Upgrade to 3.29
Update V8 to 3.29.88.17 and update makefiles to support building on
all the relevant platforms.
Bug: 17370214
Change-Id: Ia3407c157fd8d72a93e23d8318ccaf6ecf77fa4e
diff --git a/third_party/fdlibm/LICENSE b/third_party/fdlibm/LICENSE
new file mode 100644
index 0000000..b54cb52
--- /dev/null
+++ b/third_party/fdlibm/LICENSE
@@ -0,0 +1,6 @@
+Copyright (C) 1993-2004 by Sun Microsystems, Inc. All rights reserved.
+
+Developed at SunSoft, a Sun Microsystems, Inc. business.
+Permission to use, copy, modify, and distribute this
+software is freely granted, provided that this notice
+is preserved.
diff --git a/third_party/fdlibm/README.v8 b/third_party/fdlibm/README.v8
new file mode 100644
index 0000000..ea8fdb6
--- /dev/null
+++ b/third_party/fdlibm/README.v8
@@ -0,0 +1,18 @@
+Name: Freely Distributable LIBM
+Short Name: fdlibm
+URL: http://www.netlib.org/fdlibm/
+Version: 5.3
+License: Freely Distributable.
+License File: LICENSE.
+Security Critical: yes.
+License Android Compatible: yes.
+
+Description:
+This is used to provide a accurate implementation for trigonometric functions
+used in V8.
+
+Local Modifications:
+For the use in V8, fdlibm has been reduced to include only sine, cosine and
+tangent. To make inlining into generated code possible, a large portion of
+that has been translated to Javascript. The rest remains in C, but has been
+refactored and reformatted to interoperate with the rest of V8.
diff --git a/third_party/fdlibm/fdlibm.cc b/third_party/fdlibm/fdlibm.cc
new file mode 100644
index 0000000..c009cd0
--- /dev/null
+++ b/third_party/fdlibm/fdlibm.cc
@@ -0,0 +1,281 @@
+// The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunSoft, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// The original source code covered by the above license above has been
+// modified significantly by Google Inc.
+// Copyright 2014 the V8 project authors. All rights reserved.
+
+#include "src/v8.h"
+
+#include "src/double.h"
+#include "third_party/fdlibm/fdlibm.h"
+
+
+namespace v8 {
+namespace fdlibm {
+
+#ifdef _MSC_VER
+inline double scalbn(double x, int y) { return _scalb(x, y); }
+#endif // _MSC_VER
+
+const double MathConstants::constants[] = {
+ 6.36619772367581382433e-01, // invpio2 0
+ 1.57079632673412561417e+00, // pio2_1 1
+ 6.07710050650619224932e-11, // pio2_1t 2
+ 6.07710050630396597660e-11, // pio2_2 3
+ 2.02226624879595063154e-21, // pio2_2t 4
+ 2.02226624871116645580e-21, // pio2_3 5
+ 8.47842766036889956997e-32, // pio2_3t 6
+ -1.66666666666666324348e-01, // S1 7 coefficients for sin
+ 8.33333333332248946124e-03, // 8
+ -1.98412698298579493134e-04, // 9
+ 2.75573137070700676789e-06, // 10
+ -2.50507602534068634195e-08, // 11
+ 1.58969099521155010221e-10, // S6 12
+ 4.16666666666666019037e-02, // C1 13 coefficients for cos
+ -1.38888888888741095749e-03, // 14
+ 2.48015872894767294178e-05, // 15
+ -2.75573143513906633035e-07, // 16
+ 2.08757232129817482790e-09, // 17
+ -1.13596475577881948265e-11, // C6 18
+ 3.33333333333334091986e-01, // T0 19 coefficients for tan
+ 1.33333333333201242699e-01, // 20
+ 5.39682539762260521377e-02, // 21
+ 2.18694882948595424599e-02, // 22
+ 8.86323982359930005737e-03, // 23
+ 3.59207910759131235356e-03, // 24
+ 1.45620945432529025516e-03, // 25
+ 5.88041240820264096874e-04, // 26
+ 2.46463134818469906812e-04, // 27
+ 7.81794442939557092300e-05, // 28
+ 7.14072491382608190305e-05, // 29
+ -1.85586374855275456654e-05, // 30
+ 2.59073051863633712884e-05, // T12 31
+ 7.85398163397448278999e-01, // pio4 32
+ 3.06161699786838301793e-17, // pio4lo 33
