.20700371, 0.20146611, 0.19597565, 0.19053204, 0.18513499,
+ 0.17978427, 0.17447963, 0.1692209, 0.16400786, 0.15884037,
+ 0.15371831, 0.14864157, 0.14361008, 0.13862377, 0.13368265,
+ 0.12878671, 0.12393598, 0.119130544, 0.11437051, 0.10965602,
+ 0.104987256, 0.10036444, 0.095787846, 0.0912578, 0.08677467,
+ 0.0823389, 0.077950984, 0.073611505, 0.06932112, 0.06508058,
+ 0.06089077, 0.056752663, 0.0526674, 0.048636295, 0.044660863,
+ 0.040742867, 0.03688439, 0.033087887, 0.029356318,
+ 0.025693292, 0.022103304, 0.018592102, 0.015167298,
+ 0.011839478, 0.008624485, 0.005548995, 0.0026696292,
+}
diff --git a/src/math/rand/rand.go b/src/math/rand/rand.go
new file mode 100644
index 0000000..3ffb5c4
--- /dev/null
+++ b/src/math/rand/rand.go
@@ -0,0 +1,246 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package rand implements pseudo-random number generators.
+//
+// Random numbers are generated by a Source. Top-level functions, such as
+// Float64 and Int, use a default shared Source that produces a deterministic
+// sequence of values each time a program is run. Use the Seed function to
+// initialize the default Source if different behavior is required for each run.
+// The default Source is safe for concurrent use by multiple goroutines.
+package rand
+
+import "sync"
+
+// A Source represents a source of uniformly-distributed
+// pseudo-random int64 values in the range [0, 1<<63).
+type Source interface {
+ Int63() int64
+ Seed(seed int64)
+}
+
+// NewSource returns a new pseudo-random Source seeded with the given value.
+func NewSource(seed int64) Source {
+ var rng rngSource
+ rng.Seed(seed)
+ return &rng
+}
+
+// A Rand is a source of random numbers.
+type Rand struct {
+ src Source
+}
+
+// New returns a new Rand that uses random values from src
+// to generate other random values.
+func New(src Source) *Rand { return &Rand{src} }
+
+// Seed uses the provided seed value to initialize the generator to a deterministic state.
+func (r *Rand) Seed(seed int64) { r.src.Seed(seed) }
+
+// Int63 returns a non-negative pseudo-random 63-bit integer as an int64.
+func (r *Rand) Int63() int64 { return r.src.Int63() }
+
+// Uint32 returns a pseudo-random 32-bit value as a uint32.
+func (r *Rand) Uint32() uint32 { return uint32(r.Int63() >> 31) }
+
+// Int31 returns a non-negative pseudo-random 31-bit integer as an int32.
+func (r *Rand) Int31() int32 { return int32(r.Int63() >> 32) }
+
+// Int returns a non-negative pseudo-random int.
+func (r *Rand) Int() int {
+ u := uint(r.Int63())
+ return int(u << 1 >> 1) // clear sign bit if int == int32
+}
+
+// Int63n returns, as an int64, a non-negative pseudo-random number in [0,n).
+// It panics if n <= 0.
+func (r *Rand) Int63n(n int64) int64 {
+ if n <= 0 {
+ panic("invalid argument to Int63n")
+ }
+ if n&(n-1) == 0 { // n is power of two, can mask
+ return r.Int63() & (n - 1)
+ }
+ max := int64((1 << 63) - 1 - (1<<63)%uint64(n))
+ v := r.Int63()
+ for v > max {
+ v = r.Int63()
+ }
+ return v % n
+}
+
+// Int31n returns, as an int32, a non-negative pseudo-random number in [0,n).
+// It panics if n <= 0.
+func (r *Rand) Int31n(n int32) int32 {
+ if n <= 0 {
+ panic("invalid argument to Int31n")
+ }
+ if n&(n-1) == 0 { // n is power of two, can mask
+ return r.Int31() & (n - 1)
+ }
+ max := int32((1 << 31) - 1 - (1<<31)%uint32(n))
+ v := r.Int31()
+ for v > max {
+ v = r.Int31()
+ }
+ return v % n
+}
+
+// Intn returns, as an int, a non-negative pseudo-random number in [0,n).
+// It panics if n <= 0.
+func (r *Rand) Intn(n int) int {
+ if n <= 0 {
+ panic("invalid argument to Intn")
+ }
+ if n <= 1<<31-1 {
+ return int(r.Int31n(int32(n)))
+ }
+ return int(r.Int63n(int64(n)))
+}
+
+// Float64 returns, as a float64, a pseudo-random number in [0.0,1.0).
+func (r *Rand) Float64() float64 {
+ // A clearer, simpler implementation would be:
+ // return float64(r.Int63n(1<<53)) / (1<<53)
+ // However, Go 1 shipped with
+ // return float64(r.Int63()) / (1 << 63)
+ // and we want to preserve that value stream.
+ //
+ // There is one bug in the value stream: r.Int63() may be so close
+ // to 1<<63 that the division rounds up to 1.0, and we've guaranteed
+ // that the result is always less than 1.0. To fix that, we treat the
+ // range as cyclic and map 1 back to 0. This is justified by observing
+ // that while some of the values rounded down to 0, nothing was
+ // rounding up to 0, so 0 was underrepresented in the results.
+ // Mapping 1 back to zero restores some balance.
+ // (The balance is not perfect because the implementation
+ // returns denormalized numbers for very small r.Int63(),
+ // and those steal from what would normally be 0 results.)
+ // The remapping only happens 1/2⁵³ of the time, so most clients
+ // will not observe it anyway.
+ f := float64(r.Int63()) / (1 << 63)
+ if f == 1 {
+ f = 0
+ }
+ return f
+}
+
+// Float32 returns, as a float32, a pseudo-random number in [0.0,1.0).
+func (r *Rand) Float32() float32 {
+ // Same rationale as in Float64: we want to preserve the Go 1 value
+ // stream except we want to fix it not to return 1.0
+ // There is a double rounding going on here, but the argument for
+ // mapping 1 to 0 still applies: 0 was underrepresented before,
+ // so mapping 1 to 0 doesn't cause too many 0s.
+ // This only happens 1/2²⁴ of the time (plus the 1/2⁵³ of the time in Float64).
+ f := float32(r.Float64())
+ if f == 1 {
+ f = 0
+ }
+ return f
+}
+
+// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n).
+func (r *Rand) Perm(n int) []int {
+ m := make([]int, n)
+ for i := 0; i < n; i++ {
+ j := r.Intn(i + 1)
+ m[i] = m[j]
+ m[j] = i
+ }
+ return m
+}
+
+/*
+ * Top-level convenience functions
+ */
+
+var globalRand = New(&lockedSource{src: NewSource(1)})
+
+// Seed uses the provided seed value to initialize the default Source to a
+// deterministic state. If Seed is not called, the generator behaves as
+// if seeded by Seed(1).
+func Seed(seed int64) { globalRand.Seed(seed) }
+
+// Int63 returns a non-negative pseudo-random 63-bit integer as an int64
+// from the default Source.
+func Int63() int64 { return globalRand.Int63() }
+
+// Uint32 returns a pseudo-random 32-bit value as a uint32
+// from the default Source.
+func Uint32() uint32 { return globalRand.Uint32() }
+
+// Int31 returns a non-negative pseudo-random 31-bit integer as an int32
+// from the default Source.
+func Int31() int32 { return globalRand.Int31() }
+
+// Int returns a non-negative pseudo-random int from the default Source.
+func Int() int { return globalRand.Int() }
+
+// Int63n returns, as an int64, a non-negative pseudo-random number in [0,n)
+// from the default Source.
+// It panics if n <= 0.
+func Int63n(n int64) int64 { return globalRand.Int63n(n) }
+
+// Int31n returns, as an int32, a non-negative pseudo-random number in [0,n)
+// from the default Source.
+// It panics if n <= 0.
+func Int31n(n int32) int32 { return globalRand.Int31n(n) }
+
+// Intn returns, as an int, a non-negative pseudo-random number in [0,n)
+// from the default Source.
+// It panics if n <= 0.
+func Intn(n int) int { return globalRand.Intn(n) }
+
+// Float64 returns, as a float64, a pseudo-random number in [0.0,1.0)
+// from the default Source.
+func Float64() float64 { return globalRand.Float64() }
+
+// Float32 returns, as a float32, a pseudo-random number in [0.0,1.0)
+// from the default Source.
+func Float32() float32 { return globalRand.Float32() }
+
+// Perm returns, as a slice of n ints, a pseudo-random permutation of the integers [0,n)
+// from the default Source.
+func Perm(n int) []int { return globalRand.Perm(n) }
+
+// NormFloat64 returns a normally distributed float64 in the range
+// [-math.MaxFloat64, +math.MaxFloat64] with
+// standard normal distribution (mean = 0, stddev = 1)
+// from the default Source.
+// To produce a different normal distribution, callers can
+// adjust the output using:
+//
+// sample = NormFloat64() * desiredStdDev + desiredMean
+//
+func NormFloat64() float64 { return globalRand.NormFloat64() }
+
+// ExpFloat64 returns an exponentially distributed float64 in the range
+// (0, +math.MaxFloat64] with an exponential distribution whose rate parameter
+// (lambda) is 1 and whose mean is 1/lambda (1) from the default Source.
+// To produce a distribution with a different rate parameter,
+// callers can adjust the output using:
+//
+// sample = ExpFloat64() / desiredRateParameter
+//
+func ExpFloat64() float64 { return globalRand.ExpFloat64() }
+
+type lockedSource struct {
+ lk sync.Mutex
+ src Source
+}
+
+func (r *lockedSource) Int63() (n int64) {
+ r.lk.Lock()
+ n = r.src.Int63()
+ r.lk.Unlock()
+ return
+}
+
+func (r *lockedSource) Seed(seed int64) {
+ r.lk.Lock()
+ r.src.Seed(seed)
+ r.lk.Unlock()
+}
diff --git a/src/math/rand/rand_test.go b/src/math/rand/rand_test.go
new file mode 100644
index 0000000..ab0dc49
--- /dev/null
+++ b/src/math/rand/rand_test.go
@@ -0,0 +1,398 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+import (
+ "errors"
+ "fmt"
+ "math"
+ "testing"
+)
+
+const (
+ numTestSamples = 10000
+)
+
+type statsResults struct {
+ mean float64
+ stddev float64
+ closeEnough float64
+ maxError float64
+}
+
+func max(a, b float64) float64 {
+ if a > b {
+ return a
+ }
+ return b
+}
+
+func nearEqual(a, b, closeEnough, maxError float64) bool {
+ absDiff := math.Abs(a - b)
+ if absDiff < closeEnough { // Necessary when one value is zero and one value is close to zero.
