// The following is adapted from fdlibm (http://www.netlib.org/fdlibm),
//
// ====================================================
// Copyright (C) 1993-2004 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// The original source code covered by the above license above has been
// modified significantly by Google Inc.
// Copyright 2014 the V8 project authors. All rights reserved.
//
// The following is a straightforward translation of fdlibm routines
// by Raymond Toy (rtoy@google.com).
// Double constants that do not have empty lower 32 bits are found in fdlibm.cc
// and exposed through kMath as typed array. We assume the compiler to convert
// from decimal to binary accurately enough to produce the intended values.
// kMath is initialized to a Float64Array during genesis and not writable.
var kMath;
const INVPIO2 = kMath[0];
const PIO2_1 = kMath[1];
const PIO2_1T = kMath[2];
const PIO2_2 = kMath[3];
const PIO2_2T = kMath[4];
const PIO2_3 = kMath[5];
const PIO2_3T = kMath[6];
const PIO4 = kMath[32];
const PIO4LO = kMath[33];
// Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For
// precision, r is returned as two values y0 and y1 such that r = y0 + y1
// to more than double precision.
macro REMPIO2(X)
var n, y0, y1;
var hx = %_DoubleHi(X);
var ix = hx & 0x7fffffff;
if (ix < 0x4002d97c) {
// |X| ~< 3*pi/4, special case with n = +/- 1
if (hx > 0) {
var z = X - PIO2_1;
if (ix != 0x3ff921fb) {
// 33+53 bit pi is good enough
y0 = z - PIO2_1T;
y1 = (z - y0) - PIO2_1T;
} else {
// near pi/2, use 33+33+53 bit pi
z -= PIO2_2;
y0 = z - PIO2_2T;
y1 = (z - y0) - PIO2_2T;
}
n = 1;
} else {
// Negative X
var z = X + PIO2_1;
if (ix != 0x3ff921fb) {
// 33+53 bit pi is good enough
y0 = z + PIO2_1T;
y1 = (z - y0) + PIO2_1T;
} else {
// near pi/2, use 33+33+53 bit pi
z += PIO2_2;
y0 = z + PIO2_2T;
y1 = (z - y0) + PIO2_2T;
}
n = -1;
}
} else if (ix <= 0x413921fb) {
// |X| ~<= 2^19*(pi/2), medium size
var t = MathAbs(X);
n = (t * INVPIO2 + 0.5) | 0;
var r = t - n * PIO2_1;
var w = n * PIO2_1T;
// First round good to 85 bit
y0 = r - w;
if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) {
// 2nd iteration needed, good to 118
t = r;
w = n * PIO2_2;
r = t - w;
w = n * PIO2_2T - ((t - r) - w);
y0 = r - w;
if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) {
// 3rd iteration needed. 151 bits accuracy
t = r;
w = n * PIO2_3;
r = t - w;
w = n * PIO2_3T - ((t - r) - w);
y0 = r - w;
}
}
y1 = (r - y0) - w;
if (hx < 0) {
n = -n;
y0 = -y0;
y1 = -y1;
}
} else {
// Need to do full Payne-Hanek reduction here.
var r = %RemPiO2(X);
n = r[0];
y0 = r[1];
y1 = r[2];
}
endmacro
// __kernel_sin(X, Y, IY)
// kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
// Input X is assumed to be bounded by ~pi/4 in magnitude.
// Input Y is the tail of X so that x = X + Y.
//
// Algorithm
// 1. Since ieee_sin(-x) = -ieee_sin(x), we need only to consider positive x.
// 2. ieee_sin(x) is approximated by a polynomial of degree 13 on
// [0,pi/4]
// 3 13
// sin(x) ~ x + S1*x + ... + S6*x
// where
//
// |ieee_sin(x) 2 4 6 8 10 12 | -58
// |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
// | x |
//
// 3. ieee_sin(X+Y) = ieee_sin(X) + sin'(X')*Y
// ~ ieee_sin(X) + (1-X*X/2)*Y
// For better accuracy, let
// 3 2 2 2 2
// r = X *(S2+X *(S3+X *(S4+X *(S5+X *S6))))
// then 3 2
// sin(x) = X + (S1*X + (X *(r-Y/2)+Y))
//
macro KSIN(x)
kMath[7+x]
endmacro
macro RETURN_KERNELSIN(X, Y, SIGN)
var z = X * X;
var v = z * X;
var r = KSIN(1) + z * (KSIN(2) + z * (KSIN(3) +
z * (KSIN(4) + z * KSIN(5))));
return (X - ((z * (0.5 * Y - v * r) - Y) - v * KSIN(0))) SIGN;
endmacro
// __kernel_cos(X, Y)
// kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
// Input X is assumed to be bounded by ~pi/4 in magnitude.