+ 6.93147180369123816490e-01, // ln2_hi 34
+ 1.90821492927058770002e-10, // ln2_lo 35
+ 1.80143985094819840000e+16, // 2^54 36
+ 6.666666666666666666e-01, // 2/3 37
+ 6.666666666666735130e-01, // LP1 38 coefficients for log1p
+ 3.999999999940941908e-01, // 39
+ 2.857142874366239149e-01, // 40
+ 2.222219843214978396e-01, // 41
+ 1.818357216161805012e-01, // 42
+ 1.531383769920937332e-01, // 43
+ 1.479819860511658591e-01, // LP7 44
+ 7.09782712893383973096e+02, // 45 overflow threshold for expm1
+ 1.44269504088896338700e+00, // 1/ln2 46
+ -3.33333333333331316428e-02, // Q1 47 coefficients for expm1
+ 1.58730158725481460165e-03, // 48
+ -7.93650757867487942473e-05, // 49
+ 4.00821782732936239552e-06, // 50
+ -2.01099218183624371326e-07, // Q5 51
+ 710.4758600739439 // 52 overflow threshold sinh, cosh
+};
+
+
+// Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
+static const int two_over_pi[] = {
+ 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C,
+ 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649,
+ 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44,
+ 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B,
+ 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D,
+ 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
+ 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330,
+ 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08,
+ 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
+ 0x73A8C9, 0x60E27B, 0xC08C6B};
+
+static const double zero = 0.0;
+static const double two24 = 1.6777216e+07;
+static const double one = 1.0;
+static const double twon24 = 5.9604644775390625e-08;
+
+static const double PIo2[] = {
+ 1.57079625129699707031e+00, // 0x3FF921FB, 0x40000000
+ 7.54978941586159635335e-08, // 0x3E74442D, 0x00000000
+ 5.39030252995776476554e-15, // 0x3CF84698, 0x80000000
+ 3.28200341580791294123e-22, // 0x3B78CC51, 0x60000000
+ 1.27065575308067607349e-29, // 0x39F01B83, 0x80000000
+ 1.22933308981111328932e-36, // 0x387A2520, 0x40000000
+ 2.73370053816464559624e-44, // 0x36E38222, 0x80000000
+ 2.16741683877804819444e-51 // 0x3569F31D, 0x00000000
+};
+
+
+int __kernel_rem_pio2(double* x, double* y, int e0, int nx) {
+ static const int32_t jk = 3;
+ double fw;
+ int32_t jx = nx - 1;
+ int32_t jv = (e0 - 3) / 24;
+ if (jv < 0) jv = 0;
+ int32_t q0 = e0 - 24 * (jv + 1);
+ int32_t m = jx + jk;
+
+ double f[10];
+ for (int i = 0, j = jv - jx; i <= m; i++, j++) {
+ f[i] = (j < 0) ? zero : static_cast<double>(two_over_pi[j]);
+ }
+
+ double q[10];
+ for (int i = 0; i <= jk; i++) {
+ fw = 0.0;
+ for (int j = 0; j <= jx; j++) fw += x[j] * f[jx + i - j];
+ q[i] = fw;
+ }
+
+ int32_t jz = jk;
+
+recompute:
+
+ int32_t iq[10];
+ double z = q[jz];
+ for (int i = 0, j = jz; j > 0; i++, j--) {
+ fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
+ iq[i] = static_cast<int32_t>(z - two24 * fw);
+ z = q[j - 1] + fw;
+ }
+
+ z = scalbn(z, q0);
+ z -= 8.0 * std::floor(z * 0.125);
+ int32_t n = static_cast<int32_t>(z);
+ z -= static_cast<double>(n);
+ int32_t ih = 0;
+ if (q0 > 0) {
+ int32_t i = (iq[jz - 1] >> (24 - q0));
+ n += i;
+ iq[jz - 1] -= i << (24 - q0);
+ ih = iq[jz - 1] >> (23 - q0);
+ } else if (q0 == 0) {
+ ih = iq[jz - 1] >> 23;
+ } else if (z >= 0.5) {
+ ih = 2;
+ }
+
+ if (ih > 0) {
+ n += 1;
+ int32_t carry = 0;
+ for (int i = 0; i < jz; i++) {
+ int32_t j = iq[i];
+ if (carry == 0) {
+ if (j != 0) {
+ carry = 1;
+ iq[i] = 0x1000000 - j;
+ }
+ } else {
+ iq[i] = 0xffffff - j;
+ }
+ }
+ if (q0 == 1) {
+ iq[jz - 1] &= 0x7fffff;
+ } else if (q0 == 2) {
+ iq[jz - 1] &= 0x3fffff;
+ }
+ if (ih == 2) {