+ return true
+ }
+ return absDiff/max(math.Abs(a), math.Abs(b)) < maxError
+}
+
+var testSeeds = []int64{1, 1754801282, 1698661970, 1550503961}
+
+// checkSimilarDistribution returns success if the mean and stddev of the
+// two statsResults are similar.
+func (this *statsResults) checkSimilarDistribution(expected *statsResults) error {
+ if !nearEqual(this.mean, expected.mean, expected.closeEnough, expected.maxError) {
+ s := fmt.Sprintf("mean %v != %v (allowed error %v, %v)", this.mean, expected.mean, expected.closeEnough, expected.maxError)
+ fmt.Println(s)
+ return errors.New(s)
+ }
+ if !nearEqual(this.stddev, expected.stddev, 0, expected.maxError) {
+ s := fmt.Sprintf("stddev %v != %v (allowed error %v, %v)", this.stddev, expected.stddev, expected.closeEnough, expected.maxError)
+ fmt.Println(s)
+ return errors.New(s)
+ }
+ return nil
+}
+
+func getStatsResults(samples []float64) *statsResults {
+ res := new(statsResults)
+ var sum, squaresum float64
+ for _, s := range samples {
+ sum += s
+ squaresum += s * s
+ }
+ res.mean = sum / float64(len(samples))
+ res.stddev = math.Sqrt(squaresum/float64(len(samples)) - res.mean*res.mean)
+ return res
+}
+
+func checkSampleDistribution(t *testing.T, samples []float64, expected *statsResults) {
+ actual := getStatsResults(samples)
+ err := actual.checkSimilarDistribution(expected)
+ if err != nil {
+ t.Errorf(err.Error())
+ }
+}
+
+func checkSampleSliceDistributions(t *testing.T, samples []float64, nslices int, expected *statsResults) {
+ chunk := len(samples) / nslices
+ for i := 0; i < nslices; i++ {
+ low := i * chunk
+ var high int
+ if i == nslices-1 {
+ high = len(samples) - 1
+ } else {
+ high = (i + 1) * chunk
+ }
+ checkSampleDistribution(t, samples[low:high], expected)
+ }
+}
+
+//
+// Normal distribution tests
+//
+
+func generateNormalSamples(nsamples int, mean, stddev float64, seed int64) []float64 {
+ r := New(NewSource(seed))
+ samples := make([]float64, nsamples)
+ for i := range samples {
+ samples[i] = r.NormFloat64()*stddev + mean
+ }
+ return samples
+}
+
+func testNormalDistribution(t *testing.T, nsamples int, mean, stddev float64, seed int64) {
+ //fmt.Printf("testing nsamples=%v mean=%v stddev=%v seed=%v\n", nsamples, mean, stddev, seed);
+
+ samples := generateNormalSamples(nsamples, mean, stddev, seed)
+ errorScale := max(1.0, stddev) // Error scales with stddev
+ expected := &statsResults{mean, stddev, 0.10 * errorScale, 0.08 * errorScale}
+
+ // Make sure that the entire set matches the expected distribution.
+ checkSampleDistribution(t, samples, expected)
+
+ // Make sure that each half of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 2, expected)
+
+ // Make sure that each 7th of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 7, expected)
+}
+
+// Actual tests
+
+func TestStandardNormalValues(t *testing.T) {
+ for _, seed := range testSeeds {
+ testNormalDistribution(t, numTestSamples, 0, 1, seed)
+ }
+}
+
+func TestNonStandardNormalValues(t *testing.T) {
+ sdmax := 1000.0
+ mmax := 1000.0
+ if testing.Short() {
+ sdmax = 5
+ mmax = 5
+ }
+ for sd := 0.5; sd < sdmax; sd *= 2 {
+ for m := 0.5; m < mmax; m *= 2 {
+ for _, seed := range testSeeds {
+ testNormalDistribution(t, numTestSamples, m, sd, seed)
+ if testing.Short() {
+ break
+ }
+ }
+ }
+ }
+}
+
+//
+// Exponential distribution tests
+//
+
+func generateExponentialSamples(nsamples int, rate float64, seed int64) []float64 {
+ r := New(NewSource(seed))
+ samples := make([]float64, nsamples)
+ for i := range samples {
+ samples[i] = r.ExpFloat64() / rate
+ }
+ return samples
+}
+
+func testExponentialDistribution(t *testing.T, nsamples int, rate float64, seed int64) {
+ //fmt.Printf("testing nsamples=%v rate=%v seed=%v\n", nsamples, rate, seed);
+
+ mean := 1 / rate
+ stddev := mean
+
+ samples := generateExponentialSamples(nsamples, rate, seed)
+ errorScale := max(1.0, 1/rate) // Error scales with the inverse of the rate
+ expected := &statsResults{mean, stddev, 0.10 * errorScale, 0.20 * errorScale}
+
+ // Make sure that the entire set matches the expected distribution.
+ checkSampleDistribution(t, samples, expected)
+
+ // Make sure that each half of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 2, expected)
+
+ // Make sure that each 7th of the set matches the expected distribution.
+ checkSampleSliceDistributions(t, samples, 7, expected)
+}
+
+// Actual tests
+
+func TestStandardExponentialValues(t *testing.T) {
+ for _, seed := range testSeeds {
+ testExponentialDistribution(t, numTestSamples, 1, seed)
+ }
+}
+
+func TestNonStandardExponentialValues(t *testing.T) {
+ for rate := 0.05; rate < 10; rate *= 2 {
+ for _, seed := range testSeeds {
+ testExponentialDistribution(t, numTestSamples, rate, seed)
+ if testing.Short() {
+ break
+ }
+ }
+ }
+}
+
+//
+// Table generation tests
+//
+
+func initNorm() (testKn []uint32, testWn, testFn []float32) {
+ const m1 = 1 << 31
+ var (
+ dn float64 = rn
+ tn = dn
+ vn float64 = 9.91256303526217e-3
+ )
+
+ testKn = make([]uint32, 128)
+ testWn = make([]float32, 128)
+ testFn = make([]float32, 128)
+
+ q := vn / math.Exp(-0.5*dn*dn)
+ testKn[0] = uint32((dn / q) * m1)
+ testKn[1] = 0
+ testWn[0] = float32(q / m1)
+ testWn[127] = float32(dn / m1)
+ testFn[0] = 1.0
+ testFn[127] = float32(math.Exp(-0.5 * dn * dn))
+ for i := 126; i >= 1; i-- {
+ dn = math.Sqrt(-2.0 * math.Log(vn/dn+math.Exp(-0.5*dn*dn)))
+ testKn[i+1] = uint32((dn / tn) * m1)
+ tn = dn
+ testFn[i] = float32(math.Exp(-0.5 * dn * dn))
+ testWn[i] = float32(dn / m1)
+ }
+ return
+}
+
+func initExp() (testKe []uint32, testWe, testFe []float32) {
+ const m2 = 1 << 32
+ var (
+ de float64 = re
+ te = de
+ ve float64 = 3.9496598225815571993e-3
+ )
+
+ testKe = make([]uint32, 256)
+ testWe = make([]float32, 256)
+ testFe = make([]float32, 256)
+
+ q := ve / math.Exp(-de)
+ testKe[0] = uint32((de / q) * m2)
+ testKe[1] = 0
+ testWe[0] = float32(q / m2)
+ testWe[255] = float32(de / m2)
+ testFe[0] = 1.0
+ testFe[255] = float32(math.Exp(-de))
+ for i := 254; i >= 1; i-- {
+ de = -math.Log(ve/de + math.Exp(-de))
+ testKe[i+1] = uint32((de / te) * m2)
+ te = de
+ testFe[i] = float32(math.Exp(-de))
+ testWe[i] = float32(de / m2)
+ }
+ return
+}
+
+// compareUint32Slices returns the first index where the two slices
+// disagree, or <0 if the lengths are the same and all elements
+// are identical.
+func compareUint32Slices(s1, s2 []uint32) int {
+ if len(s1) != len(s2) {
+ if len(s1) > len(s2) {
+ return len(s2) + 1
+ }
+ return len(s1) + 1
+ }
+ for i := range s1 {
+ if s1[i] != s2[i] {
+ return i
+ }
+ }
+ return -1
+}
+
+// compareFloat32Slices returns the first index where the two slices
+// disagree, or <0 if the lengths are the same and all elements
+// are identical.
+func compareFloat32Slices(s1, s2 []float32) int {
+ if len(s1) != len(s2) {
+ if len(s1) > len(s2) {
+ return len(s2) + 1
+ }
+ return len(s1) + 1
+ }
+ for i := range s1 {
+ if !nearEqual(float64(s1[i]), float64(s2[i]), 0, 1e-7) {
+ return i
+ }
+ }
+ return -1
+}
+
+func TestNormTables(t *testing.T) {
+ testKn, testWn, testFn := initNorm()
+ if i := compareUint32Slices(kn[0:], testKn); i >= 0 {
+ t.Errorf("kn disagrees at index %v; %v != %v", i, kn[i], testKn[i])
+ }
+ if i := compareFloat32Slices(wn[0:], testWn); i >= 0 {
+ t.Errorf("wn disagrees at index %v; %v != %v", i, wn[i], testWn[i])
+ }
+ if i := compareFloat32Slices(fn[0:], testFn); i >= 0 {
+ t.Errorf("fn disagrees at index %v; %v != %v", i, fn[i], testFn[i])
+ }
+}
+
+func TestExpTables(t *testing.T) {
+ testKe, testWe, testFe := initExp()
+ if i := compareUint32Slices(ke[0:], testKe); i >= 0 {
+ t.Errorf("ke disagrees at index %v; %v != %v", i, ke[i], testKe[i])
+ }
+ if i := compareFloat32Slices(we[0:], testWe); i >= 0 {
+ t.Errorf("we disagrees at index %v; %v != %v", i, we[i], testWe[i])
+ }
+ if i := compareFloat32Slices(fe[0:], testFe); i >= 0 {
+ t.Errorf("fe disagrees at index %v; %v != %v", i, fe[i], testFe[i])
+ }
+}
+
+// For issue 6721, the problem came after 7533753 calls, so check 10e6.
