0f81d7698a
Port contributed by Raymond Toy <rtoy@google.com>. R=rtoy@chromium.org LOG=N BUG=v8:3481 Review URL: https://codereview.chromium.org/457643002 git-svn-id: https://v8.googlecode.com/svn/branches/bleeding_edge@23082 ce2b1a6d-e550-0410-aec6-3dcde31c8c00
519 lines
15 KiB
JavaScript
519 lines
15 KiB
JavaScript
// The following is adapted from fdlibm (http://www.netlib.org/fdlibm),
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunSoft, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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// The original source code covered by the above license above has been
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// modified significantly by Google Inc.
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// Copyright 2014 the V8 project authors. All rights reserved.
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//
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// The following is a straightforward translation of fdlibm routines
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// by Raymond Toy (rtoy@google.com).
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var kMath; // Initialized to a Float64Array during genesis and is not writable.
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const INVPIO2 = kMath[0];
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const PIO2_1 = kMath[1];
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const PIO2_1T = kMath[2];
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const PIO2_2 = kMath[3];
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const PIO2_2T = kMath[4];
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const PIO2_3 = kMath[5];
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const PIO2_3T = kMath[6];
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const PIO4 = kMath[32];
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const PIO4LO = kMath[33];
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// Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For
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// precision, r is returned as two values y0 and y1 such that r = y0 + y1
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// to more than double precision.
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macro REMPIO2(X)
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var n, y0, y1;
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var hx = %_DoubleHi(X);
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var ix = hx & 0x7fffffff;
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if (ix < 0x4002d97c) {
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// |X| ~< 3*pi/4, special case with n = +/- 1
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if (hx > 0) {
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var z = X - PIO2_1;
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if (ix != 0x3ff921fb) {
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// 33+53 bit pi is good enough
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y0 = z - PIO2_1T;
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y1 = (z - y0) - PIO2_1T;
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} else {
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// near pi/2, use 33+33+53 bit pi
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z -= PIO2_2;
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y0 = z - PIO2_2T;
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y1 = (z - y0) - PIO2_2T;
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}
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n = 1;
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} else {
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// Negative X
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var z = X + PIO2_1;
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if (ix != 0x3ff921fb) {
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// 33+53 bit pi is good enough
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y0 = z + PIO2_1T;
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y1 = (z - y0) + PIO2_1T;
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} else {
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// near pi/2, use 33+33+53 bit pi
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z += PIO2_2;
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y0 = z + PIO2_2T;
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y1 = (z - y0) + PIO2_2T;
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}
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n = -1;
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}
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} else if (ix <= 0x413921fb) {
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// |X| ~<= 2^19*(pi/2), medium size
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var t = MathAbs(X);
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n = (t * INVPIO2 + 0.5) | 0;
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var r = t - n * PIO2_1;
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var w = n * PIO2_1T;
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// First round good to 85 bit
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y0 = r - w;
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if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) {
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// 2nd iteration needed, good to 118
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t = r;
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w = n * PIO2_2;
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r = t - w;
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w = n * PIO2_2T - ((t - r) - w);
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y0 = r - w;
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if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) {
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// 3rd iteration needed. 151 bits accuracy
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t = r;
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w = n * PIO2_3;
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r = t - w;
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w = n * PIO2_3T - ((t - r) - w);
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y0 = r - w;
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}
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}
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y1 = (r - y0) - w;
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if (hx < 0) {
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n = -n;
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y0 = -y0;
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y1 = -y1;
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}
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} else {
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// Need to do full Payne-Hanek reduction here.
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var r = %RemPiO2(X);
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n = r[0];
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y0 = r[1];
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y1 = r[2];
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}
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endmacro
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// __kernel_sin(X, Y, IY)
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// kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
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// Input X is assumed to be bounded by ~pi/4 in magnitude.
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// Input Y is the tail of X so that x = X + Y.
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//
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// Algorithm
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// 1. Since ieee_sin(-x) = -ieee_sin(x), we need only to consider positive x.