+ z = one - z;
+ if (carry != 0) z -= scalbn(one, q0);
+ }
+ }
+
+ if (z == zero) {
+ int32_t j = 0;
+ for (int i = jz - 1; i >= jk; i--) j |= iq[i];
+ if (j == 0) {
+ int32_t k = 1;
+ while (iq[jk - k] == 0) k++;
+ for (int i = jz + 1; i <= jz + k; i++) {
+ f[jx + i] = static_cast<double>(two_over_pi[jv + i]);
+ for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
+ q[i] = fw;
+ }
+ jz += k;
+ goto recompute;
+ }
+ }
+
+ if (z == 0.0) {
+ jz -= 1;
+ q0 -= 24;
+ while (iq[jz] == 0) {
+ jz--;
+ q0 -= 24;
+ }
+ } else {
+ z = scalbn(z, -q0);
+ if (z >= two24) {
+ fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
+ iq[jz] = static_cast<int32_t>(z - two24 * fw);
+ jz += 1;
+ q0 += 24;
+ iq[jz] = static_cast<int32_t>(fw);
+ } else {
+ iq[jz] = static_cast<int32_t>(z);
+ }
+ }
+
+ fw = scalbn(one, q0);
+ for (int i = jz; i >= 0; i--) {
+ q[i] = fw * static_cast<double>(iq[i]);
+ fw *= twon24;
+ }
+
+ double fq[10];
+ for (int i = jz; i >= 0; i--) {
+ fw = 0.0;
+ for (int k = 0; k <= jk && k <= jz - i; k++) fw += PIo2[k] * q[i + k];
+ fq[jz - i] = fw;
+ }
+
+ fw = 0.0;
+ for (int i = jz; i >= 0; i--) fw += fq[i];
+ y[0] = (ih == 0) ? fw : -fw;
+ fw = fq[0] - fw;
+ for (int i = 1; i <= jz; i++) fw += fq[i];
+ y[1] = (ih == 0) ? fw : -fw;
+ return n & 7;
+}
+
+
+int rempio2(double x, double* y) {
+ int32_t hx = static_cast<int32_t>(internal::double_to_uint64(x) >> 32);
+ int32_t ix = hx & 0x7fffffff;
+
+ if (ix >= 0x7ff00000) {
+ *y = base::OS::nan_value();
+ return 0;
+ }
+
+ int32_t e0 = (ix >> 20) - 1046;
+ uint64_t zi = internal::double_to_uint64(x) & 0xFFFFFFFFu;
+ zi |= static_cast<uint64_t>(ix - (e0 << 20)) << 32;
+ double z = internal::uint64_to_double(zi);
+
+ double tx[3];
+ for (int i = 0; i < 2; i++) {
+ tx[i] = static_cast<double>(static_cast<int32_t>(z));
+ z = (z - tx[i]) * two24;
+ }
+ tx[2] = z;
+
+ int nx = 3;
+ while (tx[nx - 1] == zero) nx--;
+ int n = __kernel_rem_pio2(tx, y, e0, nx);
+ if (hx < 0) {
+ y[0] = -y[0];
+ y[1] = -y[1];
+ return -n;
+ }
+ return n;
+}
+}
+} // namespace v8::internal
diff --git a/third_party/fdlibm/fdlibm.h b/third_party/fdlibm/fdlibm.h
new file mode 100644
index 0000000..cadf85b
--- /dev/null
+++ b/third_party/fdlibm/fdlibm.h
@@ -0,0 +1,31 @@
+// The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunSoft, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// The original source code covered by the above license above has been
+// modified significantly by Google Inc.
+// Copyright 2014 the V8 project authors. All rights reserved.
+
+#ifndef V8_FDLIBM_H_
+#define V8_FDLIBM_H_
+
+namespace v8 {
+namespace fdlibm {
+
+int rempio2(double x, double* y);
+
+// Constants to be exposed to builtins via Float64Array.
+struct MathConstants {
+ static const double constants[53];
+};
+}
+} // namespace v8::internal
+
+#endif // V8_FDLIBM_H_
diff --git a/third_party/fdlibm/fdlibm.js b/third_party/fdlibm/fdlibm.js
new file mode 100644
index 0000000..7fd9adf
--- /dev/null
+++ b/third_party/fdlibm/fdlibm.js
@@ -0,0 +1,814 @@
+// The following is adapted from fdlibm (http://www.netlib.org/fdlibm),
+//
+// ====================================================
+// Copyright (C) 1993-2004 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunSoft, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// The original source code covered by the above license above has been
+// modified significantly by Google Inc.
+// Copyright 2014 the V8 project authors. All rights reserved.
+//
+// The following is a straightforward translation of fdlibm routines
+// by Raymond Toy (rtoy@google.com).