+func TestFloat32(t *testing.T) {
+ r := New(NewSource(1))
+ for ct := 0; ct < 10e6; ct++ {
+ f := r.Float32()
+ if f >= 1 {
+ t.Fatal("Float32() should be in range [0,1). ct:", ct, "f:", f)
+ }
+ }
+}
+
+// Benchmarks
+
+func BenchmarkInt63Threadsafe(b *testing.B) {
+ for n := b.N; n > 0; n-- {
+ Int63()
+ }
+}
+
+func BenchmarkInt63Unthreadsafe(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Int63()
+ }
+}
+
+func BenchmarkIntn1000(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Intn(1000)
+ }
+}
+
+func BenchmarkInt63n1000(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Int63n(1000)
+ }
+}
+
+func BenchmarkInt31n1000(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Int31n(1000)
+ }
+}
+
+func BenchmarkFloat32(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Float32()
+ }
+}
+
+func BenchmarkFloat64(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Float64()
+ }
+}
+
+func BenchmarkPerm3(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Perm(3)
+ }
+}
+
+func BenchmarkPerm30(b *testing.B) {
+ r := New(NewSource(1))
+ for n := b.N; n > 0; n-- {
+ r.Perm(30)
+ }
+}
diff --git a/src/math/rand/regress_test.go b/src/math/rand/regress_test.go
new file mode 100644
index 0000000..2b012af
--- /dev/null
+++ b/src/math/rand/regress_test.go
@@ -0,0 +1,355 @@
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Test that random number sequences generated by a specific seed
+// do not change from version to version.
+//
+// Do NOT make changes to the golden outputs. If bugs need to be fixed
+// in the underlying code, find ways to fix them that do not affect the
+// outputs.
+
+package rand_test
+
+import (
+ "flag"
+ "fmt"
+ . "math/rand"
+ "reflect"
+ "testing"
+)
+
+var printgolden = flag.Bool("printgolden", false, "print golden results for regression test")
+
+func TestRegress(t *testing.T) {
+ var int32s = []int32{1, 10, 32, 1 << 20, 1<<20 + 1, 1000000000, 1 << 30, 1<<31 - 2, 1<<31 - 1}
+ var int64s = []int64{1, 10, 32, 1 << 20, 1<<20 + 1, 1000000000, 1 << 30, 1<<31 - 2, 1<<31 - 1, 1000000000000000000, 1 << 60, 1<<63 - 2, 1<<63 - 1}
+ var permSizes = []int{0, 1, 5, 8, 9, 10, 16}
+ r := New(NewSource(0))
+
+ rv := reflect.ValueOf(r)
+ n := rv.NumMethod()
+ p := 0
+ if *printgolden {
+ fmt.Printf("var regressGolden = []interface{}{\n")
+ }
+ for i := 0; i < n; i++ {
+ m := rv.Type().Method(i)
+ mv := rv.Method(i)
+ mt := mv.Type()
+ if mt.NumOut() == 0 {
+ continue
+ }
+ if mt.NumOut() != 1 {
+ t.Fatalf("unexpected result count for r.%s", m.Name)
+ }
+ r.Seed(0)
+ for repeat := 0; repeat < 20; repeat++ {
+ var args []reflect.Value
+ var argstr string
+ if mt.NumIn() == 1 {
+ var x interface{}
+ switch mt.In(0).Kind() {
+ default:
+ t.Fatalf("unexpected argument type for r.%s", m.Name)
+
+ case reflect.Int:
+ if m.Name == "Perm" {
+ x = permSizes[repeat%len(permSizes)]
+ break
+ }
+ big := int64s[repeat%len(int64s)]
+ if int64(int(big)) != big {
+ r.Int63n(big) // what would happen on 64-bit machine, to keep stream in sync
+ if *printgolden {
+ fmt.Printf("\tskipped, // must run printgolden on 64-bit machine\n")
+ }
+ p++
+ continue
+ }
+ x = int(big)
+
+ case reflect.Int32:
+ x = int32s[repeat%len(int32s)]
+
+ case reflect.Int64:
+ x = int64s[repeat%len(int64s)]
+ }
+ argstr = fmt.Sprint(x)
+ args = append(args, reflect.ValueOf(x))
+ }
+ out := mv.Call(args)[0].Interface()
+ if m.Name == "Int" || m.Name == "Intn" {
+ out = int64(out.(int))
+ }
+ if *printgolden {
+ var val string
+ big := int64(1 << 60)
+ if int64(int(big)) != big && (m.Name == "Int" || m.Name == "Intn") {
+ // 32-bit machine cannot print 64-bit results
+ val = "truncated"
+ } else if reflect.TypeOf(out).Kind() == reflect.Slice {
+ val = fmt.Sprintf("%#v", out)
+ } else {
+ val = fmt.Sprintf("%T(%v)", out, out)
+ }
+ fmt.Printf("\t%s, // %s(%s)\n", val, m.Name, argstr)
+ } else {
+ want := regressGolden[p]
+ if m.Name == "Int" {
+ want = int64(int(uint(want.(int64)) << 1 >> 1))
+ }
+ if !reflect.DeepEqual(out, want) {
+ t.Errorf("r.%s(%s) = %v, want %v", m.Name, argstr, out, want)
+ }
+ }
+ p++
+ }
+ }
+ if *printgolden {
+ fmt.Printf("}\n")
+ }
+}
+
+var regressGolden = []interface{}{
+ float64(4.668112973579268), // ExpFloat64()
+ float64(0.1601593871172866), // ExpFloat64()
+ float64(3.0465834105636), // ExpFloat64()
+ float64(0.06385839451671879), // ExpFloat64()
+ float64(1.8578917487258961), // ExpFloat64()
+ float64(0.784676123472182), // ExpFloat64()
+ float64(0.11225477361256932), // ExpFloat64()
+ float64(0.20173283329802255), // ExpFloat64()
+ float64(0.3468619496201105), // ExpFloat64()
+ float64(0.35601103454384536), // ExpFloat64()
+ float64(0.888376329507869), // ExpFloat64()
+ float64(1.4081362450365698), // ExpFloat64()
+ float64(1.0077753823151994), // ExpFloat64()
+ float64(0.23594100766227588), // ExpFloat64()
+ float64(2.777245612300007), // ExpFloat64()
+ float64(0.5202997830662377), // ExpFloat64()
+ float64(1.2842705247770294), // ExpFloat64()
+ float64(0.030307408362776206), // ExpFloat64()
+ float64(2.204156824853721), // ExpFloat64()
+ float64(2.09891923895058), // ExpFloat64()
+ float32(0.94519615), // Float32()
+ float32(0.24496509), // Float32()
+ float32(0.65595627), // Float32()
+ float32(0.05434384), // Float32()
+ float32(0.3675872), // Float32()
+ float32(0.28948045), // Float32()
+ float32(0.1924386), // Float32()
+ float32(0.65533215), // Float32()
+ float32(0.8971697), // Float32()
+ float32(0.16735445), // Float32()
+ float32(0.28858566), // Float32()
+ float32(0.9026048), // Float32()
+ float32(0.84978026), // Float32()
+ float32(0.2730468), // Float32()
+ float32(0.6090802), // Float32()
+ float32(0.253656), // Float32()
+ float32(0.7746542), // Float32()
+ float32(0.017480763), // Float32()
+ float32(0.78707397), // Float32()
+ float32(0.7993937), // Float32()
+ float64(0.9451961492941164), // Float64()
+ float64(0.24496508529377975), // Float64()
+ float64(0.6559562651954052), // Float64()
+ float64(0.05434383959970039), // Float64()
+ float64(0.36758720663245853), // Float64()
+ float64(0.2894804331565928), // Float64()
+ float64(0.19243860967493215), // Float64()
+ float64(0.6553321508148324), // Float64()
+ float64(0.897169713149801), // Float64()
+ float64(0.16735444255905835), // Float64()
+ float64(0.2885856518054551), // Float64()
+ float64(0.9026048462705047), // Float64()
+ float64(0.8497802817628735), // Float64()
+ float64(0.2730468047134829), // Float64()
+ float64(0.6090801919903561), // Float64()
+ float64(0.25365600644283687), // Float64()
+ float64(0.7746542391859803), // Float64()
+ float64(0.017480762156647272), // Float64()
+ float64(0.7870739563039942), // Float64()
+ float64(0.7993936979594545), // Float64()
+ int64(8717895732742165505), // Int()
+ int64(2259404117704393152), // Int()
+ int64(6050128673802995827), // Int()
+ int64(501233450539197794), // Int()
+ int64(3390393562759376202), // Int()
+ int64(2669985732393126063), // Int()
+ int64(1774932891286980153), // Int()
+ int64(6044372234677422456), // Int()
+ int64(8274930044578894929), // Int()
+ int64(1543572285742637646), // Int()
+ int64(2661732831099943416), // Int()
+ int64(8325060299420976708), // Int()
+ int64(7837839688282259259), // Int()
+ int64(2518412263346885298), // Int()
+ int64(5617773211005988520), // Int()
+ int64(2339563716805116249), // Int()
+ int64(7144924247938981575), // Int()
+ int64(161231572858529631), // Int()
+ int64(7259475919510918339), // Int()
+ int64(7373105480197164748), // Int()
+ int32(2029793274), // Int31()
+ int32(526058514), // Int31()
+ int32(1408655353), // Int31()
+ int32(116702506), // Int31()
+ int32(789387515), // Int31()
+ int32(621654496), // Int31()
+ int32(413258767), // Int31()
+ int32(1407315077), // Int31()
+ int32(1926657288), // Int31()
+ int32(359390928), // Int31()
+ int32(619732968), // Int31()
+ int32(1938329147), // Int31()
+ int32(1824889259), // Int31()
+ int32(586363548), // Int31()
+ int32(1307989752), // Int31()
+ int32(544722126), // Int31()
+ int32(1663557311), // Int31()
+ int32(37539650), // Int31()
+ int32(1690228450), // Int31()
+ int32(1716684894), // Int31()
+ int32(0), // Int31n(1)
+ int32(4), // Int31n(10)
+ int32(25), // Int31n(32)
+ int32(310570), // Int31n(1048576)
+ int32(857611), // Int31n(1048577)
+ int32(621654496), // Int31n(1000000000)
+ int32(413258767), // Int31n(1073741824)
+ int32(1407315077), // Int31n(2147483646)
+ int32(1926657288), // Int31n(2147483647)
+ int32(0), // Int31n(1)
+ int32(8), // Int31n(10)
+ int32(27), // Int31n(32)
+ int32(367019), // Int31n(1048576)
+ int32(209005), // Int31n(1048577)
+ int32(307989752), // Int31n(1000000000)
+ int32(544722126), // Int31n(1073741824)
+ int32(1663557311), // Int31n(2147483646)
+ int32(37539650), // Int31n(2147483647)
+ int32(0), // Int31n(1)
+ int32(4), // Int31n(10)
+ int64(8717895732742165505), // Int63()
+ int64(2259404117704393152), // Int63()
+ int64(6050128673802995827), // Int63()
+ int64(501233450539197794), // Int63()
+ int64(3390393562759376202), // Int63()
+ int64(2669985732393126063), // Int63()
+ int64(1774932891286980153), // Int63()
+ int64(6044372234677422456), // Int63()
+ int64(8274930044578894929), // Int63()
+ int64(1543572285742637646), // Int63()
+ int64(2661732831099943416), // Int63()
+ int64(8325060299420976708), // Int63()
+ int64(7837839688282259259), // Int63()
+ int64(2518412263346885298), // Int63()
+ int64(5617773211005988520), // Int63()
+ int64(2339563716805116249), // Int63()
+ int64(7144924247938981575), // Int63()
+ int64(161231572858529631), // Int63()
+ int64(7259475919510918339), // Int63()
+ int64(7373105480197164748), // Int63()
+ int64(0), // Int63n(1)
+ int64(2), // Int63n(10)
+ int64(19), // Int63n(32)
+ int64(959842), // Int63n(1048576)
+ int64(688912), // Int63n(1048577)
+ int64(393126063), // Int63n(1000000000)
+ int64(89212473), // Int63n(1073741824)
+ int64(834026388), // Int63n(2147483646)
+ int64(1577188963), // Int63n(2147483647)
+ int64(543572285742637646), // Int63n(1000000000000000000)
+ int64(355889821886249464), // Int63n(1152921504606846976)
+ int64(8325060299420976708), // Int63n(9223372036854775806)
+ int64(7837839688282259259), // Int63n(9223372036854775807)
+ int64(0), // Int63n(1)
+ int64(0), // Int63n(10)
+ int64(25), // Int63n(32)
+ int64(679623), // Int63n(1048576)
+ int64(882178), // Int63n(1048577)
+ int64(510918339), // Int63n(1000000000)
+ int64(782454476), // Int63n(1073741824)
+ int64(0), // Intn(1)
+ int64(4), // Intn(10)
+ int64(25), // Intn(32)
+ int64(310570), // Intn(1048576)
+ int64(857611), // Intn(1048577)
+ int64(621654496), // Intn(1000000000)
