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Unified Diff: third_party/fdlibm/fdlibm.js

Issue 638553003: Move fdlibm in src/third_party. (Closed) Base URL: https://v8.googlecode.com/svn/branches/bleeding_edge
Patch Set: fix gn Created 6 years, 2 months ago
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Index: third_party/fdlibm/fdlibm.js
diff --git a/third_party/fdlibm/fdlibm.js b/third_party/fdlibm/fdlibm.js
deleted file mode 100644
index 08c6f5e7207112ac80c5f420f98990e56b7468b0..0000000000000000000000000000000000000000
--- a/third_party/fdlibm/fdlibm.js
+++ /dev/null
@@ -1,814 +0,0 @@
-// 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 >= 0x3fe59429) { // |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;
-}
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