// Input Y is the tail of X so that x = X + Y.
//
// Algorithm
// 1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x.
// 2. ieee_cos(x) is approximated by a polynomial of degree 14 on
// [0,pi/4]
// 4 14
// cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
// where the remez error is
//
// | 2 4 6 8 10 12 14 | -58
// |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
// | |
//
// 4 6 8 10 12 14
// 3. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
// ieee_cos(x) = 1 - x*x/2 + r
// since ieee_cos(X+Y) ~ ieee_cos(X) - ieee_sin(X)*Y
// ~ ieee_cos(X) - X*Y,
// a correction term is necessary in ieee_cos(x) and hence
// cos(X+Y) = 1 - (X*X/2 - (r - X*Y))
// For better accuracy when x > 0.3, let qx = |x|/4 with
// the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
// Then
// cos(X+Y) = (1-qx) - ((X*X/2-qx) - (r-X*Y)).
// Note that 1-qx and (X*X/2-qx) is EXACT here, and the
// magnitude of the latter is at least a quarter of X*X/2,
// thus, reducing the rounding error in the subtraction.
//
macro KCOS(x)
kMath[13+x]
endmacro
macro RETURN_KERNELCOS(X, Y, SIGN)
var ix = %_DoubleHi(X) & 0x7fffffff;
var z = X * X;
var r = z * (KCOS(0) + z * (KCOS(1) + z * (KCOS(2)+
z * (KCOS(3) + z * (KCOS(4) + z * KCOS(5))))));
if (ix < 0x3fd33333) { // |x| ~< 0.3
return (1 - (0.5 * z - (z * r - X * Y))) SIGN;
} else {
var qx;
if (ix > 0x3fe90000) { // |x| > 0.78125
qx = 0.28125;
} else {
qx = %_ConstructDouble(%_DoubleHi(0.25 * X), 0);
}
var hz = 0.5 * z - qx;
return (1 - qx - (hz - (z * r - X * Y))) SIGN;
}
endmacro
// kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
// Input x is assumed to be bounded by ~pi/4 in magnitude.
// Input y is the tail of x.
// Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1)
// is returned.
//
// Algorithm
// 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x.
// 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
// 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on
// [0,0.67434]
// 3 27
// tan(x) ~ x + T1*x + ... + T13*x
// where
//
// |ieee_tan(x) 2 4 26 | -59.2
// |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
// | x |
//
// Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y
// ~ ieee_tan(x) + (1+x*x)*y
// Therefore, for better accuracy in computing ieee_tan(x+y), let
// 3 2 2 2 2
// r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
// then
// 3 2
// tan(x+y) = x + (T1*x + (x *(r+y)+y))
//
// 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
// tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y))
// = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y)))
//
// Set returnTan to 1 for tan; -1 for cot. Anything else is illegal
// and will cause incorrect results.
