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// 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 2016 the V8 project authors. All rights reserved.
#include "src/base/ieee754.h"
#include <cmath>
#include <limits>
#include "src/base/build_config.h"
#include "src/base/macros.h"
#include "src/base/overflowing-math.h"
namespace v8 {
namespace base {
namespace ieee754 {
namespace {
/* Disable "potential divide by 0" warning in Visual Studio compiler. */
#if V8_CC_MSVC
#pragma warning(disable : 4723)
#endif
/*
* The original fdlibm code used statements like:
* n0 = ((*(int*)&one)>>29)^1; * index of high word *
* ix0 = *(n0+(int*)&x); * high word of x *
* ix1 = *((1-n0)+(int*)&x); * low word of x *
* to dig two 32 bit words out of the 64 bit IEEE floating point
* value. That is non-ANSI, and, moreover, the gcc instruction
* scheduler gets it wrong. We instead use the following macros.
* Unlike the original code, we determine the endianness at compile
* time, not at run time; I don't see much benefit to selecting
* endianness at run time.
*/
/* Get two 32 bit ints from a double. */
#define EXTRACT_WORDS(ix0, ix1, d) \
do { \
uint64_t bits = bit_cast<uint64_t>(d); \
(ix0) = bits >> 32; \
(ix1) = bits & 0xFFFFFFFFu; \
} while (false)
/* Get the more significant 32 bit int from a double. */
#define GET_HIGH_WORD(i, d) \
do { \
uint64_t bits = bit_cast<uint64_t>(d); \
(i) = bits >> 32; \
} while (false)
/* Get the less significant 32 bit int from a double. */
#define GET_LOW_WORD(i, d) \
do { \
uint64_t bits = bit_cast<uint64_t>(d); \
(i) = bits & 0xFFFFFFFFu; \
} while (false)
/* Set a double from two 32 bit ints. */
#define INSERT_WORDS(d, ix0, ix1) \
do { \
uint64_t bits = 0; \
bits |= static_cast<uint64_t>(ix0) << 32; \
bits |= static_cast<uint32_t>(ix1); \
(d) = bit_cast<double>(bits); \
} while (false)
/* Set the more significant 32 bits of a double from an int. */
#define SET_HIGH_WORD(d, v) \
do { \
uint64_t bits = bit_cast<uint64_t>(d); \
bits &= 0x0000'0000'FFFF'FFFF; \
bits |= static_cast<uint64_t>(v) << 32; \
(d) = bit_cast<double>(bits); \
} while (false)
/* Set the less significant 32 bits of a double from an int. */
#define SET_LOW_WORD(d, v) \
do { \
uint64_t bits = bit_cast<uint64_t>(d); \
bits &= 0xFFFF'FFFF'0000'0000; \
bits |= static_cast<uint32_t>(v); \
(d) = bit_cast<double>(bits); \
} while (false)
int32_t __ieee754_rem_pio2(double x, double* y) V8_WARN_UNUSED_RESULT;
double __kernel_cos(double x, double y) V8_WARN_UNUSED_RESULT;
int __kernel_rem_pio2(double* x, double* y, int e0, int nx, int prec,
const int32_t* ipio2) V8_WARN_UNUSED_RESULT;
double __kernel_sin(double x, double y, int iy) V8_WARN_UNUSED_RESULT;
/* __ieee754_rem_pio2(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __kernel_rem_pio2()
*/
int32_t __ieee754_rem_pio2(double x, double *y) {
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*/
static const int32_t two_over_pi[] = {
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C,
0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649,
0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44,
0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B,
0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D,
0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330,
0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08,
0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
0x73A8C9, 0x60E27B, 0xC08C6B,
};
static const int32_t npio2_hw[] = {
0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
0x404858EB, 0x404921FB,
};
/*
* invpio2: 53 bits of 2/pi
* pio2_1: first 33 bit of pi/2
* pio2_1t: pi/2 - pio2_1
* pio2_2: second 33 bit of pi/2
* pio2_2t: pi/2 - (pio2_1+pio2_2)
* pio2_3: third 33 bit of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const double
zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
double z, w, t, r, fn;
double tx[3];
int32_t e0, i, j, nx, n, ix, hx;
uint32_t low;
z = 0;
GET_HIGH_WORD(hx, x); /* high word of x */
ix = hx & 0x7FFFFFFF;
if (ix <= 0x3FE921FB) { /* |x| ~<= pi/4 , no need for reduction */
y[0] = x;
y[1] = 0;
return 0;
}
if (ix < 0x4002D97C) { /* |x| < 3pi/4, special case with n=+-1 */
if (hx > 0) {
z = x - pio2_1;
if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */
y[0] = z - pio2_1t;
y[1] = (z - y[0]) - pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z - y[0]) - pio2_2t;
}
return 1;
} else { /* negative x */
z = x + pio2_1;
if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */
y[0] = z + pio2_1t;
y[1] = (z - y[0]) + pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z - y[0]) + pio2_2t;
}
return -1;
}
}
if (ix <= 0x413921FB) { /* |x| ~<= 2^19*(pi/2), medium size */
t = fabs(x);
n = static_cast<int32_t>(t * invpio2 + half);
fn = static_cast<double>(n);
r = t - fn * pio2_1;
w = fn * pio2_1t; /* 1st round good to 85 bit */
if (n < 32 && ix != npio2_hw[n - 1]) {
y[0] = r - w; /* quick check no cancellation */
} else {
uint32_t high;
j = ix >> 20;
y[0] = r - w;
GET_HIGH_WORD(high, y[0]);
i = j - ((high >> 20) & 0x7FF);
if (i > 16) { /* 2nd iteration needed, good to 118 */
t = r;
w = fn * pio2_2;
r = t - w;
w = fn * pio2_2t - ((t - r) - w);
y[0] = r - w;
GET_HIGH_WORD(high, y[0]);
i = j - ((high >> 20) & 0x7FF);
if (i > 49) { /* 3rd iteration need, 151 bits acc */
t = r; /* will cover all possible cases */
w = fn * pio2_3;
r = t - w;
w = fn * pio2_3t - ((t - r) - w);
y[0] = r - w;
}
}
}
y[1] = (r - y[0]) - w;
if (hx < 0) {
y[0] = -y[0];
y[1] = -y[1];
return -n;
} else {
return n;
}
}
/*
* all other (large) arguments
*/
if (ix >= 0x7FF00000) { /* x is inf or NaN */
y[0] = y[1] = x - x;
return 0;
}
/* set z = scalbn(|x|,ilogb(x)-23) */
GET_LOW_WORD(low, x);
SET_LOW_WORD(z, low);
e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
SET_HIGH_WORD(z, ix - static_cast<int32_t>(static_cast<uint32_t>(e0) << 20));
for (i = 0; i < 2; i++) {
tx[i] = static_cast<double>(static_cast<int32_t>(z));
z = (z - tx[i]) * two24;
}
tx[2] = z;
nx = 3;
while (tx[nx - 1] == zero) nx--; /* skip zero term */
n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
if (hx < 0) {
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
return n;
}
/* __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.
*
* Algorithm
* 1. Since cos(-x) = cos(x), we need only to consider positive x.
* 2. if x < 2^-27 (hx<0x3E400000 0), return 1 with inexact if x!=0.
* 3. 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
* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
* | |
*
* 4 6 8 10 12 14
* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
* cos(x) = 1 - x*x/2 + r
* since cos(x+y) ~ cos(x) - sin(x)*y
* ~ cos(x) - x*y,
* a correction term is necessary in 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.
*/
V8_INLINE double __kernel_cos(double x, double y) {
static const double
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
double a, iz, z, r, qx;
int32_t ix;
GET_HIGH_WORD(ix, x);
ix &= 0x7FFFFFFF; /* ix = |x|'s high word*/
if (ix < 0x3E400000) { /* if x < 2**27 */
if (static_cast<int>(x) == 0) return one; /* generate inexact */
}
z = x * x;
r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
if (ix < 0x3FD33333) { /* if |x| < 0.3 */
return one - (0.5 * z - (z * r - x * y));
} else {
if (ix > 0x3FE90000) { /* x > 0.78125 */
qx = 0.28125;
} else {
INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
}
iz = 0.5 * z - qx;
a = one - qx;
return a - (iz - (z * r - x * y));
}
}
/* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
* double x[],y[]; int e0,nx,prec; int ipio2[];
*
* __kernel_rem_pio2 return the last three digits of N with
* y = x - N*pi/2
* so that |y| < pi/2.
