| // 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). |
| |
| // rempio2result is used as a container for return values of %RemPiO2. It is |
| // initialized to a two-element Float64Array during genesis. |
| |
| (function(global, utils) { |
| |
| "use strict"; |
| |
| %CheckIsBootstrapping(); |
| |
| // ------------------------------------------------------------------- |
| // Imports |
| |
| var GlobalFloat64Array = global.Float64Array; |
| var GlobalMath = global.Math; |
| var MathAbs; |
| var MathExp; |
| var NaN = %GetRootNaN(); |
| var rempio2result; |
| |
| utils.Import(function(from) { |
| MathAbs = from.MathAbs; |
| MathExp = from.MathExp; |
| }); |
| |
| utils.CreateDoubleResultArray = function(global) { |
| rempio2result = new GlobalFloat64Array(2); |
| }; |
| |
| // ------------------------------------------------------------------- |
| |
| define INVPIO2 = 6.36619772367581382433e-01; |
| define PIO2_1 = 1.57079632673412561417; |
| define PIO2_1T = 6.07710050650619224932e-11; |
| define PIO2_2 = 6.07710050630396597660e-11; |
| define PIO2_2T = 2.02226624879595063154e-21; |
| define PIO2_3 = 2.02226624871116645580e-21; |
| define PIO2_3T = 8.47842766036889956997e-32; |
| define PIO4 = 7.85398163397448278999e-01; |
| define PIO4LO = 3.06161699786838301793e-17; |
| |
| // 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. |
| n = %RemPiO2(X, rempio2result); |
| y0 = rempio2result[0]; |
| y1 = rempio2result[1]; |
| } |
| 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)) |
| // |
| define S1 = -1.66666666666666324348e-01; |
| define S2 = 8.33333333332248946124e-03; |
| define S3 = -1.98412698298579493134e-04; |
| define S4 = 2.75573137070700676789e-06; |
| define S5 = -2.50507602534068634195e-08; |
| define S6 = 1.58969099521155010221e-10; |
| |
| macro RETURN_KERNELSIN(X, Y, SIGN) |
| var z = X * X; |
| var v = z * X; |
| var r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); |
| return (X - ((z * (0.5 * Y - v * r) - Y) - v * S1)) 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. |
| // |
| define C1 = 4.16666666666666019037e-02; |
| define C2 = -1.38888888888741095749e-03; |
| define C3 = 2.48015872894767294178e-05; |
| define C4 = -2.75573143513906633035e-07; |
| define C5 = 2.08757232129817482790e-09; |
| define C6 = -1.13596475577881948265e-11; |
| |
| macro RETURN_KERNELCOS(X, Y, SIGN) |
| var ix = %_DoubleHi(X) & 0x7fffffff; |
| var z = X * X; |
| var r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); |
| 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. |
| // |
| define T00 = 3.33333333333334091986e-01; |
| define T01 = 1.33333333333201242699e-01; |
| define T02 = 5.39682539762260521377e-02; |
| define T03 = 2.18694882948595424599e-02; |
| define T04 = 8.86323982359930005737e-03; |
| define T05 = 3.59207910759131235356e-03; |
| define T06 = 1.45620945432529025516e-03; |
| define T07 = 5.88041240820264096874e-04; |
| define T08 = 2.46463134818469906812e-04; |
| define T09 = 7.81794442939557092300e-05; |
| define T10 = 7.14072491382608190305e-05; |