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// 2. ieee_sin(x) is approximated by a polynomial of degree 13 on
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// [0,pi/4]
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// 3 13
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// sin(x) ~ x + S1*x + ... + S6*x
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// where
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//
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// |ieee_sin(x) 2 4 6 8 10 12 | -58
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// |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
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// | x |
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//
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// 3. ieee_sin(X+Y) = ieee_sin(X) + sin'(X')*Y
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// ~ ieee_sin(X) + (1-X*X/2)*Y
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// For better accuracy, let
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// 3 2 2 2 2
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// r = X *(S2+X *(S3+X *(S4+X *(S5+X *S6))))
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// then 3 2
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// sin(x) = X + (S1*X + (X *(r-Y/2)+Y))
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//
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macro KSIN(x)
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kMath[7+x]
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endmacro
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macro RETURN_KERNELSIN(X, Y, SIGN)
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var z = X * X;
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var v = z * X;
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var r = KSIN(1) + z * (KSIN(2) + z * (KSIN(3) +
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z * (KSIN(4) + z * KSIN(5))));
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return (X - ((z * (0.5 * Y - v * r) - Y) - v * KSIN(0))) SIGN;
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endmacro
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// __kernel_cos(X, Y)
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// kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
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// Input X is assumed to be bounded by ~pi/4 in magnitude.
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// Input Y is the tail of X so that x = X + Y.
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//
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// Algorithm
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// 1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x.
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// 2. ieee_cos(x) is approximated by a polynomial of degree 14 on
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// [0,pi/4]
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// 4 14
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// cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
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// where the remez error is
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//
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// | 2 4 6 8 10 12 14 | -58
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// |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
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// | |
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//
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// 4 6 8 10 12 14
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// 3. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
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// ieee_cos(x) = 1 - x*x/2 + r
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// since ieee_cos(X+Y) ~ ieee_cos(X) - ieee_sin(X)*Y
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// ~ ieee_cos(X) - X*Y,
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// a correction term is necessary in ieee_cos(x) and hence
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// cos(X+Y) = 1 - (X*X/2 - (r - X*Y))
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// For better accuracy when x > 0.3, let qx = |x|/4 with
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// the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
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// Then
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// cos(X+Y) = (1-qx) - ((X*X/2-qx) - (r-X*Y)).
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// Note that 1-qx and (X*X/2-qx) is EXACT here, and the
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// magnitude of the latter is at least a quarter of X*X/2,
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// thus, reducing the rounding error in the subtraction.
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//
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macro KCOS(x)
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kMath[13+x]
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endmacro
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macro RETURN_KERNELCOS(X, Y, SIGN)
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var ix = %_DoubleHi(X) & 0x7fffffff;
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var z = X * X;
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var r = z * (KCOS(0) + z * (KCOS(1) + z * (KCOS(2)+
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z * (KCOS(3) + z * (KCOS(4) + z * KCOS(5))))));
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if (ix < 0x3fd33333) { // |x| ~< 0.3
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return (1 - (0.5 * z - (z * r - X * Y))) SIGN;
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} else {
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var qx;
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if (ix > 0x3fe90000) { // |x| > 0.78125
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qx = 0.28125;
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} else {
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qx = %_ConstructDouble(%_DoubleHi(0.25 * X), 0);
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}
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var hz = 0.5 * z - qx;
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return (1 - qx - (hz - (z * r - X * Y))) SIGN;
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}
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endmacro
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// kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
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// Input x is assumed to be bounded by ~pi/4 in magnitude.
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// Input y is the tail of x.
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// Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1)
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// is returned.
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//
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// Algorithm
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// 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x.
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// 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
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// 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on
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// [0,0.67434]
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// 3 27
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// tan(x) ~ x + T1*x + ... + T13*x
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// where
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//
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// |ieee_tan(x) 2 4 26 | -59.2
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// |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
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// | x |
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//
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// Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y
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// ~ ieee_tan(x) + (1+x*x)*y
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// Therefore, for better accuracy in computing ieee_tan(x+y), let
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// 3 2 2 2 2
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// r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
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// then
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// 3 2
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// tan(x+y) = x + (T1*x + (x *(r+y)+y))
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//
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// 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
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// tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y))
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// = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y)))
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//
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// Set returnTan to 1 for tan; -1 for cot. Anything else is illegal
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// and will cause incorrect results.