//
macro KTAN(x)
kMath[19+x]
endmacro
function KernelTan(x, y, returnTan) {
var z;
var w;
var hx = %_DoubleHi(x);
var ix = hx & 0x7fffffff;
if (ix < 0x3e300000) { // |x| < 2^-28
if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) {
// x == 0 && returnTan = -1
return 1 / MathAbs(x);
} else {
if (returnTan == 1) {
return x;
} else {
// Compute -1/(x + y) carefully
var w = x + y;
var z = %_ConstructDouble(%_DoubleHi(w), 0);
var v = y - (z - x);
var a = -1 / w;
var t = %_ConstructDouble(%_DoubleHi(a), 0);
var s = 1 + t * z;
return t + a * (s + t * v);
}
}
}
if (ix >= 0x3fe59428) { // |x| > .6744
if (x < 0) {
x = -x;
y = -y;
}
z = PIO4 - x;
w = PIO4LO - y;
x = z + w;
y = 0;
}
z = x * x;
w = z * z;
// Break x^5 * (T1 + x^2*T2 + ...) into
// x^5 * (T1 + x^4*T3 + ... + x^20*T11) +
// x^5 * (x^2 * (T2 + x^4*T4 + ... + x^22*T12))
var r = KTAN(1) + w * (KTAN(3) + w * (KTAN(5) +
w * (KTAN(7) + w * (KTAN(9) + w * KTAN(11)))));
var v = z * (KTAN(2) + w * (KTAN(4) + w * (KTAN(6) +
w * (KTAN(8) + w * (KTAN(10) + w * KTAN(12))))));
var s = z * x;
r = y + z * (s * (r + v) + y);
r = r + KTAN(0) * s;
w = x + r;
if (ix >= 0x3fe59428) {
return (1 - ((hx >> 30) & 2)) *
(returnTan - 2.0 * (x - (w * w / (w + returnTan) - r)));
}
if (returnTan == 1) {
return w;
} else {
z = %_ConstructDouble(%_DoubleHi(w), 0);
v = r - (z - x);
var a = -1 / w;
var t = %_ConstructDouble(%_DoubleHi(a), 0);
s = 1 + t * z;
return t + a * (s + t * v);
}
}
function MathSinSlow(x) {
REMPIO2(x);
var sign = 1 - (n & 2);
if (n & 1) {
RETURN_KERNELCOS(y0, y1, * sign);
} else {
RETURN_KERNELSIN(y0, y1, * sign);
}
}
function MathCosSlow(x) {
REMPIO2(x);
if (n & 1) {
var sign = (n & 2) - 1;
RETURN_KERNELSIN(y0, y1, * sign);
} else {
var sign = 1 - (n & 2);
RETURN_KERNELCOS(y0, y1, * sign);
}
}
// ECMA 262 - 15.8.2.16
function MathSin(x) {
x = x * 1; // Convert to number.
if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
// |x| < pi/4, approximately. No reduction needed.
RETURN_KERNELSIN(x, 0, /* empty */);
}
return MathSinSlow(x);
}
// ECMA 262 - 15.8.2.7
function MathCos(x) {
x = x * 1; // Convert to number.
if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
// |x| < pi/4, approximately. No reduction needed.
RETURN_KERNELCOS(x, 0, /* empty */);
}
return MathCosSlow(x);
}
// ECMA 262 - 15.8.2.18
function MathTan(x) {
x = x * 1; // Convert to number.
if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) {
// |x| < pi/4, approximately. No reduction needed.
return KernelTan(x, 0, 1);
}
REMPIO2(x);
return KernelTan(y0, y1, (n & 1) ? -1 : 1);
}
// ES6 draft 09-27-13, section 20.2.2.20.
// Math.log1p
//
// Method :
// 1. Argument Reduction: find k and f such that
// 1+x = 2^k * (1+f),
// where sqrt(2)/2 < 1+f < sqrt(2) .
//
// Note. If k=0, then f=x is exact. However, if k!=0, then f
// may not be representable exactly. In that case, a correction
// term is need. Let u=1+x rounded. Let c = (1+x)-u, then
// log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
// and add back the correction term c/u.
// (Note: when x > 2**53, one can simply return log(x))
//
// 2. Approximation of log1p(f).
// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
// = 2s + s*R
// We use a special Reme algorithm on [0,0.1716] to generate
// a polynomial of degree 14 to approximate R The maximum error
// of this polynomial approximation is bounded by 2**-58.45. In
// other words,
// 2 4 6 8 10 12 14
// R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
// (the values of Lp1 to Lp7 are listed in the program)
// and
// | 2 14 | -58.45
// | Lp1*s +...+Lp7*s - R(z) | <= 2
// | |
// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
// In order to guarantee error in log below 1ulp, we compute log
// by
// log1p(f) = f - (hfsq - s*(hfsq+R)).
//
// 3. Finally, log1p(x) = k*ln2 + log1p(f).
// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
// Here ln2 is split into two floating point number:
// ln2_hi + ln2_lo,
// where n*ln2_hi is always exact for |n| < 2000.