| define T11 = -1.85586374855275456654e-05; |
| define T12 = 2.59073051863633712884e-05; |
| |
| 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 = T01 + w * (T03 + w * (T05 + |
| w * (T07 + w * (T09 + w * T11)))); |
| var v = z * (T02 + w * (T04 + w * (T06 + |
| w * (T08 + w * (T10 + w * T12))))); |
| var s = z * x; |
| r = y + z * (s * (r + v) + y); |
| r = r + T00 * 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; // 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; // 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. |
| // |
| define LN2_HI = 6.93147180369123816490e-01; |
| define LN2_LO = 1.90821492927058770002e-10; |
| define TWO_THIRD = 6.666666666666666666e-01; |
| define LP1 = 6.666666666666735130e-01; |
| define LP2 = 3.999999999940941908e-01; |
| define LP3 = 2.857142874366239149e-01; |
| define LP4 = 2.222219843214978396e-01; |
| define LP5 = 1.818357216161805012e-01; |
| define LP6 = 1.531383769920937332e-01; |
| define LP7 = 1.479819860511658591e-01; |
| |
| // 2^54 |
| define TWO54 = 18014398509481984; |
| |
| 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 * (LP1 + z * (LP2 + z * (LP3 + z * (LP4 + |
| z * (LP5 + z * (LP6 + z * LP7)))))); |
| 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 |
| // |
| define KEXPM1_OVERFLOW = 7.09782712893383973096e+02; |
| define INVLN2 = 1.44269504088896338700; |
| define EXPM1_1 = -3.33333333333331316428e-02; |
| define EXPM1_2 = 1.58730158725481460165e-03; |
| define EXPM1_3 = -7.93650757867487942473e-05; |
| define EXPM1_4 = 4.00821782732936239552e-06; |
| define EXPM1_5 = -2.01099218183624371326e-07; |
| |
| 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 * (EXPM1_1 + hxs * (EXPM1_2 + hxs * |
| (EXPM1_3 + hxs * (EXPM1_4 + hxs * EXPM1_5)))); |
| 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. |
| // |
| define KSINH_OVERFLOW = 710.4758600739439; |
| define TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half |
| define 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. |
| // |
| define KCOSH_OVERFLOW = 710.4758600739439; |
| |
| 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; |
| } |
| |
| // ES6 draft 09-27-13, section 20.2.2.33. |
| // Math.tanh(x) |
| // Method : |
| // x -x |
| // e - e |
| // 0. tanh(x) is defined to be ----------- |
| // x -x |
| // e + e |
| // 1. reduce x to non-negative by tanh(-x) = -tanh(x). |
| // 2. 0 <= x <= 2**-55 : tanh(x) := x*(one+x) |
| // -t |
| // 2**-55 < x <= 1 : tanh(x) := -----; t = expm1(-2x) |
| // t + 2 |
| // 2 |
| // 1 <= x <= 22.0 : tanh(x) := 1- ----- ; t = expm1(2x) |
| // t + 2 |
| // 22.0 < x <= INF : tanh(x) := 1. |
| // |
| // Special cases: |
| // tanh(NaN) is NaN; |
| // only tanh(0) = 0 is exact for finite argument. |
| // |
| |
| define TWO_M55 = 2.77555756156289135105e-17; // 2^-55, empty lower half |
| |
| function MathTanh(x) { |
| x = x * 1; // Convert to number. |
| // x is Infinity or NaN |
| if (!NUMBER_IS_FINITE(x)) { |
| if (x > 0) return 1; |
| if (x < 0) return -1; |
| return x; |
| } |
| |
| var ax = MathAbs(x); |