+
+// Double constants that do not have empty lower 32 bits are found in fdlibm.cc
+// and exposed through kMath as typed array. We assume the compiler to convert
+// from decimal to binary accurately enough to produce the intended values.
+// kMath is initialized to a Float64Array during genesis and not writable.
+var kMath;
+
+const INVPIO2 = kMath[0];
+const PIO2_1 = kMath[1];
+const PIO2_1T = kMath[2];
+const PIO2_2 = kMath[3];
+const PIO2_2T = kMath[4];
+const PIO2_3 = kMath[5];
+const PIO2_3T = kMath[6];
+const PIO4 = kMath[32];
+const PIO4LO = kMath[33];
+
+// Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For
+// precision, r is returned as two values y0 and y1 such that r = y0 + y1
+// to more than double precision.
+macro REMPIO2(X)
+ var n, y0, y1;
+ var hx = %_DoubleHi(X);
+ var ix = hx & 0x7fffffff;
+
+ if (ix < 0x4002d97c) {
+ // |X| ~< 3*pi/4, special case with n = +/- 1
+ if (hx > 0) {
+ var z = X - PIO2_1;
+ if (ix != 0x3ff921fb) {
+ // 33+53 bit pi is good enough
+ y0 = z - PIO2_1T;
+ y1 = (z - y0) - PIO2_1T;
+ } else {
+ // near pi/2, use 33+33+53 bit pi
+ z -= PIO2_2;
+ y0 = z - PIO2_2T;
+ y1 = (z - y0) - PIO2_2T;
+ }
+ n = 1;
+ } else {
+ // Negative X
+ var z = X + PIO2_1;
+ if (ix != 0x3ff921fb) {
+ // 33+53 bit pi is good enough
+ y0 = z + PIO2_1T;
+ y1 = (z - y0) + PIO2_1T;
+ } else {
+ // near pi/2, use 33+33+53 bit pi
+ z += PIO2_2;
+ y0 = z + PIO2_2T;
+ y1 = (z - y0) + PIO2_2T;
+ }
+ n = -1;
+ }
+ } else if (ix <= 0x413921fb) {
+ // |X| ~<= 2^19*(pi/2), medium size
+ var t = MathAbs(X);
+ n = (t * INVPIO2 + 0.5) | 0;
+ var r = t - n * PIO2_1;
+ var w = n * PIO2_1T;
+ // First round good to 85 bit
+ y0 = r - w;
+ if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) {
+ // 2nd iteration needed, good to 118
+ t = r;
+ w = n * PIO2_2;
+ r = t - w;
+ w = n * PIO2_2T - ((t - r) - w);
+ y0 = r - w;
+ if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) {
+ // 3rd iteration needed. 151 bits accuracy
+ t = r;
+ w = n * PIO2_3;
+ r = t - w;
+ w = n * PIO2_3T - ((t - r) - w);
+ y0 = r - w;
+ }
+ }
+ y1 = (r - y0) - w;
+ if (hx < 0) {
+ n = -n;
+ y0 = -y0;
+ y1 = -y1;
+ }
+ } else {
+ // Need to do full Payne-Hanek reduction here.
+ var r = %RemPiO2(X);
+ n = r[0];
+ y0 = r[1];
+ y1 = r[2];
+ }
+endmacro
+
+
+// __kernel_sin(X, Y, IY)
+// kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
+// Input X is assumed to be bounded by ~pi/4 in magnitude.
+// Input Y is the tail of X so that x = X + Y.
+//
+// Algorithm
+// 1. Since ieee_sin(-x) = -ieee_sin(x), we need only to consider positive x.
+// 2. ieee_sin(x) is approximated by a polynomial of degree 13 on
+// [0,pi/4]
+// 3 13
+// sin(x) ~ x + S1*x + ... + S6*x
+// where
+//
+// |ieee_sin(x) 2 4 6 8 10 12 | -58
+// |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
+// | x |
+//
+// 3. ieee_sin(X+Y) = ieee_sin(X) + sin'(X')*Y
+// ~ ieee_sin(X) + (1-X*X/2)*Y
+// For better accuracy, let
+// 3 2 2 2 2
+// r = X *(S2+X *(S3+X *(S4+X *(S5+X *S6))))
+// then 3 2
+// sin(x) = X + (S1*X + (X *(r-Y/2)+Y))
+//
+macro KSIN(x)
+kMath[7+x]
+endmacro
+
+macro RETURN_KERNELSIN(X, Y, SIGN)
+ var z = X * X;
+ var v = z * X;
+ var r = KSIN(1) + z * (KSIN(2) + z * (KSIN(3) +
+ z * (KSIN(4) + z * KSIN(5))));
+ return (X - ((z * (0.5 * Y - v * r) - Y) - v * KSIN(0))) SIGN;
+endmacro
+
+// __kernel_cos(X, Y)
+// kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
+// Input X is assumed to be bounded by ~pi/4 in magnitude.