+ int64(413258767), // Intn(1073741824)
+ int64(1407315077), // Intn(2147483646)
+ int64(1926657288), // Intn(2147483647)
+ int64(543572285742637646), // Intn(1000000000000000000)
+ int64(355889821886249464), // Intn(1152921504606846976)
+ int64(8325060299420976708), // Intn(9223372036854775806)
+ int64(7837839688282259259), // Intn(9223372036854775807)
+ int64(0), // Intn(1)
+ int64(2), // Intn(10)
+ int64(14), // Intn(32)
+ int64(515775), // Intn(1048576)
+ int64(839455), // Intn(1048577)
+ int64(690228450), // Intn(1000000000)
+ int64(642943070), // Intn(1073741824)
+ float64(-0.28158587086436215), // NormFloat64()
+ float64(0.570933095808067), // NormFloat64()
+ float64(-1.6920196326157044), // NormFloat64()
+ float64(0.1996229111693099), // NormFloat64()
+ float64(1.9195199291234621), // NormFloat64()
+ float64(0.8954838794918353), // NormFloat64()
+ float64(0.41457072128813166), // NormFloat64()
+ float64(-0.48700161491544713), // NormFloat64()
+ float64(-0.1684059662402393), // NormFloat64()
+ float64(0.37056410998929545), // NormFloat64()
+ float64(1.0156889027029008), // NormFloat64()
+ float64(-0.5174422210625114), // NormFloat64()
+ float64(-0.5565834214413804), // NormFloat64()
+ float64(0.778320596648391), // NormFloat64()
+ float64(-1.8970718197702225), // NormFloat64()
+ float64(0.5229525761688676), // NormFloat64()
+ float64(-1.5515595563231523), // NormFloat64()
+ float64(0.0182029289376123), // NormFloat64()
+ float64(-0.6820951356608795), // NormFloat64()
+ float64(-0.5987943422687668), // NormFloat64()
+ []int{}, // Perm(0)
+ []int{0}, // Perm(1)
+ []int{0, 4, 1, 3, 2}, // Perm(5)
+ []int{3, 1, 0, 4, 7, 5, 2, 6}, // Perm(8)
+ []int{5, 0, 3, 6, 7, 4, 2, 1, 8}, // Perm(9)
+ []int{4, 5, 0, 2, 6, 9, 3, 1, 8, 7}, // Perm(10)
+ []int{14, 2, 0, 8, 3, 5, 13, 12, 1, 4, 6, 7, 11, 9, 15, 10}, // Perm(16)
+ []int{}, // Perm(0)
+ []int{0}, // Perm(1)
+ []int{3, 0, 1, 2, 4}, // Perm(5)
+ []int{5, 1, 2, 0, 4, 7, 3, 6}, // Perm(8)
+ []int{4, 0, 6, 8, 1, 5, 2, 7, 3}, // Perm(9)
+ []int{8, 6, 1, 7, 5, 4, 3, 2, 9, 0}, // Perm(10)
+ []int{0, 3, 13, 2, 15, 4, 10, 1, 8, 14, 7, 6, 12, 9, 5, 11}, // Perm(16)
+ []int{}, // Perm(0)
+ []int{0}, // Perm(1)
+ []int{0, 4, 2, 1, 3}, // Perm(5)
+ []int{2, 1, 7, 0, 6, 3, 4, 5}, // Perm(8)
+ []int{8, 7, 5, 3, 4, 6, 0, 1, 2}, // Perm(9)
+ []int{1, 0, 2, 5, 7, 6, 9, 8, 3, 4}, // Perm(10)
+ uint32(4059586549), // Uint32()
+ uint32(1052117029), // Uint32()
+ uint32(2817310706), // Uint32()
+ uint32(233405013), // Uint32()
+ uint32(1578775030), // Uint32()
+ uint32(1243308993), // Uint32()
+ uint32(826517535), // Uint32()
+ uint32(2814630155), // Uint32()
+ uint32(3853314576), // Uint32()
+ uint32(718781857), // Uint32()
+ uint32(1239465936), // Uint32()
+ uint32(3876658295), // Uint32()
+ uint32(3649778518), // Uint32()
+ uint32(1172727096), // Uint32()
+ uint32(2615979505), // Uint32()
+ uint32(1089444252), // Uint32()
+ uint32(3327114623), // Uint32()
+ uint32(75079301), // Uint32()
+ uint32(3380456901), // Uint32()
+ uint32(3433369789), // Uint32()
+}
diff --git a/src/math/rand/rng.go b/src/math/rand/rng.go
new file mode 100644
index 0000000..947c49f
--- /dev/null
+++ b/src/math/rand/rng.go
@@ -0,0 +1,246 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package rand
+
+/*
+ * Uniform distribution
+ *
+ * algorithm by
+ * DP Mitchell and JA Reeds
+ */
+
+const (
+ _LEN = 607
+ _TAP = 273
+ _MAX = 1 << 63
+ _MASK = _MAX - 1
+ _A = 48271
+ _M = (1 << 31) - 1
+ _Q = 44488
+ _R = 3399
+)
+
+var (
+ // cooked random numbers
+ // the state of the rng
+ // after 780e10 iterations
+ rng_cooked [_LEN]int64 = [...]int64{
+ 5041579894721019882, 4646389086726545243, 1395769623340756751, 5333664234075297259,
+ 2875692520355975054, 9033628115061424579, 7143218595135194537, 4812947590706362721,
+ 7937252194349799378, 5307299880338848416, 8209348851763925077, 2115741599318814044,
+ 4593015457530856296, 8140875735541888011, 3319429241265089026, 8619815648190321034,
+ 1727074043483619500, 113108499721038619, 4569519971459345583, 5062833859075314731,
+ 2387618771259064424, 2716131344356686112, 6559392774825876886, 7650093201692370310,
+ 7684323884043752161, 257867835996031390, 6593456519409015164, 271327514973697897,
+ 2789386447340118284, 1065192797246149621, 3344507881999356393, 4459797941780066633,
+ 7465081662728599889, 1014950805555097187, 4449440729345990775, 3481109366438502643,
+ 2418672789110888383, 5796562887576294778, 4484266064449540171, 3738982361971787048,
+ 4523597184512354423, 10530508058128498, 8633833783282346118, 2625309929628791628,
+ 8660405965245884302, 10162832508971942, 6540714680961817391, 7031802312784620857,
+ 6240911277345944669, 831864355460801054, 8004434137542152891, 2116287251661052151,
+ 2202309800992166967, 9161020366945053561, 4069299552407763864, 4936383537992622449,
+ 457351505131524928, 342195045928179354, 2847771682816600509, 2068020115986376518,
+ 4368649989588021065, 887231587095185257, 5563591506886576496, 6816225200251950296,
+ 5616972787034086048, 8471809303394836566, 1686575021641186857, 4045484338074262002,
+ 4244156215201778923, 7848217333783577387, 5632136521049761902, 833283142057835272,
+ 9029726508369077193, 3243583134664087292, 4316371101804477087, 8937849979965997980,
+ 6446940406810434101, 1679342092332374735, 6050638460742422078, 6993520719509581582,
+ 7640877852514293609, 5881353426285907985, 812786550756860885, 4541845584483343330,
+ 2725470216277009086, 4980675660146853729, 5210769080603236061, 8894283318990530821,
+ 6326442804750084282, 1495812843684243920, 7069751578799128019, 7370257291860230865,
+ 6756929275356942261, 4706794511633873654, 7824520467827898663, 8549875090542453214,
+ 33650829478596156, 1328918435751322643, 7297902601803624459, 1011190183918857495,
+ 2238025036817854944, 5147159997473910359, 896512091560522982, 2659470849286379941,
+ 6097729358393448602, 1731725986304753684, 4106255841983812711, 8327155210721535508,
+ 8477511620686074402, 5803876044675762232, 8435417780860221662, 5988852856651071244,
+ 4715837297103951910, 7566171971264485114, 505808562678895611, 5070098180695063370,
+ 842110666775871513, 572156825025677802, 1791881013492340891, 3393267094866038768,
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+ 4707864159563588614, 5640248530963493951, 5982507712689997893, 3315098242282210105,
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+ 7062410411742090847, 741381002980207668, 6027994129690250817, 2497829994150063930,
+ 6251390334426228834, 1368930247903518833, 8809096399316380241, 6492004350391900708,
+ 2462145737463489636, 404828418920299174, 4153026434231690595, 261785715255475940,
+ 5464715384600071357, 592710404378763017, 6764129236657751224, 8513655718539357449,
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+ 7150122561309371392, 368107899140673753, 3115186834558311558, 2915636353584281051,
+ 4782583894627718279, 6718292300699989587, 8387085186914375220, 3387513132024756289,
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+ 246994305896273627, 866417324803099287, 6473547110565816071, 3092379936208876896,
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+ 8585842181454472135, 6137678347805511274, 2070447184436970006, 5708223427705576541,
+ 5999657892458244504, 4358391411789012426, 325123008708389849, 6837621693887290924,
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+ 1044646352569048800, 9106614159853161675, 829256115228593269, 4919284369102997000,
+ 2681532557646850893, 3681559472488511871, 5307999518958214035, 6334130388442829274,
+ 2658708232916537604, 1163313865052186287, 581945337509520675, 3648778920718647903,
+ 4423673246306544414, 1620799783996955743, 220828013409515943, 8150384699999389761,
+ 4287360518296753003, 4590000184845883843, 5513660857261085186, 6964829100392774275,
+ 478991688350776035, 8746140185685648781, 228500091334420247, 1356187007457302238,
+ 3019253992034194581, 3152601605678500003, 430152752706002213, 5559581553696971176,
+ 4916432985369275664, 663574931734554391, 3420773838927732076, 2868348622579915573,
+ 1999319134044418520, 3328689518636282723, 2587672709781371173, 1517255313529399333,
+ 3092343956317362483, 3662252519007064108, 972445599196498113, 7664865435875959367,
+ 1708913533482282562, 6917817162668868494, 3217629022545312900, 2570043027221707107,
+ 8739788839543624613, 2488075924621352812, 4694002395387436668, 4559628481798514356,
+ 2997203966153298104, 1282559373026354493, 240113143146674385, 8665713329246516443,
+ 628141331766346752, 4571950817186770476, 1472811188152235408, 7596648026010355826,
+ 6091219417754424743, 7834161864828164065, 7103445518877254909, 4390861237357459201,
+ 4442653864240571734, 8903482404847331368, 622261699494173647, 6037261250297213248,
+ 504404948065709118, 7275215526217113061, 1011176780856001400, 2194750105623461063,
+ 2623071828615234808, 5157313728073836108, 3738405111966602044, 2539767524076729570,
+ 2467284396349269342, 5256026990536851868, 7841086888628396109, 6640857538655893162,
+ 1202087339038317498, 2113514992440715978, 7534350895342931403, 4925284734898484745,
+ 5145623771477493805, 8225140880134972332, 2719520354384050532, 9132346697815513771,
+ 4332154495710163773, 7137789594094346916, 6994721091344268833, 6667228574869048934,
+ 655440045726677499, 59934747298466858, 6124974028078036405, 8957774780655365418,
+ 2332206071942466437, 1701056712286369627, 3154897383618636503, 1637766181387607527,
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+ 4287946071480840813, 8362686366770308971, 6486469209321732151, 3617727845841796026,
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+ 1326024180520890843, 7537449876596048829, 5464680203499696154, 3189671183162196045,
+ 6346751753565857109, 241159987320630307, 3095793449658682053, 8978332846736310159,
+ 2902794662273147216, 7208698530190629697, 7276901792339343736, 1732385229314443140,
+ 4133292154170828382, 2918308698224194548, 1519461397937144458, 5293934712616591764,
+ 4922828954023452664, 2879211533496425641, 5896236396443472108, 8465043815351752425,
+ 7329020396871624740, 8915471717014488588, 2944902635677463047, 7052079073493465134,
+ 8382142935188824023, 9103922860780351547, 4152330101494654406,
+ }
+)
+
+type rngSource struct {
+ tap int // index into vec
+ feed int // index into vec
+ vec [_LEN]int64 // current feedback register
+}
+
+// seed rng x[n+1] = 48271 * x[n] mod (2**31 - 1)
+func seedrand(x int32) int32 {
+ hi := x / _Q
+ lo := x % _Q
+ x = _A*lo - _R*hi
+ if x < 0 {
+ x += _M
+ }
+ return x
+}
+
+// Seed uses the provided seed value to initialize the generator to a deterministic state.