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//
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macro KTAN(x)
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kMath[19+x]
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endmacro
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function KernelTan(x, y, returnTan) {
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var z;
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var w;
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var hx = %_DoubleHi(x);
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var ix = hx & 0x7fffffff;
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if (ix < 0x3e300000) { // |x| < 2^-28
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if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) {
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// x == 0 && returnTan = -1
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return 1 / MathAbs(x);
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} else {
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if (returnTan == 1) {
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return x;
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} else {
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// Compute -1/(x + y) carefully
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var w = x + y;
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var z = %_ConstructDouble(%_DoubleHi(w), 0);
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var v = y - (z - x);
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var a = -1 / w;
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var t = %_ConstructDouble(%_DoubleHi(a), 0);
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var s = 1 + t * z;
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return t + a * (s + t * v);
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}
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}
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}
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if (ix >= 0x3fe59429) { // |x| > .6744
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if (x < 0) {
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x = -x;
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y = -y;
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}
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z = PIO4 - x;
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w = PIO4LO - y;
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x = z + w;
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y = 0;
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}
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z = x * x;
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w = z * z;
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// Break x^5 * (T1 + x^2*T2 + ...) into
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// x^5 * (T1 + x^4*T3 + ... + x^20*T11) +
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// x^5 * (x^2 * (T2 + x^4*T4 + ... + x^22*T12))
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var r = KTAN(1) + w * (KTAN(3) + w * (KTAN(5) +
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w * (KTAN(7) + w * (KTAN(9) + w * KTAN(11)))));
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var v = z * (KTAN(2) + w * (KTAN(4) + w * (KTAN(6) +
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w * (KTAN(8) + w * (KTAN(10) + w * KTAN(12))))));
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var s = z * x;
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r = y + z * (s * (r + v) + y);
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r = r + KTAN(0) * s;
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w = x + r;
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if (ix >= 0x3fe59428) {
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return (1 - ((hx >> 30) & 2)) *
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(returnTan - 2.0 * (x - (w * w / (w + returnTan) - r)));
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}
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if (returnTan == 1) {
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return w;
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} else {
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z = %_ConstructDouble(%_DoubleHi(w), 0);
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v = r - (z - x);
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var a = -1 / w;
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var t = %_ConstructDouble(%_DoubleHi(a), 0);
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s = 1 + t * z;
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return t + a * (s + t * v);
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}
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}
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function MathSinSlow(x) {
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REMPIO2(x);
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var sign = 1 - (n & 2);
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if (n & 1) {
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RETURN_KERNELCOS(y0, y1, * sign);
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} else {
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RETURN_KERNELSIN(y0, y1, * sign);
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}
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}
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function MathCosSlow(x) {
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REMPIO2(x);
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if (n & 1) {
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var sign = (n & 2) - 1;
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RETURN_KERNELSIN(y0, y1, * sign);
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} else {
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var sign = 1 - (n & 2);
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RETURN_KERNELCOS(y0, y1, * sign);
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}
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}
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// ECMA 262 - 15.8.2.16
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function MathSin(x) {
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x = x * 1; // Convert to number.
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if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
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// |x| < pi/4, approximately. No reduction needed.
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RETURN_KERNELSIN(x, 0, /* empty */);
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}
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return MathSinSlow(x);
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}
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// ECMA 262 - 15.8.2.7
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function MathCos(x) {
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x = x * 1; // Convert to number.
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if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
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// |x| < pi/4, approximately. No reduction needed.
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RETURN_KERNELCOS(x, 0, /* empty */);
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}
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return MathCosSlow(x);
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}
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// ECMA 262 - 15.8.2.18
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function MathTan(x) {
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x = x * 1; // Convert to number.
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if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
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// |x| < pi/4, approximately. No reduction needed.
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return KernelTan(x, 0, 1);
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}
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REMPIO2(x);
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return KernelTan(y0, y1, (n & 1) ? -1 : 1);
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}
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// ES6 draft 09-27-13, section 20.2.2.20.
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// Math.log1p
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//
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// Method :
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// 1. Argument Reduction: find k and f such that
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// 1+x = 2^k * (1+f),
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// where sqrt(2)/2 < 1+f < sqrt(2) .
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//
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// Note. If k=0, then f=x is exact. However, if k!=0, then f
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// may not be representable exactly. In that case, a correction
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// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
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// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
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// and add back the correction term c/u.
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// (Note: when x > 2**53, one can simply return log(x))
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//
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// 2. Approximation of log1p(f).