+// Input Y is the tail of X so that x = X + Y.
+//
+// Algorithm
+// 1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x.
+// 2. ieee_cos(x) is approximated by a polynomial of degree 14 on
+// [0,pi/4]
+// 4 14
+// cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
+// where the remez error is
+//
+// | 2 4 6 8 10 12 14 | -58
+// |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
+// | |
+//
+// 4 6 8 10 12 14
+// 3. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
+// ieee_cos(x) = 1 - x*x/2 + r
+// since ieee_cos(X+Y) ~ ieee_cos(X) - ieee_sin(X)*Y
+// ~ ieee_cos(X) - X*Y,
+// a correction term is necessary in ieee_cos(x) and hence
+// cos(X+Y) = 1 - (X*X/2 - (r - X*Y))
+// For better accuracy when x > 0.3, let qx = |x|/4 with
+// the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
+// Then
+// cos(X+Y) = (1-qx) - ((X*X/2-qx) - (r-X*Y)).
+// Note that 1-qx and (X*X/2-qx) is EXACT here, and the
+// magnitude of the latter is at least a quarter of X*X/2,
+// thus, reducing the rounding error in the subtraction.
+//
+macro KCOS(x)
+kMath[13+x]
+endmacro
+
+macro RETURN_KERNELCOS(X, Y, SIGN)
+ var ix = %_DoubleHi(X) & 0x7fffffff;
+ var z = X * X;
+ var r = z * (KCOS(0) + z * (KCOS(1) + z * (KCOS(2)+
+ z * (KCOS(3) + z * (KCOS(4) + z * KCOS(5))))));
+ if (ix < 0x3fd33333) { // |x| ~< 0.3
+ return (1 - (0.5 * z - (z * r - X * Y))) SIGN;
+ } else {
+ var qx;
+ if (ix > 0x3fe90000) { // |x| > 0.78125
+ qx = 0.28125;
+ } else {
+ qx = %_ConstructDouble(%_DoubleHi(0.25 * X), 0);
+ }
+ var hz = 0.5 * z - qx;
+ return (1 - qx - (hz - (z * r - X * Y))) SIGN;
+ }
+endmacro
+
+
+// kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
+// Input x is assumed to be bounded by ~pi/4 in magnitude.
+// Input y is the tail of x.
+// Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1)
+// is returned.
+//
+// Algorithm
+// 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x.
+// 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
+// 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on
+// [0,0.67434]
+// 3 27
+// tan(x) ~ x + T1*x + ... + T13*x
+// where
+//
+// |ieee_tan(x) 2 4 26 | -59.2
+// |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
+// | x |
+//
+// Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y
+// ~ ieee_tan(x) + (1+x*x)*y
+// Therefore, for better accuracy in computing ieee_tan(x+y), let
+// 3 2 2 2 2
+// r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
+// then
+// 3 2
+// tan(x+y) = x + (T1*x + (x *(r+y)+y))
+//
+// 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
+// tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y))
+// = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y)))
+//
+// Set returnTan to 1 for tan; -1 for cot. Anything else is illegal
+// and will cause incorrect results.