+func (rng *rngSource) Seed(seed int64) {
+ rng.tap = 0
+ rng.feed = _LEN - _TAP
+
+ seed = seed % _M
+ if seed < 0 {
+ seed += _M
+ }
+ if seed == 0 {
+ seed = 89482311
+ }
+
+ x := int32(seed)
+ for i := -20; i < _LEN; i++ {
+ x = seedrand(x)
+ if i >= 0 {
+ var u int64
+ u = int64(x) << 40
+ x = seedrand(x)
+ u ^= int64(x) << 20
+ x = seedrand(x)
+ u ^= int64(x)
+ u ^= rng_cooked[i]
+ rng.vec[i] = u & _MASK
+ }
+ }
+}
+
+// Int63 returns a non-negative pseudo-random 63-bit integer as an int64.
+func (rng *rngSource) Int63() int64 {
+ rng.tap--
+ if rng.tap < 0 {
+ rng.tap += _LEN
+ }
+
+ rng.feed--
+ if rng.feed < 0 {
+ rng.feed += _LEN
+ }
+
+ x := (rng.vec[rng.feed] + rng.vec[rng.tap]) & _MASK
+ rng.vec[rng.feed] = x
+ return x
+}
diff --git a/src/math/rand/zipf.go b/src/math/rand/zipf.go
new file mode 100644
index 0000000..8db2c6f
--- /dev/null
+++ b/src/math/rand/zipf.go
@@ -0,0 +1,75 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// W.Hormann, G.Derflinger:
+// "Rejection-Inversion to Generate Variates
+// from Monotone Discrete Distributions"
+// http://eeyore.wu-wien.ac.at/papers/96-04-04.wh-der.ps.gz
+
+package rand
+
+import "math"
+
+// A Zipf generates Zipf distributed variates.
+type Zipf struct {
+ r *Rand
+ imax float64
+ v float64
+ q float64
+ s float64
+ oneminusQ float64
+ oneminusQinv float64
+ hxm float64
+ hx0minusHxm float64
+}
+
+func (z *Zipf) h(x float64) float64 {
+ return math.Exp(z.oneminusQ*math.Log(z.v+x)) * z.oneminusQinv
+}
+
+func (z *Zipf) hinv(x float64) float64 {
+ return math.Exp(z.oneminusQinv*math.Log(z.oneminusQ*x)) - z.v
+}
+
+// NewZipf returns a Zipf generating variates p(k) on [0, imax]
+// proportional to (v+k)**(-s) where s>1 and k>=0, and v>=1.
+func NewZipf(r *Rand, s float64, v float64, imax uint64) *Zipf {
+ z := new(Zipf)
+ if s <= 1.0 || v < 1 {
+ return nil
+ }
+ z.r = r
+ z.imax = float64(imax)
+ z.v = v
+ z.q = s
+ z.oneminusQ = 1.0 - z.q
+ z.oneminusQinv = 1.0 / z.oneminusQ
+ z.hxm = z.h(z.imax + 0.5)
+ z.hx0minusHxm = z.h(0.5) - math.Exp(math.Log(z.v)*(-z.q)) - z.hxm
+ z.s = 1 - z.hinv(z.h(1.5)-math.Exp(-z.q*math.Log(z.v+1.0)))
+ return z
+}
+
+// Uint64 returns a value drawn from the Zipf distribution described
+// by the Zipf object.
+func (z *Zipf) Uint64() uint64 {
+ if z == nil {
+ panic("rand: nil Zipf")
+ }
+ k := 0.0
+
+ for {
+ r := z.r.Float64() // r on [0,1]
+ ur := z.hxm + r*z.hx0minusHxm
+ x := z.hinv(ur)
+ k = math.Floor(x + 0.5)
+ if k-x <= z.s {
+ break
+ }
+ if ur >= z.h(k+0.5)-math.Exp(-math.Log(k+z.v)*z.q) {
+ break
+ }
+ }
+ return uint64(k)
+}
diff --git a/src/math/remainder.go b/src/math/remainder.go
new file mode 100644
index 0000000..9a4e415
--- /dev/null
+++ b/src/math/remainder.go
@@ -0,0 +1,85 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code and the comment below are from
+// FreeBSD's /usr/src/lib/msun/src/e_remainder.c and came
+// with this notice. The go code is a simplified version of
+// the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_remainder(x,y)
+// Return :
+// returns x REM y = x - [x/y]*y as if in infinite
+// precision arithmetic, where [x/y] is the (infinite bit)
+// integer nearest x/y (in half way cases, choose the even one).
+// Method :
+// Based on Mod() returning x - [x/y]chopped * y exactly.
+
+// Remainder returns the IEEE 754 floating-point remainder of x/y.
+//
+// Special cases are:
+// Remainder(±Inf, y) = NaN
+// Remainder(NaN, y) = NaN
+// Remainder(x, 0) = NaN
+// Remainder(x, ±Inf) = x
+// Remainder(x, NaN) = NaN
+func Remainder(x, y float64) float64
+
+func remainder(x, y float64) float64 {
+ const (
+ Tiny = 4.45014771701440276618e-308 // 0x0020000000000000
+ HalfMax = MaxFloat64 / 2
+ )
+ // special cases
+ switch {
+ case IsNaN(x) || IsNaN(y) || IsInf(x, 0) || y == 0:
+ return NaN()
+ case IsInf(y, 0):
+ return x
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ if y < 0 {
+ y = -y
+ }
+ if x == y {
+ return 0
+ }
+ if y <= HalfMax {
+ x = Mod(x, y+y) // now x < 2y
+ }
+ if y < Tiny {
+ if x+x > y {
+ x -= y
+ if x+x >= y {
+ x -= y
+ }
+ }
+ } else {
+ yHalf := 0.5 * y
+ if x > yHalf {
+ x -= y
+ if x >= yHalf {
+ x -= y
+ }
+ }
+ }
+ if sign {
+ x = -x
+ }
+ return x
+}
diff --git a/src/math/remainder_386.s b/src/math/remainder_386.s
new file mode 100644
index 0000000..318fa2c
--- /dev/null
+++ b/src/math/remainder_386.s
@@ -0,0 +1,17 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Remainder(x, y float64) float64
+TEXT ·Remainder(SB),NOSPLIT,$0
+ FMOVD y+8(FP), F0 // F0=y
+ FMOVD x+0(FP), F0 // F0=x, F1=y
+ FPREM1 // F0=reduced_x, F1=y
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=x-q*y
+ FMOVDP F0, ret+16(FP)
+ RET
diff --git a/src/math/remainder_amd64.s b/src/math/remainder_amd64.s
new file mode 100644
index 0000000..f7fda99
--- /dev/null
+++ b/src/math/remainder_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Remainder(SB),NOSPLIT,$0
+ JMP ·remainder(SB)
diff --git a/src/math/remainder_amd64p32.s b/src/math/remainder_amd64p32.s
new file mode 100644
index 0000000..cd5cf55
--- /dev/null
+++ b/src/math/remainder_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "remainder_amd64.s"
diff --git a/src/math/remainder_arm.s b/src/math/remainder_arm.s
new file mode 100644
index 0000000..1ae597a
--- /dev/null
+++ b/src/math/remainder_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Remainder(SB),NOSPLIT,$0
+ B ·remainder(SB)
diff --git a/src/math/signbit.go b/src/math/signbit.go
new file mode 100644
index 0000000..670cc1a
--- /dev/null
+++ b/src/math/signbit.go
@@ -0,0 +1,10 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Signbit returns true if x is negative or negative zero.