| var z; |
| // |x| < 22 |
| if (ax < 22) { |
| if (ax < TWO_M55) { |
| // |x| < 2^-55, tanh(small) = small. |
| return x; |
| } |
| if (ax >= 1) { |
| // |x| >= 1 |
| var t = MathExpm1(2 * ax); |
| z = 1 - 2 / (t + 2); |
| } else { |
| var t = MathExpm1(-2 * ax); |
| z = -t / (t + 2); |
| } |
| } else { |
| // |x| > 22, return +/- 1 |
| z = 1; |
| } |
| return (x >= 0) ? z : -z; |
| } |
| |
| // ES6 draft 09-27-13, section 20.2.2.21. |
| // Return the base 10 logarithm of x |
| // |
| // Method : |
| // Let log10_2hi = leading 40 bits of log10(2) and |
| // log10_2lo = log10(2) - log10_2hi, |
| // ivln10 = 1/log(10) rounded. |
| // Then |
| // n = ilogb(x), |
| // if(n<0) n = n+1; |
| // x = scalbn(x,-n); |
| // log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x)) |
| // |
| // Note 1: |
| // To guarantee log10(10**n)=n, where 10**n is normal, the rounding |
| // mode must set to Round-to-Nearest. |
| // Note 2: |
| // [1/log(10)] rounded to 53 bits has error .198 ulps; |
| // log10 is monotonic at all binary break points. |
| // |
| // Special cases: |
| // log10(x) is NaN if x < 0; |
| // log10(+INF) is +INF; log10(0) is -INF; |
| // log10(NaN) is that NaN; |
| // log10(10**N) = N for N=0,1,...,22. |
| // |
| |
| define IVLN10 = 4.34294481903251816668e-01; |
| define LOG10_2HI = 3.01029995663611771306e-01; |
| define LOG10_2LO = 3.69423907715893078616e-13; |
| |
| function MathLog10(x) { |
| x = x * 1; // Convert to number. |
| var hx = %_DoubleHi(x); |
| var lx = %_DoubleLo(x); |
| var k = 0; |
| |
| if (hx < 0x00100000) { |
| // x < 2^-1022 |
| // log10(+/- 0) = -Infinity. |
| if (((hx & 0x7fffffff) | lx) === 0) return -INFINITY; |
| // log10 of negative number is NaN. |
| if (hx < 0) return NaN; |
| // Subnormal number. Scale up x. |
| k -= 54; |
| x *= TWO54; |
| hx = %_DoubleHi(x); |
| lx = %_DoubleLo(x); |
| } |
| |
| // Infinity or NaN. |
| if (hx >= 0x7ff00000) return x; |
| |
| k += (hx >> 20) - 1023; |
| var i = (k & 0x80000000) >>> 31; |
| hx = (hx & 0x000fffff) | ((0x3ff - i) << 20); |
| var y = k + i; |
| x = %_ConstructDouble(hx, lx); |
| |
| var z = y * LOG10_2LO + IVLN10 * %_MathLogRT(x); |
| return z + y * LOG10_2HI; |
| } |
| |
| |
| // ES6 draft 09-27-13, section 20.2.2.22. |
| // Return the base 2 logarithm of x |
| // |
| // fdlibm does not have an explicit log2 function, but fdlibm's pow |
| // function does implement an accurate log2 function as part of the |
| // pow implementation. This extracts the core parts of that as a |
| // separate log2 function. |
| |
| // Method: |
| // Compute log2(x) in two pieces: |
| // log2(x) = w1 + w2 |
| // where w1 has 53-24 = 29 bits of trailing zeroes. |
| |
| define DP_H = 5.84962487220764160156e-01; |
| define DP_L = 1.35003920212974897128e-08; |
| |
| // Polynomial coefficients for (3/2)*(log2(x) - 2*s - 2/3*s^3) |
| define LOG2_1 = 5.99999999999994648725e-01; |
| define LOG2_2 = 4.28571428578550184252e-01; |
| define LOG2_3 = 3.33333329818377432918e-01; |
| define LOG2_4 = 2.72728123808534006489e-01; |
| define LOG2_5 = 2.30660745775561754067e-01; |