+//
+macro KTAN(x)
+kMath[19+x]
+endmacro
+
+function KernelTan(x, y, returnTan) {
+ var z;
+ var w;
+ var hx = %_DoubleHi(x);
+ var ix = hx & 0x7fffffff;
+
+ if (ix < 0x3e300000) { // |x| < 2^-28
+ if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) {
+ // x == 0 && returnTan = -1
+ return 1 / MathAbs(x);
+ } else {
+ if (returnTan == 1) {
+ return x;
+ } else {
+ // Compute -1/(x + y) carefully
+ var w = x + y;
+ var z = %_ConstructDouble(%_DoubleHi(w), 0);
+ var v = y - (z - x);
+ var a = -1 / w;
+ var t = %_ConstructDouble(%_DoubleHi(a), 0);
+ var s = 1 + t * z;
+ return t + a * (s + t * v);
+ }
+ }
+ }
+ if (ix >= 0x3fe59428) { // |x| > .6744
+ if (x < 0) {
+ x = -x;
+ y = -y;
+ }
+ z = PIO4 - x;
+ w = PIO4LO - y;
+ x = z + w;
+ y = 0;
+ }
+ z = x * x;
+ w = z * z;
+
+ // Break x^5 * (T1 + x^2*T2 + ...) into
+ // x^5 * (T1 + x^4*T3 + ... + x^20*T11) +
+ // x^5 * (x^2 * (T2 + x^4*T4 + ... + x^22*T12))
+ var r = KTAN(1) + w * (KTAN(3) + w * (KTAN(5) +
+ w * (KTAN(7) + w * (KTAN(9) + w * KTAN(11)))));
+ var v = z * (KTAN(2) + w * (KTAN(4) + w * (KTAN(6) +
+ w * (KTAN(8) + w * (KTAN(10) + w * KTAN(12))))));
+ var s = z * x;
+ r = y + z * (s * (r + v) + y);
+ r = r + KTAN(0) * s;
+ w = x + r;
+ if (ix >= 0x3fe59428) {
+ return (1 - ((hx >> 30) & 2)) *
+ (returnTan - 2.0 * (x - (w * w / (w + returnTan) - r)));
+ }
+ if (returnTan == 1) {
+ return w;
+ } else {
+ z = %_ConstructDouble(%_DoubleHi(w), 0);
+ v = r - (z - x);
+ var a = -1 / w;
+ var t = %_ConstructDouble(%_DoubleHi(a), 0);
+ s = 1 + t * z;
+ return t + a * (s + t * v);
+ }
+}
+
+function MathSinSlow(x) {
+ REMPIO2(x);
+ var sign = 1 - (n & 2);
+ if (n & 1) {
+ RETURN_KERNELCOS(y0, y1, * sign);
+ } else {
+ RETURN_KERNELSIN(y0, y1, * sign);
+ }
+}
+
+function MathCosSlow(x) {
+ REMPIO2(x);
+ if (n & 1) {
+ var sign = (n & 2) - 1;
+ RETURN_KERNELSIN(y0, y1, * sign);
+ } else {
+ var sign = 1 - (n & 2);
+ RETURN_KERNELCOS(y0, y1, * sign);
+ }
+}
+
+// ECMA 262 - 15.8.2.16
+function MathSin(x) {
+ x = x * 1; // Convert to number.
+ if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
+ // |x| < pi/4, approximately. No reduction needed.
+ RETURN_KERNELSIN(x, 0, /* empty */);
+ }
+ return MathSinSlow(x);
+}
+
+// ECMA 262 - 15.8.2.7
+function MathCos(x) {
+ x = x * 1; // Convert to number.
+ if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
+ // |x| < pi/4, approximately. No reduction needed.
+ RETURN_KERNELCOS(x, 0, /* empty */);
+ }
+ return MathCosSlow(x);
+}
+
+// ECMA 262 - 15.8.2.18
+function MathTan(x) {
+ x = x * 1; // Convert to number.
+ if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
+ // |x| < pi/4, approximately. No reduction needed.
+ return KernelTan(x, 0, 1);
+ }
+ REMPIO2(x);
+ return KernelTan(y0, y1, (n & 1) ? -1 : 1);
+}
+
+// ES6 draft 09-27-13, section 20.2.2.20.
+// Math.log1p
+//
+// 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:
+// Constants are found in fdlibm.cc. We assume the C++ compiler to convert
+// from decimal to binary accurately enough to produce the intended values.
+//
+// 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.
+//
+const LN2_HI = kMath[34];
+const LN2_LO = kMath[35];
+const TWO54 = kMath[36];
+const TWO_THIRD = kMath[37];
+macro KLOG1P(x)
+(kMath[38+x])
+endmacro
+
+function MathLog1p(x) {
+ x = x * 1; // Convert to number.
+ var hx = %_DoubleHi(x);
+ var ax = hx & 0x7fffffff;
+ var k = 1;
+ var f = x;
+ var hu = 1;
+ var c = 0;
+ var u = x;
+
+ if (hx < 0x3fda827a) {
+ // x < 0.41422
+ if (ax >= 0x3ff00000) { // |x| >= 1
+ if (x === -1) {
+ return -INFINITY; // log1p(-1) = -inf
+ } else {
+ return NAN; // log1p(x<-1) = NaN
+ }
+ } else if (ax < 0x3c900000) {
+ // For |x| < 2^-54 we can return x.
+ return x;
+ } else if (ax < 0x3e200000) {
+ // For |x| < 2^-29 we can use a simple two-term Taylor series.