+func Signbit(x float64) bool {
+ return Float64bits(x)&(1<<63) != 0
+}
diff --git a/src/math/sin.go b/src/math/sin.go
new file mode 100644
index 0000000..ed85f21
--- /dev/null
+++ b/src/math/sin.go
@@ -0,0 +1,224 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point sine and cosine.
+*/
+
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+//
+// sin.c
+//
+// Circular sine
+//
+// SYNOPSIS:
+//
+// double x, y, sin();
+// y = sin( x );
+//
+// DESCRIPTION:
+//
+// Range reduction is into intervals of pi/4. The reduction error is nearly
+// eliminated by contriving an extended precision modular arithmetic.
+//
+// Two polynomial approximating functions are employed.
+// Between 0 and pi/4 the sine is approximated by
+// x + x**3 P(x**2).
+// Between pi/4 and pi/2 the cosine is represented as
+// 1 - x**2 Q(x**2).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC 0, 10 150000 3.0e-17 7.8e-18
+// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
+//
+// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss
+// is not gradual, but jumps suddenly to about 1 part in 10e7. Results may
+// be meaningless for x > 2**49 = 5.6e14.
+//
+// cos.c
+//
+// Circular cosine
+//
+// SYNOPSIS:
+//
+// double x, y, cos();
+// y = cos( x );
+//
+// DESCRIPTION:
+//
+// Range reduction is into intervals of pi/4. The reduction error is nearly
+// eliminated by contriving an extended precision modular arithmetic.
+//
+// Two polynomial approximating functions are employed.
+// Between 0 and pi/4 the cosine is approximated by
+// 1 - x**2 Q(x**2).
+// Between pi/4 and pi/2 the sine is represented as
+// x + x**3 P(x**2).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
+// DEC 0,+1.07e9 17000 3.0e-17 7.2e-18
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// sin coefficients
+var _sin = [...]float64{
+ 1.58962301576546568060E-10, // 0x3de5d8fd1fd19ccd
+ -2.50507477628578072866E-8, // 0xbe5ae5e5a9291f5d
+ 2.75573136213857245213E-6, // 0x3ec71de3567d48a1
+ -1.98412698295895385996E-4, // 0xbf2a01a019bfdf03
+ 8.33333333332211858878E-3, // 0x3f8111111110f7d0
+ -1.66666666666666307295E-1, // 0xbfc5555555555548
+}
+
+// cos coefficients
+var _cos = [...]float64{
+ -1.13585365213876817300E-11, // 0xbda8fa49a0861a9b
+ 2.08757008419747316778E-9, // 0x3e21ee9d7b4e3f05
+ -2.75573141792967388112E-7, // 0xbe927e4f7eac4bc6
+ 2.48015872888517045348E-5, // 0x3efa01a019c844f5
+ -1.38888888888730564116E-3, // 0xbf56c16c16c14f91
+ 4.16666666666665929218E-2, // 0x3fa555555555554b
+}
+
+// Cos returns the cosine of the radian argument x.
+//
+// Special cases are:
+// Cos(±Inf) = NaN
+// Cos(NaN) = NaN
+func Cos(x float64) float64
+
+func cos(x float64) float64 {
+ const (
+ PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
+ M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
+ )
+ // special cases
+ switch {
+ case IsNaN(x) || IsInf(x, 0):
+ return NaN()
+ }
+
+ // make argument positive
+ sign := false
+ if x < 0 {
+ x = -x
+ }
+
+ j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y := float64(j) // integer part of x/(Pi/4), as float
+
+ // map zeros to origin
+ if j&1 == 1 {
+ j += 1
+ y += 1
+ }
+ j &= 7 // octant modulo 2Pi radians (360 degrees)
+ if j > 3 {
+ j -= 4
+ sign = !sign
+ }
+ if j > 1 {
+ sign = !sign
+ }
+
+ z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+ zz := z * z
+ if j == 1 || j == 2 {
+ y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+ } else {
+ y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+ }
+ if sign {
+ y = -y
+ }
+ return y
+}
+
+// Sin returns the sine of the radian argument x.
+//
+// Special cases are:
+// Sin(±0) = ±0
+// Sin(±Inf) = NaN
+// Sin(NaN) = NaN
+func Sin(x float64) float64
+
+func sin(x float64) float64 {
+ const (
+ PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
+ M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x):
+ return x // return ±0 || NaN()
+ case IsInf(x, 0):
+ return NaN()
+ }
+
+ // make argument positive but save the sign
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y := float64(j) // integer part of x/(Pi/4), as float
+
+ // map zeros to origin
+ if j&1 == 1 {
+ j += 1
+ y += 1
+ }
+ j &= 7 // octant modulo 2Pi radians (360 degrees)
+ // reflect in x axis
+ if j > 3 {
+ sign = !sign
+ j -= 4
+ }
+
+ z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+ zz := z * z
+ if j == 1 || j == 2 {
+ y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+ } else {
+ y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+ }
+ if sign {
+ y = -y
+ }
+ return y
+}
diff --git a/src/math/sin_386.s b/src/math/sin_386.s
new file mode 100644
index 0000000..ccc8e64
--- /dev/null
+++ b/src/math/sin_386.s
@@ -0,0 +1,47 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Cos(x float64) float64
+TEXT ·Cos(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FCOS // F0=cos(x) if -2**63 < x < 2**63
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE 3(PC) // jump if x outside range
+ FMOVDP F0, ret+8(FP)
+ RET
+ FLDPI // F0=Pi, F1=x
+ FADDD F0, F0 // F0=2*Pi, F1=x
+ FXCHD F0, F1 // F0=x, F1=2*Pi
+ FPREM1 // F0=reduced_x, F1=2*Pi
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=reduced_x
+ FCOS // F0=cos(reduced_x)
+ FMOVDP F0, ret+8(FP)
+ RET
+
+// func Sin(x float64) float64
+TEXT ·Sin(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSIN // F0=sin(x) if -2**63 < x < 2**63
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE 3(PC) // jump if x outside range
+ FMOVDP F0, ret+8(FP)
+ RET
+ FLDPI // F0=Pi, F1=x
+ FADDD F0, F0 // F0=2*Pi, F1=x
+ FXCHD F0, F1 // F0=x, F1=2*Pi
+ FPREM1 // F0=reduced_x, F1=2*Pi
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=reduced_x
+ FSIN // F0=sin(reduced_x)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/sin_amd64.s b/src/math/sin_amd64.s
new file mode 100644
index 0000000..0c33cec
--- /dev/null
+++ b/src/math/sin_amd64.s
@@ -0,0 +1,11 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Sin(SB),NOSPLIT,$0
+ JMP ·sin(SB)
+
+TEXT ·Cos(SB),NOSPLIT,$0
+ JMP ·cos(SB)
diff --git a/src/math/sin_amd64p32.s b/src/math/sin_amd64p32.s
new file mode 100644
index 0000000..9f93eba
--- /dev/null
+++ b/src/math/sin_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "sin_amd64.s"
diff --git a/src/math/sin_arm.s b/src/math/sin_arm.s
new file mode 100644
index 0000000..467af3d
--- /dev/null
+++ b/src/math/sin_arm.s
@@ -0,0 +1,11 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Sin(SB),NOSPLIT,$0
+ B ·sin(SB)
+
+TEXT ·Cos(SB),NOSPLIT,$0
+ B ·cos(SB)
diff --git a/src/math/sincos.go b/src/math/sincos.go
new file mode 100644
index 0000000..7180303
--- /dev/null
+++ b/src/math/sincos.go
@@ -0,0 +1,69 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Coefficients _sin[] and _cos[] are found in pkg/math/sin.go.
+
+// Sincos returns Sin(x), Cos(x).
+//
+// Special cases are:
+// Sincos(±0) = ±0, 1
+// Sincos(±Inf) = NaN, NaN
+// Sincos(NaN) = NaN, NaN
+func Sincos(x float64) (sin, cos float64)
+
+func sincos(x float64) (sin, cos float64) {
+ const (
+ PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
+ M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
+ )
+ // special cases
+ switch {
+ case x == 0:
+ return x, 1 // return ±0.0, 1.0
+ case IsNaN(x) || IsInf(x, 0):
+ return NaN(), NaN()
+ }
+
+ // make argument positive
+ sinSign, cosSign := false, false
+ if x < 0 {
+ x = -x
+ sinSign = true
+ }
+
+ j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y := float64(j) // integer part of x/(Pi/4), as float
+
+ if j&1 == 1 { // map zeros to origin
+ j += 1
+ y += 1
+ }
+ j &= 7 // octant modulo 2Pi radians (360 degrees)
+ if j > 3 { // reflect in x axis
+ j -= 4
+ sinSign, cosSign = !sinSign, !cosSign
+ }
+ if j > 1 {
+ cosSign = !cosSign
+ }
+
+ z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+ zz := z * z
+ cos = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+ sin = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+ if j == 1 || j == 2 {
+ sin, cos = cos, sin
+ }
+ if cosSign {
+ cos = -cos
+ }
+ if sinSign {
+ sin = -sin
+ }
+ return
+}
diff --git a/src/math/sincos_386.s b/src/math/sincos_386.s
new file mode 100644
index 0000000..83af501
--- /dev/null
+++ b/src/math/sincos_386.s
@@ -0,0 +1,28 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sincos(x float64) (sin, cos float64)
+TEXT ·Sincos(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FSINCOS // F0=cos(x), F1=sin(x) if -2**63 < x < 2**63
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE 4(PC) // jump if x outside range
+ FMOVDP F0, cos+16(FP) // F0=sin(x)
+ FMOVDP F0, sin+8(FP)
+ RET
+ FLDPI // F0=Pi, F1=x
+ FADDD F0, F0 // F0=2*Pi, F1=x
+ FXCHD F0, F1 // F0=x, F1=2*Pi
+ FPREM1 // F0=reduced_x, F1=2*Pi
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=reduced_x
+ FSINCOS // F0=cos(reduced_x), F1=sin(reduced_x)
+ FMOVDP F0, cos+16(FP) // F0=sin(reduced_x)
+ FMOVDP F0, sin+8(FP)
+ RET
diff --git a/src/math/sincos_amd64.s b/src/math/sincos_amd64.s
new file mode 100644
index 0000000..59bf55f
--- /dev/null
+++ b/src/math/sincos_amd64.s
@@ -0,0 +1,142 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// The method is based on a paper by Naoki Shibata: "Efficient evaluation
+// methods of elementary functions suitable for SIMD computation", Proc.
+// of International Supercomputing Conference 2010 (ISC'10), pp. 25 -- 32
+// (May 2010). The paper is available at
+// http://www.springerlink.com/content/340228x165742104/
+//
+// The original code and the constants below are from the author's
+// implementation available at http://freshmeat.net/projects/sleef.
+// The README file says, "The software is in public domain.
+// You can use the software without any obligation."
+//
+// This code is a simplified version of the original.