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// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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// = 2s + s*R
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// We use a special Reme algorithm on [0,0.1716] to generate
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// a polynomial of degree 14 to approximate R The maximum error
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// of this polynomial approximation is bounded by 2**-58.45. In
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// other words,
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// 2 4 6 8 10 12 14
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// R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
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// (the values of Lp1 to Lp7 are listed in the program)
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// and
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// | 2 14 | -58.45
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// | Lp1*s +...+Lp7*s - R(z) | <= 2
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// | |
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// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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// In order to guarantee error in log below 1ulp, we compute log
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// by
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// log1p(f) = f - (hfsq - s*(hfsq+R)).
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//
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// 3. Finally, log1p(x) = k*ln2 + log1p(f).
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// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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// Here ln2 is split into two floating point number:
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// ln2_hi + ln2_lo,
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// where n*ln2_hi is always exact for |n| < 2000.
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//
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// Special cases:
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// log1p(x) is NaN with signal if x < -1 (including -INF) ;
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// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
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// log1p(NaN) is that NaN with no signal.
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//
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// Accuracy:
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// according to an error analysis, the error is always less than
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// 1 ulp (unit in the last place).
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//
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// Constants:
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// The hexadecimal values are the intended ones for the following
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// constants. The decimal values may be used, provided that the
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// compiler will convert from decimal to binary accurately enough
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// to produce the hexadecimal values shown.
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//
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// Note: Assuming log() return accurate answer, the following
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// algorithm can be used to compute log1p(x) to within a few ULP:
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//
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// u = 1+x;
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// if (u==1.0) return x ; else
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// return log(u)*(x/(u-1.0));
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//
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// See HP-15C Advanced Functions Handbook, p.193.
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//
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const LN2_HI = kMath[34];
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const LN2_LO = kMath[35];
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const TWO54 = kMath[36];
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const TWO_THIRD = kMath[37];
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macro KLOGP1(x)
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(kMath[38+x])
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endmacro
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function MathLog1p(x) {
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x = x * 1; // Convert to number.
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var hx = %_DoubleHi(x);
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var ax = hx & 0x7fffffff;
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var k = 1;
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var f = x;
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var hu = 1;
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var c = 0;
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var u = x;
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if (hx < 0x3fda827a) {
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// x < 0.41422
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if (ax >= 0x3ff00000) { // |x| >= 1
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if (x === -1) {
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return -INFINITY; // log1p(-1) = -inf
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} else {
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return NAN; // log1p(x<-1) = NaN
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}
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} else if (ax < 0x3c900000) {
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// For |x| < 2^-54 we can return x.
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return x;
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} else if (ax < 0x3e200000) {
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// For |x| < 2^-29 we can use a simple two-term Taylor series.
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return x - x * x * 0.5;
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}
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if ((hx > 0) || (hx <= -0x402D413D)) { // (int) 0xbfd2bec3 = -0x402d413d
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// -.2929 < x < 0.41422
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k = 0;
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}
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}
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// Handle Infinity and NAN
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if (hx >= 0x7ff00000) return x;
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if (k !== 0) {
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if (hx < 0x43400000) {
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// x < 2^53
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u = 1 + x;
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hu = %_DoubleHi(u);
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k = (hu >> 20) - 1023;
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c = (k > 0) ? 1 - (u - x) : x - (u - 1);
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c = c / u;
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} else {
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hu = %_DoubleHi(u);
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k = (hu >> 20) - 1023;
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}
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hu = hu & 0xfffff;
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if (hu < 0x6a09e) {
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u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u)); // Normalize u.
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} else {
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++k;
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u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u)); // Normalize u/2.
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hu = (0x00100000 - hu) >> 2;
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}
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f = u - 1;
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}
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var hfsq = 0.5 * f * f;
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if (hu === 0) {
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// |f| < 2^-20;
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if (f === 0) {
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if (k === 0) {
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return 0.0;
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} else {
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return k * LN2_HI + (c + k * LN2_LO);
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}
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}
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var R = hfsq * (1 - TWO_THIRD * f);
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if (k === 0) {
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return f - R;
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} else {
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return k * LN2_HI - ((R - (k * LN2_LO + c)) - f);
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}
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}
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var s = f / (2 + f);
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var z = s * s;
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var R = z * (KLOGP1(0) + z * (KLOGP1(1) + z *
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(KLOGP1(2) + z * (KLOGP1(3) + z *
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(KLOGP1(4) + z * (KLOGP1(5) + z * KLOGP1(6)))))));
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if (k === 0) {
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return f - (hfsq - s * (hfsq + R));
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} else {
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return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f);
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}
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}
|