+ return x - x * x * 0.5;
+ }
+
+ if ((hx > 0) || (hx <= -0x402D413D)) { // (int) 0xbfd2bec3 = -0x402d413d
+ // -.2929 < x < 0.41422
+ k = 0;
+ }
+ }
+
+ // Handle Infinity and NAN
+ if (hx >= 0x7ff00000) return x;
+
+ if (k !== 0) {
+ if (hx < 0x43400000) {
+ // x < 2^53
+ u = 1 + x;
+ hu = %_DoubleHi(u);
+ k = (hu >> 20) - 1023;
+ c = (k > 0) ? 1 - (u - x) : x - (u - 1);
+ c = c / u;
+ } else {
+ hu = %_DoubleHi(u);
+ k = (hu >> 20) - 1023;
+ }
+ hu = hu & 0xfffff;
+ if (hu < 0x6a09e) {
+ u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u)); // Normalize u.
+ } else {
+ ++k;
+ u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u)); // Normalize u/2.
+ hu = (0x00100000 - hu) >> 2;
+ }
+ f = u - 1;
+ }
+
+ var hfsq = 0.5 * f * f;
+ if (hu === 0) {
+ // |f| < 2^-20;
+ if (f === 0) {
+ if (k === 0) {
+ return 0.0;
+ } else {
+ return k * LN2_HI + (c + k * LN2_LO);
+ }
+ }
+ var R = hfsq * (1 - TWO_THIRD * f);
+ if (k === 0) {
+ return f - R;
+ } else {
+ return k * LN2_HI - ((R - (k * LN2_LO + c)) - f);
+ }
+ }
+
+ var s = f / (2 + f);
+ var z = s * s;
+ var R = z * (KLOG1P(0) + z * (KLOG1P(1) + z *
+ (KLOG1P(2) + z * (KLOG1P(3) + z *
+ (KLOG1P(4) + z * (KLOG1P(5) + z * KLOG1P(6)))))));
+ if (k === 0) {
+ return f - (hfsq - s * (hfsq + R));
+ } else {
+ return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f);
+ }
+}
+
+// ES6 draft 09-27-13, section 20.2.2.14.
+// Math.expm1
+// 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 Remes 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
+//
+const KEXPM1_OVERFLOW = kMath[45];
+const INVLN2 = kMath[46];
+macro KEXPM1(x)
+(kMath[47+x])
+endmacro
+
+function MathExpm1(x) {
+ x = x * 1; // Convert to number.
+ var y;
+ var hi;
+ var lo;
+ var k;
+ var t;
+ var c;
+
+ var hx = %_DoubleHi(x);
+ var xsb = hx & 0x80000000; // Sign bit of x
+ var y = (xsb === 0) ? x : -x; // y = |x|
+ hx &= 0x7fffffff; // High word of |x|
+
+ // Filter out huge and non-finite argument
+ if (hx >= 0x4043687a) { // if |x| ~=> 56 * ln2
+ if (hx >= 0x40862e42) { // if |x| >= 709.78
+ if (hx >= 0x7ff00000) {
+ // expm1(inf) = inf; expm1(-inf) = -1; expm1(nan) = nan;
+ return (x === -INFINITY) ? -1 : x;
+ }
+ if (x > KEXPM1_OVERFLOW) return INFINITY; // Overflow
+ }
+ if (xsb != 0) return -1; // x < -56 * ln2, return -1.
+ }
+
+ // Argument reduction
+ if (hx > 0x3fd62e42) { // if |x| > 0.5 * ln2
+ if (hx < 0x3ff0a2b2) { // and |x| < 1.5 * ln2
+ if (xsb === 0) {
+ hi = x - LN2_HI;
+ lo = LN2_LO;
+ k = 1;
+ } else {
+ hi = x + LN2_HI;
+ lo = -LN2_LO;
+ k = -1;
+ }
+ } else {
+ k = (INVLN2 * x + ((xsb === 0) ? 0.5 : -0.5)) | 0;
+ t = k;
+ // t * ln2_hi is exact here.
+ hi = x - t * LN2_HI;
+ lo = t * LN2_LO;
+ }
+ x = hi - lo;
+ c = (hi - x) - lo;
+ } else if (hx < 0x3c900000) {
+ // When |x| < 2^-54, we can return x.
+ return x;
+ } else {
+ // Fall through.