//
// Special cases:
// log1p(x) is NaN with signal if x < -1 (including -INF) ;
// log1p(+INF) is +INF; log1p(-1) is -INF with signal;
// log1p(NaN) is that NaN with no signal.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Constants:
// Constants are found in fdlibm.cc. We assume the C++ compiler to convert
// from decimal to binary accurately enough to produce the intended values.
//
// Note: Assuming log() return accurate answer, the following
// algorithm can be used to compute log1p(x) to within a few ULP:
//
// u = 1+x;
// if (u==1.0) return x ; else
// return log(u)*(x/(u-1.0));
//
// See HP-15C Advanced Functions Handbook, p.193.
//
const LN2_HI = kMath[34];
const LN2_LO = kMath[35];
const TWO54 = kMath[36];
const TWO_THIRD = kMath[37];
macro KLOG1P(x)
(kMath[38+x])
endmacro
function MathLog1p(x) {
x = x * 1; // Convert to number.
var hx = %_DoubleHi(x);
var ax = hx & 0x7fffffff;
var k = 1;
var f = x;
var hu = 1;
var c = 0;
var u = x;
if (hx < 0x3fda827a) {
// x < 0.41422
if (ax >= 0x3ff00000) { // |x| >= 1
if (x === -1) {
return -INFINITY; // log1p(-1) = -inf
} else {
return NAN; // log1p(x<-1) = NaN
}
} else if (ax < 0x3c900000) {
// For |x| < 2^-54 we can return x.
return x;
} else if (ax < 0x3e200000) {
// For |x| < 2^-29 we can use a simple two-term Taylor series.
return x - x * x * 0.5;
}
if ((hx > 0) || (hx <= -0x402D413D)) { // (int) 0xbfd2bec3 = -0x402d413d
// -.2929 < x < 0.41422
k = 0;
}
}
// Handle Infinity and NAN
if (hx >= 0x7ff00000) return x;
if (k !== 0) {
if (hx < 0x43400000) {
// x < 2^53
u = 1 + x;
hu = %_DoubleHi(u);
k = (hu >> 20) - 1023;
c = (k > 0) ? 1 - (u - x) : x - (u - 1);
c = c / u;
} else {
hu = %_DoubleHi(u);
k = (hu >> 20) - 1023;
}
hu = hu & 0xfffff;
if (hu < 0x6a09e) {
u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u)); // Normalize u.
} else {
++k;
u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u)); // Normalize u/2.
hu = (0x00100000 - hu) >> 2;
}
f = u - 1;
}
var hfsq = 0.5 * f * f;
if (hu === 0) {
// |f| < 2^-20;
if (f === 0) {
if (k === 0) {
return 0.0;
} else {
return k * LN2_HI + (c + k * LN2_LO);
}
}
var R = hfsq * (1 - TWO_THIRD * f);
if (k === 0) {
return f - R;
} else {
return k * LN2_HI - ((R - (k * LN2_LO + c)) - f);
}
}
var s = f / (2 + f);
var z = s * s;
var R = z * (KLOG1P(0) + z * (KLOG1P(1) + z *
(KLOG1P(2) + z * (KLOG1P(3) + z *
(KLOG1P(4) + z * (KLOG1P(5) + z * KLOG1P(6)))))));
if (k === 0) {
return f - (hfsq - s * (hfsq + R));
} else {
return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f);
}
}
// ES6 draft 09-27-13, section 20.2.2.14.
// Math.expm1
// Returns exp(x)-1, the exponential of x minus 1.
//
// Method
// 1. Argument reduction:
// Given x, find r and integer k such that
//
// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
//
// Here a correction term c will be computed to compensate
// the error in r when rounded to a floating-point number.
