// The following is adapted from fdlibm (http://www.netlib.org/fdlibm),
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// The original source code covered by the above license above has been
// modified significantly by Google Inc.
// Copyright 2014 the V8 project authors. All rights reserved.
//
// The following is a straightforward translation of fdlibm routines for
// sin, cos, and tan, by Raymond Toy (rtoy@google.com).
var kTrig; // Initialized to a Float64Array during genesis and is not writable.
const INVPIO2 = kTrig[0];
const PIO2_1 = kTrig[1];
const PIO2_1T = kTrig[2];
const PIO2_2 = kTrig[3];
const PIO2_2T = kTrig[4];
const PIO2_3 = kTrig[5];
const PIO2_3T = kTrig[6];
const PIO4 = kTrig[32];
const PIO4LO = kTrig[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)
kTrig[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)
kTrig[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)
kTrig[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);
}