+
+#define PosOne 0x3FF0000000000000
+#define PosInf 0x7FF0000000000000
+#define NaN 0x7FF8000000000001
+#define PI4A 0.7853981554508209228515625 // pi/4 split into three parts
+#define PI4B 0.794662735614792836713604629039764404296875e-8
+#define PI4C 0.306161699786838294306516483068750264552437361480769e-16
+#define M4PI 1.273239544735162542821171882678754627704620361328125 // 4/pi
+#define T0 1.0
+#define T1 -8.33333333333333333333333e-02 // (-1.0/12)
+#define T2 2.77777777777777777777778e-03 // (+1.0/360)
+#define T3 -4.96031746031746031746032e-05 // (-1.0/20160)
+#define T4 5.51146384479717813051146e-07 // (+1.0/1814400)
+
+// func Sincos(d float64) (sin, cos float64)
+TEXT ·Sincos(SB),NOSPLIT,$0
+ // test for special cases
+ MOVQ $~(1<<63), DX // sign bit mask
+ MOVQ x+0(FP), BX
+ ANDQ BX, DX
+ JEQ isZero
+ MOVQ $PosInf, AX
+ CMPQ AX, DX
+ JLE isInfOrNaN
+ // Reduce argument
+ MOVQ BX, X7 // x7= d
+ MOVQ DX, X0 // x0= |d|
+ MOVSD $M4PI, X2
+ MULSD X0, X2
+ CVTTSD2SQ X2, BX // bx= q
+ MOVQ $1, AX
+ ANDQ BX, AX
+ ADDQ BX, AX
+ CVTSQ2SD AX, X2
+ MOVSD $PI4A, X3
+ MULSD X2, X3
+ SUBSD X3, X0
+ MOVSD $PI4B, X3
+ MULSD X2, X3
+ SUBSD X3, X0
+ MOVSD $PI4C, X3
+ MULSD X2, X3
+ SUBSD X3, X0
+ MULSD $0.125, X0 // x0= x, x7= d, bx= q
+ // Evaluate Taylor series
+ MULSD X0, X0
+ MOVSD $T4, X2
+ MULSD X0, X2
+ ADDSD $T3, X2
+ MULSD X0, X2
+ ADDSD $T2, X2
+ MULSD X0, X2
+ ADDSD $T1, X2
+ MULSD X0, X2
+ ADDSD $T0, X2
+ MULSD X2, X0 // x0= x, x7= d, bx= q
+ // Apply double angle formula
+ MOVSD $4.0, X2
+ SUBSD X0, X2
+ MULSD X2, X0
+ MOVSD $4.0, X2
+ SUBSD X0, X2
+ MULSD X2, X0
+ MOVSD $4.0, X2
+ SUBSD X0, X2
+ MULSD X2, X0
+ MULSD $0.5, X0 // x0= x, x7= d, bx= q
+ // sin = sqrt((2 - x) * x)
+ MOVSD $2.0, X2
+ SUBSD X0, X2
+ MULSD X0, X2
+ SQRTSD X2, X2 // x0= x, x2= z, x7= d, bx= q
+ // cos = 1 - x
+ MOVSD $1.0, X1
+ SUBSD X0, X1 // x1= x, x2= z, x7= d, bx= q
+ // if ((q + 1) & 2) != 0 { sin, cos = cos, sin }
+ MOVQ $1, DX
+ ADDQ BX, DX
+ ANDQ $2, DX
+ SHRQ $1, DX
+ SUBQ $1, DX
+ MOVQ DX, X3
+ // sin = (y & z) | (^y & x)
+ MOVAPD X2, X0
+ ANDPD X3, X0 // x0= sin
+ MOVAPD X3, X4
+ ANDNPD X1, X4
+ ORPD X4, X0 // x0= sin, x1= x, x2= z, x3= y, x7= d, bx= q
+ // cos = (y & x) | (^y & z)
+ ANDPD X3, X1 // x1= cos
+ ANDNPD X2, X3
+ ORPD X3, X1 // x0= sin, x1= cos, x7= d, bx= q
+ // if ((q & 4) != 0) != (d < 0) { sin = -sin }
+ MOVQ BX, AX
+ MOVQ $61, CX
+ SHLQ CX, AX
+ MOVQ AX, X3
+ XORPD X7, X3
+ MOVQ $(1<<63), AX
+ MOVQ AX, X2 // x2= -0.0
+ ANDPD X2, X3
+ ORPD X3, X0 // x0= sin, x1= cos, x2= -0.0, bx= q
+ // if ((q + 2) & 4) != 0 { cos = -cos }
+ MOVQ $2, AX
+ ADDQ AX, BX
+ MOVQ $61, CX
+ SHLQ CX, BX
+ MOVQ BX, X3
+ ANDPD X2, X3
+ ORPD X3, X1 // x0= sin, x1= cos
+ // return (sin, cos)
+ MOVSD X0, sin+8(FP)
+ MOVSD X1, cos+16(FP)
+ RET
+isZero: // return (±0.0, 1.0)
+ MOVQ BX, sin+8(FP)
+ MOVQ $PosOne, AX
+ MOVQ AX, cos+16(FP)
+ RET
+isInfOrNaN: // return (NaN, NaN)
+ MOVQ $NaN, AX
+ MOVQ AX, sin+8(FP)
+ MOVQ AX, cos+16(FP)
+ RET
diff --git a/src/math/sincos_amd64p32.s b/src/math/sincos_amd64p32.s
new file mode 100644
index 0000000..360e94d
--- /dev/null
+++ b/src/math/sincos_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "sincos_amd64.s"
diff --git a/src/math/sincos_arm.s b/src/math/sincos_arm.s
new file mode 100644
index 0000000..9fe0482
--- /dev/null
+++ b/src/math/sincos_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Sincos(SB),NOSPLIT,$0
+ B ·sincos(SB)
diff --git a/src/math/sinh.go b/src/math/sinh.go
new file mode 100644
index 0000000..139b911
--- /dev/null
+++ b/src/math/sinh.go
@@ -0,0 +1,77 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point hyperbolic sine and cosine.
+
+ The exponential func is called for arguments
+ greater in magnitude than 0.5.
+
+ A series is used for arguments smaller in magnitude than 0.5.
+
+ Cosh(x) is computed from the exponential func for
+ all arguments.
+*/
+
+// Sinh returns the hyperbolic sine of x.
+//
+// Special cases are:
+// Sinh(±0) = ±0
+// Sinh(±Inf) = ±Inf
+// Sinh(NaN) = NaN
+func Sinh(x float64) float64 {
+ // The coefficients are #2029 from Hart & Cheney. (20.36D)
+ const (
+ P0 = -0.6307673640497716991184787251e+6
+ P1 = -0.8991272022039509355398013511e+5
+ P2 = -0.2894211355989563807284660366e+4
+ P3 = -0.2630563213397497062819489e+2
+ Q0 = -0.6307673640497716991212077277e+6
+ Q1 = 0.1521517378790019070696485176e+5
+ Q2 = -0.173678953558233699533450911e+3
+ )
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ var temp float64
+ switch true {
+ case x > 21:
+ temp = Exp(x) / 2
+
+ case x > 0.5:
+ temp = (Exp(x) - Exp(-x)) / 2
+
+ default:
+ sq := x * x
+ temp = (((P3*sq+P2)*sq+P1)*sq + P0) * x
+ temp = temp / (((sq+Q2)*sq+Q1)*sq + Q0)
+ }
+
+ if sign {
+ temp = -temp
+ }
+ return temp
+}
+
+// Cosh returns the hyperbolic cosine of x.
+//
+// Special cases are:
+// Cosh(±0) = 1
+// Cosh(±Inf) = +Inf
+// Cosh(NaN) = NaN
+func Cosh(x float64) float64 {
+ if x < 0 {
+ x = -x
+ }
+ if x > 21 {
+ return Exp(x) / 2
+ }
+ return (Exp(x) + Exp(-x)) / 2
+}
diff --git a/src/math/sqrt.go b/src/math/sqrt.go
new file mode 100644
index 0000000..fdc8699
--- /dev/null
+++ b/src/math/sqrt.go
@@ -0,0 +1,143 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_sqrt(x)
+// Return correctly rounded sqrt.
+// -----------------------------------------
+// | Use the hardware sqrt if you have one |
+// -----------------------------------------
+// Method:
+// Bit by bit method using integer arithmetic. (Slow, but portable)
+// 1. Normalization
+// Scale x to y in [1,4) with even powers of 2:
+// find an integer k such that 1 <= (y=x*2**(2k)) < 4, then
+// sqrt(x) = 2**k * sqrt(y)
+// 2. Bit by bit computation
+// Let q = sqrt(y) truncated to i bit after binary point (q = 1),
+// i 0
+// i+1 2
+// s = 2*q , and y = 2 * ( y - q ). (1)
+// i i i i
+//
+// To compute q from q , one checks whether
+// i+1 i
+//
+// -(i+1) 2
+// (q + 2 ) <= y. (2)
+// i
+// -(i+1)
+// If (2) is false, then q = q ; otherwise q = q + 2 .
+// i+1 i i+1 i
+//
+// With some algebraic manipulation, it is not difficult to see
+// that (2) is equivalent to
+// -(i+1)
+// s + 2 <= y (3)
+// i i
+//
+// The advantage of (3) is that s and y can be computed by
+// i i
+// the following recurrence formula:
+// if (3) is false
+//
+// s = s , y = y ; (4)
+// i+1 i i+1 i
+//
+// otherwise,
+// -i -(i+1)
+// s = s + 2 , y = y - s - 2 (5)
+// i+1 i i+1 i i
+//
+// One may easily use induction to prove (4) and (5).
+// Note. Since the left hand side of (3) contain only i+2 bits,
+// it does not necessary to do a full (53-bit) comparison
+// in (3).
+// 3. Final rounding
+// After generating the 53 bits result, we compute one more bit.
+// Together with the remainder, we can decide whether the
+// result is exact, bigger than 1/2ulp, or less than 1/2ulp
+// (it will never equal to 1/2ulp).
+// The rounding mode can be detected by checking whether
+// huge + tiny is equal to huge, and whether huge - tiny is
+// equal to huge for some floating point number "huge" and "tiny".
+//
+//
+// Notes: Rounding mode detection omitted. The constants "mask", "shift",
+// and "bias" are found in src/math/bits.go
+
+// Sqrt returns the square root of x.