//
// 2. Approximating expm1(r) by a special rational function on
// the interval [0,0.34658]:
// Since
// r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
// we define R1(r*r) by
// r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
// That is,
// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
// = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
// We use a special Remes algorithm on [0,0.347] to generate
// a polynomial of degree 5 in r*r to approximate R1. The
// maximum error of this polynomial approximation is bounded
// by 2**-61. In other words,
// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
// where Q1 = -1.6666666666666567384E-2,
// Q2 = 3.9682539681370365873E-4,
// Q3 = -9.9206344733435987357E-6,
// Q4 = 2.5051361420808517002E-7,
// Q5 = -6.2843505682382617102E-9;
// (where z=r*r, and the values of Q1 to Q5 are listed below)
// with error bounded by
// | 5 | -61
// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
// | |
//
// expm1(r) = exp(r)-1 is then computed by the following
// specific way which minimize the accumulation rounding error:
// 2 3
// r r [ 3 - (R1 + R1*r/2) ]
// expm1(r) = r + --- + --- * [--------------------]
// 2 2 [ 6 - r*(3 - R1*r/2) ]
//
// To compensate the error in the argument reduction, we use
// expm1(r+c) = expm1(r) + c + expm1(r)*c
// ~ expm1(r) + c + r*c
// Thus c+r*c will be added in as the correction terms for
// expm1(r+c). Now rearrange the term to avoid optimization
// screw up:
// ( 2 2 )
// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
// ( )
//
// = r - E
// 3. Scale back to obtain expm1(x):
// From step 1, we have
// expm1(x) = either 2^k*[expm1(r)+1] - 1
// = or 2^k*[expm1(r) + (1-2^-k)]
// 4. Implementation notes:
// (A). To save one multiplication, we scale the coefficient Qi
// to Qi*2^i, and replace z by (x^2)/2.
// (B). To achieve maximum accuracy, we compute expm1(x) by
// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
// (ii) if k=0, return r-E
// (iii) if k=-1, return 0.5*(r-E)-0.5
// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
// else return 1.0+2.0*(r-E);
// (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
// (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
// (vii) return 2^k(1-((E+2^-k)-r))
//
// Special cases:
// expm1(INF) is INF, expm1(NaN) is NaN;
// expm1(-INF) is -1, and
// for finite argument, only expm1(0)=0 is exact.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Misc. info.
// For IEEE double
// if x > 7.09782712893383973096e+02 then expm1(x) overflow
//
const KEXPM1_OVERFLOW = kMath[45];
const INVLN2 = kMath[46];
macro KEXPM1(x)
(kMath[47+x])
endmacro
function MathExpm1(x) {
x = x * 1; // Convert to number.
var y;
var hi;
var lo;
var k;
var t;
var c;
var hx = %_DoubleHi(x);
var xsb = hx & 0x80000000; // Sign bit of x
var y = (xsb === 0) ? x : -x; // y = |x|
hx &= 0x7fffffff; // High word of |x|
// Filter out huge and non-finite argument
if (hx >= 0x4043687a) { // if |x| ~=> 56 * ln2
if (hx >= 0x40862e42) { // if |x| >= 709.78
if (hx >= 0x7ff00000) {
// expm1(inf) = inf; expm1(-inf) = -1; expm1(nan) = nan;
return (x === -INFINITY) ? -1 : x;
}
if (x > KEXPM1_OVERFLOW) return INFINITY; // Overflow
}
if (xsb != 0) return -1; // x < -56 * ln2, return -1.
}
// Argument reduction
if (hx > 0x3fd62e42) { // if |x| > 0.5 * ln2
if (hx < 0x3ff0a2b2) { // and |x| < 1.5 * ln2
if (xsb === 0) {
hi = x - LN2_HI;
lo = LN2_LO;
k = 1;
} else {
hi = x + LN2_HI;
lo = -LN2_LO;
k = -1;
}
} else {
k = (INVLN2 * x + ((xsb === 0) ? 0.5 : -0.5)) | 0;
t = k;
// t * ln2_hi is exact here.
hi = x - t * LN2_HI;
lo = t * LN2_LO;
}
x = hi - lo;
c = (hi - x) - lo;
} else if (hx < 0x3c900000) {
// When |x| < 2^-54, we can return x.
return x;
} else {
// Fall through.