+ k = 0;
+ }
+
+ // x is now in primary range
+ var hfx = 0.5 * x;
+ var hxs = x * hfx;
+ var r1 = 1 + hxs * (KEXPM1(0) + hxs * (KEXPM1(1) + hxs *
+ (KEXPM1(2) + hxs * (KEXPM1(3) + hxs * KEXPM1(4)))));
+ t = 3 - r1 * hfx;
+ var e = hxs * ((r1 - t) / (6 - x * t));
+ if (k === 0) { // c is 0
+ return x - (x*e - hxs);
+ } else {
+ e = (x * (e - c) - c);
+ e -= hxs;
+ if (k === -1) return 0.5 * (x - e) - 0.5;
+ if (k === 1) {
+ if (x < -0.25) return -2 * (e - (x + 0.5));
+ return 1 + 2 * (x - e);
+ }
+
+ if (k <= -2 || k > 56) {
+ // suffice to return exp(x) + 1
+ y = 1 - (e - x);
+ // Add k to y's exponent
+ y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
+ return y - 1;
+ }
+ if (k < 20) {
+ // t = 1 - 2^k
+ t = %_ConstructDouble(0x3ff00000 - (0x200000 >> k), 0);
+ y = t - (e - x);
+ // Add k to y's exponent
+ y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
+ } else {
+ // t = 2^-k
+ t = %_ConstructDouble((0x3ff - k) << 20, 0);
+ y = x - (e + t);
+ y += 1;
+ // Add k to y's exponent
+ y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
+ }
+ }
+ return y;
+}
+
+
+// ES6 draft 09-27-13, section 20.2.2.30.
+// Math.sinh
+// Method :
+// mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
+// 1. Replace x by |x| (sinh(-x) = -sinh(x)).
+// 2.
+// E + E/(E+1)
+// 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
+// 2
+//
+// 22 <= x <= lnovft : sinh(x) := exp(x)/2
+// lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
+// ln2ovft < x : sinh(x) := x*shuge (overflow)
+//
+// Special cases:
+// sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
+// only sinh(0)=0 is exact for finite x.
+//
+const KSINH_OVERFLOW = kMath[52];
+const TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half
+const LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half
+
+function MathSinh(x) {
+ x = x * 1; // Convert to number.
+ var h = (x < 0) ? -0.5 : 0.5;
+ // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
+ var ax = MathAbs(x);
+ if (ax < 22) {
+ // For |x| < 2^-28, sinh(x) = x
+ if (ax < TWO_M28) return x;
+ var t = MathExpm1(ax);
+ if (ax < 1) return h * (2 * t - t * t / (t + 1));
+ return h * (t + t / (t + 1));
+ }
+ // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
+ if (ax < LOG_MAXD) return h * MathExp(ax);
+ // |x| in [log(maxdouble), overflowthreshold]
+ // overflowthreshold = 710.4758600739426
+ if (ax <= KSINH_OVERFLOW) {
+ var w = MathExp(0.5 * ax);
+ var t = h * w;
+ return t * w;
+ }
+ // |x| > overflowthreshold or is NaN.
+ // Return Infinity of the appropriate sign or NaN.
+ return x * INFINITY;
+}
+
+
+// ES6 draft 09-27-13, section 20.2.2.12.
+// Math.cosh
+// Method :
+// mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
+// 1. Replace x by |x| (cosh(x) = cosh(-x)).
+// 2.
+// [ exp(x) - 1 ]^2
+// 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
+// 2*exp(x)
+//
+// exp(x) + 1/exp(x)
+// ln2/2 <= x <= 22 : cosh(x) := -------------------
+// 2
+// 22 <= x <= lnovft : cosh(x) := exp(x)/2
+// lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
+// ln2ovft < x : cosh(x) := huge*huge (overflow)
+//
+// Special cases:
+// cosh(x) is |x| if x is +INF, -INF, or NaN.
+// only cosh(0)=1 is exact for finite x.
+//
+const KCOSH_OVERFLOW = kMath[52];
+
+function MathCosh(x) {
+ x = x * 1; // Convert to number.
+ var ix = %_DoubleHi(x) & 0x7fffffff;
+ // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
+ if (ix < 0x3fd62e43) {
+ var t = MathExpm1(MathAbs(x));
+ var w = 1 + t;
+ // For |x| < 2^-55, cosh(x) = 1
+ if (ix < 0x3c800000) return w;
+ return 1 + (t * t) / (w + w);
+ }
+ // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
+ if (ix < 0x40360000) {
+ var t = MathExp(MathAbs(x));
+ return 0.5 * t + 0.5 / t;
+ }
+ // |x| in [22, log(maxdouble)], return half*exp(|x|)
+ if (ix < 0x40862e42) return 0.5 * MathExp(MathAbs(x));
+ // |x| in [log(maxdouble), overflowthreshold]
+ if (MathAbs(x) <= KCOSH_OVERFLOW) {
+ var w = MathExp(0.5 * MathAbs(x));
+ var t = 0.5 * w;
+ return t * w;
+ }
+ if (NUMBER_IS_NAN(x)) return x;
+ // |x| > overflowthreshold.
+ return INFINITY;
+}