| define LOG2_6 = 2.06975017800338417784e-01; |
| |
| // cp = 2/(3*ln(2)). Note that cp_h + cp_l is cp, but with more accuracy. |
| define CP = 9.61796693925975554329e-01; |
| define CP_H = 9.61796700954437255859e-01; |
| define CP_L = -7.02846165095275826516e-09; |
| // 2^53 |
| define TWO53 = 9007199254740992; |
| |
| function MathLog2(x) { |
| x = x * 1; // Convert to number. |
| var ax = MathAbs(x); |
| var hx = %_DoubleHi(x); |
| var lx = %_DoubleLo(x); |
| var ix = hx & 0x7fffffff; |
| |
| // Handle special cases. |
| // log2(+/- 0) = -Infinity |
| if ((ix | lx) == 0) return -INFINITY; |
| |
| // log(x) = NaN, if x < 0 |
| if (hx < 0) return NaN; |
| |
| // log2(Infinity) = Infinity, log2(NaN) = NaN |
| if (ix >= 0x7ff00000) return x; |
| |
| var n = 0; |
| |
| // Take care of subnormal number. |
| if (ix < 0x00100000) { |
| ax *= TWO53; |
| n -= 53; |
| ix = %_DoubleHi(ax); |
| } |
| |
| n += (ix >> 20) - 0x3ff; |
| var j = ix & 0x000fffff; |
| |
| // Determine interval. |
| ix = j | 0x3ff00000; // normalize ix. |
| |
| var bp = 1; |
| var dp_h = 0; |
| var dp_l = 0; |
| if (j > 0x3988e) { // |x| > sqrt(3/2) |
| if (j < 0xbb67a) { // |x| < sqrt(3) |
| bp = 1.5; |
| dp_h = DP_H; |
| dp_l = DP_L; |
| } else { |
| n += 1; |
| ix -= 0x00100000; |
| } |
| } |
| |
| ax = %_ConstructDouble(ix, %_DoubleLo(ax)); |
| |
| // Compute ss = s_h + s_l = (x - 1)/(x+1) or (x - 1.5)/(x + 1.5) |
| var u = ax - bp; |
| var v = 1 / (ax + bp); |
| var ss = u * v; |
| var s_h = %_ConstructDouble(%_DoubleHi(ss), 0); |
| |
| // t_h = ax + bp[k] High |
| var t_h = %_ConstructDouble(%_DoubleHi(ax + bp), 0) |
| var t_l = ax - (t_h - bp); |
| var s_l = v * ((u - s_h * t_h) - s_h * t_l); |
| |
| // Compute log2(ax) |
| var s2 = ss * ss; |
| var r = s2 * s2 * (LOG2_1 + s2 * (LOG2_2 + s2 * (LOG2_3 + s2 * ( |
| LOG2_4 + s2 * (LOG2_5 + s2 * LOG2_6))))); |
| r += s_l * (s_h + ss); |
| s2 = s_h * s_h; |
| t_h = %_ConstructDouble(%_DoubleHi(3.0 + s2 + r), 0); |
| t_l = r - ((t_h - 3.0) - s2); |
| // u + v = ss * (1 + ...) |
| u = s_h * t_h; |
| v = s_l * t_h + t_l * ss; |
| |
| // 2 / (3 * log(2)) * (ss + ...) |
| var p_h = %_ConstructDouble(%_DoubleHi(u + v), 0); |
| var p_l = v - (p_h - u); |
| var z_h = CP_H * p_h; |
| var z_l = CP_L * p_h + p_l * CP + dp_l; |
| |
| // log2(ax) = (ss + ...) * 2 / (3 * log(2)) = n + dp_h + z_h + z_l |
| var t = n; |
| var t1 = %_ConstructDouble(%_DoubleHi(((z_h + z_l) + dp_h) + t), 0); |
| var t2 = z_l - (((t1 - t) - dp_h) - z_h); |
| |
| // t1 + t2 = log2(ax), sum up because we do not care about extra precision. |
| return t1 + t2; |
| } |
| |
| //------------------------------------------------------------------- |
| |
| utils.InstallFunctions(GlobalMath, DONT_ENUM, [ |
| "cos", MathCos, |
| "sin", MathSin, |
| "tan", MathTan, |
| "sinh", MathSinh, |
| "cosh", MathCosh, |
| "tanh", MathTanh, |
| "log10", MathLog10, |
| "log2", MathLog2, |
| "log1p", MathLog1p, |
| "expm1", MathExpm1 |
| ]); |
| |
| %SetForceInlineFlag(MathSin); |
| %SetForceInlineFlag(MathCos); |
| |
| }) |