k = 0;
}
// x is now in primary range
var hfx = 0.5 * x;
var hxs = x * hfx;
var r1 = 1 + hxs * (KEXPM1(0) + hxs * (KEXPM1(1) + hxs *
(KEXPM1(2) + hxs * (KEXPM1(3) + hxs * KEXPM1(4)))));
t = 3 - r1 * hfx;
var e = hxs * ((r1 - t) / (6 - x * t));
if (k === 0) { // c is 0
return x - (x*e - hxs);
} else {
e = (x * (e - c) - c);
e -= hxs;
if (k === -1) return 0.5 * (x - e) - 0.5;
if (k === 1) {
if (x < -0.25) return -2 * (e - (x + 0.5));
return 1 + 2 * (x - e);
}
if (k <= -2 || k > 56) {
// suffice to return exp(x) + 1
y = 1 - (e - x);
// Add k to y's exponent
y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
return y - 1;
}
if (k < 20) {
// t = 1 - 2^k
t = %_ConstructDouble(0x3ff00000 - (0x200000 >> k), 0);
y = t - (e - x);
// Add k to y's exponent
y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
} else {
// t = 2^-k
t = %_ConstructDouble((0x3ff - k) << 20, 0);
y = x - (e + t);
y += 1;
// Add k to y's exponent
y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y));
}
}
return y;
}
// ES6 draft 09-27-13, section 20.2.2.30.
// Math.sinh
// Method :
// mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
// 1. Replace x by |x| (sinh(-x) = -sinh(x)).
// 2.
// E + E/(E+1)
// 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
// 2
//
// 22 <= x <= lnovft : sinh(x) := exp(x)/2
// lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
// ln2ovft < x : sinh(x) := x*shuge (overflow)
//
// Special cases:
// sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
// only sinh(0)=0 is exact for finite x.
//
const KSINH_OVERFLOW = kMath[52];
const TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half
const LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half
function MathSinh(x) {
x = x * 1; // Convert to number.
var h = (x < 0) ? -0.5 : 0.5;
// |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
var ax = MathAbs(x);
if (ax < 22) {
// For |x| < 2^-28, sinh(x) = x
if (ax < TWO_M28) return x;
var t = MathExpm1(ax);
if (ax < 1) return h * (2 * t - t * t / (t + 1));
return h * (t + t / (t + 1));
}
// |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
if (ax < LOG_MAXD) return h * MathExp(ax);
// |x| in [log(maxdouble), overflowthreshold]
// overflowthreshold = 710.4758600739426
if (ax <= KSINH_OVERFLOW) {
var w = MathExp(0.5 * ax);
var t = h * w;
return t * w;
}
// |x| > overflowthreshold or is NaN.
// Return Infinity of the appropriate sign or NaN.
return x * INFINITY;
}
// ES6 draft 09-27-13, section 20.2.2.12.
// Math.cosh
// Method :
// mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
// 1. Replace x by |x| (cosh(x) = cosh(-x)).
// 2.
// [ exp(x) - 1 ]^2
// 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
// 2*exp(x)
//
// exp(x) + 1/exp(x)
// ln2/2 <= x <= 22 : cosh(x) := -------------------
// 2
// 22 <= x <= lnovft : cosh(x) := exp(x)/2
// lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
// ln2ovft < x : cosh(x) := huge*huge (overflow)
//
// Special cases:
// cosh(x) is |x| if x is +INF, -INF, or NaN.
// only cosh(0)=1 is exact for finite x.
//
const KCOSH_OVERFLOW = kMath[52];
function MathCosh(x) {
x = x * 1; // Convert to number.
var ix = %_DoubleHi(x) & 0x7fffffff;
// |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
if (ix < 0x3fd62e43) {
var t = MathExpm1(MathAbs(x));
var w = 1 + t;
// For |x| < 2^-55, cosh(x) = 1
if (ix < 0x3c800000) return w;
return 1 + (t * t) / (w + w);
}
// |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
if (ix < 0x40360000) {
var t = MathExp(MathAbs(x));
return 0.5 * t + 0.5 / t;
}
// |x| in [22, log(maxdouble)], return half*exp(|x|)
if (ix < 0x40862e42) return 0.5 * MathExp(MathAbs(x));
// |x| in [log(maxdouble), overflowthreshold]
if (MathAbs(x) <= KCOSH_OVERFLOW) {
var w = MathExp(0.5 * MathAbs(x));
var t = 0.5 * w;
return t * w;
}
if (NUMBER_IS_NAN(x)) return x;
// |x| > overflowthreshold.
return INFINITY;
}