+//
+// Special cases are:
+// Sqrt(+Inf) = +Inf
+// Sqrt(±0) = ±0
+// Sqrt(x < 0) = NaN
+// Sqrt(NaN) = NaN
+func Sqrt(x float64) float64
+
+func sqrt(x float64) float64 {
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x) || IsInf(x, 1):
+ return x
+ case x < 0:
+ return NaN()
+ }
+ ix := Float64bits(x)
+ // normalize x
+ exp := int((ix >> shift) & mask)
+ if exp == 0 { // subnormal x
+ for ix&1<<shift == 0 {
+ ix <<= 1
+ exp--
+ }
+ exp++
+ }
+ exp -= bias // unbias exponent
+ ix &^= mask << shift
+ ix |= 1 << shift
+ if exp&1 == 1 { // odd exp, double x to make it even
+ ix <<= 1
+ }
+ exp >>= 1 // exp = exp/2, exponent of square root
+ // generate sqrt(x) bit by bit
+ ix <<= 1
+ var q, s uint64 // q = sqrt(x)
+ r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
+ for r != 0 {
+ t := s + r
+ if t <= ix {
+ s = t + r
+ ix -= t
+ q += r
+ }
+ ix <<= 1
+ r >>= 1
+ }
+ // final rounding
+ if ix != 0 { // remainder, result not exact
+ q += q & 1 // round according to extra bit
+ }
+ ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
+ return Float64frombits(ix)
+}
+
+func sqrtC(f float64, r *float64) {
+ *r = sqrt(f)
+}
diff --git a/src/math/sqrt_386.s b/src/math/sqrt_386.s
new file mode 100644
index 0000000..5234a1e
--- /dev/null
+++ b/src/math/sqrt_386.s
@@ -0,0 +1,12 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ FMOVD x+0(FP),F0
+ FSQRT
+ FMOVDP F0,ret+8(FP)
+ RET
diff --git a/src/math/sqrt_amd64.s b/src/math/sqrt_amd64.s
new file mode 100644
index 0000000..443d83f
--- /dev/null
+++ b/src/math/sqrt_amd64.s
@@ -0,0 +1,11 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ SQRTSD x+0(FP), X0
+ MOVSD X0, ret+8(FP)
+ RET
diff --git a/src/math/sqrt_amd64p32.s b/src/math/sqrt_amd64p32.s
new file mode 100644
index 0000000..d83a286
--- /dev/null
+++ b/src/math/sqrt_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "sqrt_amd64.s"
diff --git a/src/math/sqrt_arm.s b/src/math/sqrt_arm.s
new file mode 100644
index 0000000..4f9dc2e
--- /dev/null
+++ b/src/math/sqrt_arm.s
@@ -0,0 +1,12 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Sqrt(x float64) float64
+TEXT ·Sqrt(SB),NOSPLIT,$0
+ MOVD x+0(FP),F0
+ SQRTD F0,F0
+ MOVD F0,ret+8(FP)
+ RET
diff --git a/src/math/tan.go b/src/math/tan.go
new file mode 100644
index 0000000..285eff1
--- /dev/null
+++ b/src/math/tan.go
@@ -0,0 +1,130 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point tangent.
+*/
+
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+//
+// tan.c
+//
+// Circular tangent
+//
+// SYNOPSIS:
+//
+// double x, y, tan();
+// y = tan( x );
+//
+// DESCRIPTION:
+//
+// Returns the circular tangent of the radian argument x.
+//
+// Range reduction is modulo pi/4. A rational function
+// x + x**3 P(x**2)/Q(x**2)
+// is employed in the basic interval [0, pi/4].
+//
+// ACCURACY:
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC +-1.07e9 44000 4.1e-17 1.0e-17
+// IEEE +-1.07e9 30000 2.9e-16 8.1e-17
+//
+// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss
+// is not gradual, but jumps suddenly to about 1 part in 10e7. Results may
+// be meaningless for x > 2**49 = 5.6e14.
+// [Accuracy loss statement from sin.go comments.]
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// tan coefficients
+var _tanP = [...]float64{
+ -1.30936939181383777646E4, // 0xc0c992d8d24f3f38
+ 1.15351664838587416140E6, // 0x413199eca5fc9ddd
+ -1.79565251976484877988E7, // 0xc1711fead3299176
+}
+var _tanQ = [...]float64{
+ 1.00000000000000000000E0,
+ 1.36812963470692954678E4, //0x40cab8a5eeb36572
+ -1.32089234440210967447E6, //0xc13427bc582abc96
+ 2.50083801823357915839E7, //0x4177d98fc2ead8ef
+ -5.38695755929454629881E7, //0xc189afe03cbe5a31
+}
+
+// Tan returns the tangent of the radian argument x.
+//
+// Special cases are:
+// Tan(±0) = ±0
+// Tan(±Inf) = NaN
+// Tan(NaN) = NaN
+func Tan(x float64) float64
+
+func tan(x float64) float64 {
+ const (
+ PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts
+ PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000,
+ PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
+ M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x):
+ return x // return ±0 || NaN()
+ case IsInf(x, 0):
+ return NaN()
+ }
+
+ // make argument positive but save the sign
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+
+ j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
+ y := float64(j) // integer part of x/(Pi/4), as float
+
+ /* map zeros and singularities to origin */
+ if j&1 == 1 {
+ j += 1
+ y += 1
+ }
+
+ z := ((x - y*PI4A) - y*PI4B) - y*PI4C
+ zz := z * z
+
+ if zz > 1e-14 {
+ y = z + z*(zz*(((_tanP[0]*zz)+_tanP[1])*zz+_tanP[2])/((((zz+_tanQ[1])*zz+_tanQ[2])*zz+_tanQ[3])*zz+_tanQ[4]))
+ } else {
+ y = z
+ }
+ if j&2 == 2 {
+ y = -1 / y
+ }
+ if sign {
+ y = -y
+ }
+ return y
+}
diff --git a/src/math/tan_386.s b/src/math/tan_386.s
new file mode 100644
index 0000000..f1bdae1
--- /dev/null
+++ b/src/math/tan_386.s
@@ -0,0 +1,28 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+// func Tan(x float64) float64
+TEXT ·Tan(SB),NOSPLIT,$0
+ FMOVD x+0(FP), F0 // F0=x
+ FPTAN // F0=1, F1=tan(x) if -2**63 < x < 2**63
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE 4(PC) // jump if x outside range
+ FMOVDP F0, F0 // F0=tan(x)
+ FMOVDP F0, ret+8(FP)
+ RET
+ FLDPI // F0=Pi, F1=x
+ FADDD F0, F0 // F0=2*Pi, F1=x
+ FXCHD F0, F1 // F0=x, F1=2*Pi
+ FPREM1 // F0=reduced_x, F1=2*Pi
+ FSTSW AX // AX=status word
+ ANDW $0x0400, AX
+ JNE -3(PC) // jump if reduction incomplete
+ FMOVDP F0, F1 // F0=reduced_x
+ FPTAN // F0=1, F1=tan(reduced_x)
+ FMOVDP F0, F0 // F0=tan(reduced_x)
+ FMOVDP F0, ret+8(FP)
+ RET
diff --git a/src/math/tan_amd64.s b/src/math/tan_amd64.s
new file mode 100644
index 0000000..39aa080
--- /dev/null
+++ b/src/math/tan_amd64.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Tan(SB),NOSPLIT,$0
+ JMP ·tan(SB)
diff --git a/src/math/tan_amd64p32.s b/src/math/tan_amd64p32.s
new file mode 100644
index 0000000..9b3f70d
--- /dev/null
+++ b/src/math/tan_amd64p32.s
@@ -0,0 +1,5 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "tan_amd64.s"
diff --git a/src/math/tan_arm.s b/src/math/tan_arm.s
new file mode 100644
index 0000000..36c7c12
--- /dev/null
+++ b/src/math/tan_arm.s
@@ -0,0 +1,8 @@
+// Copyright 2011 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+#include "textflag.h"
+
+TEXT ·Tan(SB),NOSPLIT,$0
+ B ·tan(SB)
diff --git a/src/math/tanh.go b/src/math/tanh.go
new file mode 100644
index 0000000..cf0ffa1
--- /dev/null
+++ b/src/math/tanh.go
@@ -0,0 +1,97 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+// tanh.c
+//
+// Hyperbolic tangent
+//
+// SYNOPSIS:
+//
+// double x, y, tanh();
+//
+// y = tanh( x );
+//
+// DESCRIPTION:
+//
+// Returns hyperbolic tangent of argument in the range MINLOG to MAXLOG.
+// MAXLOG = 8.8029691931113054295988e+01 = log(2**127)
+// MINLOG = -8.872283911167299960540e+01 = log(2**-128)
+//
+// A rational function is used for |x| < 0.625. The form
+// x + x**3 P(x)/Q(x) of Cody & Waite is employed.
+// Otherwise,
+// tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
+//
+// ACCURACY:
+//
+// Relative error:
+// arithmetic domain # trials peak rms
+// IEEE -2,2 30000 2.5e-16 5.8e-17
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+//
+
+var tanhP = [...]float64{
+ -9.64399179425052238628E-1,
+ -9.92877231001918586564E1,
+ -1.61468768441708447952E3,
+}
+var tanhQ = [...]float64{
+ 1.12811678491632931402E2,
+ 2.23548839060100448583E3,
+ 4.84406305325125486048E3,
+}
+
+// Tanh returns the hyperbolic tangent of x.
+//
+// Special cases are:
+// Tanh(±0) = ±0
+// Tanh(±Inf) = ±1
+// Tanh(NaN) = NaN
+func Tanh(x float64) float64 {
+ const MAXLOG = 8.8029691931113054295988e+01 // log(2**127)
+ z := Abs(x)
+ switch {
+ case z > 0.5*MAXLOG:
+ if x < 0 {
+ return -1
+ }
+ return 1
+ case z >= 0.625:
+ s := Exp(2 * z)
+ z = 1 - 2/(s+1)
+ if x < 0 {
+ z = -z
+ }
+ default:
+ if x == 0 {
+ return x
+ }
+ s := x * x
+ z = x + x*s*((tanhP[0]*s+tanhP[1])*s+tanhP[2])/(((s+tanhQ[0])*s+tanhQ[1])*s+tanhQ[2])
+ }
+ return z
+}
diff --git a/src/math/unsafe.go b/src/math/unsafe.go
new file mode 100644
index 0000000..5ae6742
--- /dev/null
+++ b/src/math/unsafe.go
@@ -0,0 +1,21 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+import "unsafe"
+
+// Float32bits returns the IEEE 754 binary representation of f.
+func Float32bits(f float32) uint32 { return *(*uint32)(unsafe.Pointer(&f)) }
+
+// Float32frombits returns the floating point number corresponding
+// to the IEEE 754 binary representation b.
+func Float32frombits(b uint32) float32 { return *(*float32)(unsafe.Pointer(&b)) }
+
+// Float64bits returns the IEEE 754 binary representation of f.
+func Float64bits(f float64) uint64 { return *(*uint64)(unsafe.Pointer(&f)) }
+
+// Float64frombits returns the floating point number corresponding
+// the IEEE 754 binary representation b.
+func Float64frombits(b uint64) float64 { return *(*float64)(unsafe.Pointer(&b)) }