*
* The method is to compute the integer (mod 8) and fraction parts of
* (2/pi)*x without doing the full multiplication. In general we
* skip the part of the product that are known to be a huge integer (
* more accurately, = 0 mod 8 ). Thus the number of operations are
* independent of the exponent of the input.
*
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
*
* Input parameters:
* x[] The input value (must be positive) is broken into nx
* pieces of 24-bit integers in double precision format.
* x[i] will be the i-th 24 bit of x. The scaled exponent
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
* match x's up to 24 bits.
*
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
* e0 = ilogb(z)-23
* z = scalbn(z,-e0)
* for i = 0,1,2
* x[i] = floor(z)
* z = (z-x[i])*2**24
*
*
* y[] output result in an array of double precision numbers.
* The dimension of y[] is:
* 24-bit precision 1
* 53-bit precision 2
* 64-bit precision 2
* 113-bit precision 3
* The actual value is the sum of them. Thus for 113-bit
* precison, one may have to do something like:
*
* long double t,w,r_head, r_tail;
* t = (long double)y[2] + (long double)y[1];
* w = (long double)y[0];
* r_head = t+w;
* r_tail = w - (r_head - t);
*
* e0 The exponent of x[0]
*
* nx dimension of x[]
*
* prec an integer indicating the precision:
* 0 24 bits (single)
* 1 53 bits (double)
* 2 64 bits (extended)
* 3 113 bits (quad)
*
* ipio2[]
* integer array, contains the (24*i)-th to (24*i+23)-th
* bit of 2/pi after binary point. The corresponding
* floating value is
*
* ipio2[i] * 2^(-24(i+1)).
*
* External function:
* double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
* jk jk+1 is the initial number of terms of ipio2[] needed
* in the computation. The recommended value is 2,3,4,
* 6 for single, double, extended,and quad.
*
* jz local integer variable indicating the number of
* terms of ipio2[] used.
*
* jx nx - 1
*
* jv index for pointing to the suitable ipio2[] for the
* computation. In general, we want
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
* is an integer. Thus
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
* Hence jv = max(0,(e0-3)/24).
*
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
*
* q[] double array with integral value, representing the
* 24-bits chunk of the product of x and 2/pi.
*
* q0 the corresponding exponent of q[0]. Note that the
* exponent for q[i] would be q0-24*i.
*
* PIo2[] double precision array, obtained by cutting pi/2
* into 24 bits chunks.
*
* f[] ipio2[] in floating point
*
* iq[] integer array by breaking up q[] in 24-bits chunk.
*
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
*
* ih integer. If >0 it indicates q[] is >= 0.5, hence
* it also indicates the *sign* of the result.
*
*/
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
const int32_t *ipio2) {
/* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
static const int init_jk[] = {2, 3, 4, 6}; /* initial value for jk */
static const double PIo2[] = {
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};
static const double
zero = 0.0,
one = 1.0,
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
double z, fw, f[20], fq[20], q[20];
/* initialize jk*/
jk = init_jk[prec];
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx - 1;
jv = (e0 - 3) / 24;
if (jv < 0) jv = 0;
q0 = e0 - 24 * (jv + 1);
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
j = jv - jx;
m = jx + jk;
for (i = 0; i <= m; i++, j++) {
f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]);
}
/* compute q[0],q[1],...q[jk] */
for (i = 0; i <= jk; i++) {
for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
iq[i] = static_cast<int32_t>(z - two24 * fw);
z = q[j - 1] + fw;
}
/* compute n */
z = scalbn(z, q0); /* actual value of z */
z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
n = static_cast<int32_t>(z);
z -= static_cast<double>(n);
ih = 0;
if (q0 > 0) { /* need iq[jz-1] to determine n */
i = (iq[jz - 1] >> (24 - q0));
n += i;
iq[jz - 1] -= i << (24 - q0);
ih = iq[jz - 1] >> (23 - q0);
} else if (q0 == 0) {
ih = iq[jz - 1] >> 23;
} else if (z >= 0.5) {
ih = 2;
}
if (ih > 0) { /* q > 0.5 */
n += 1;
carry = 0;
for (i = 0; i < jz; i++) { /* compute 1-q */
j = iq[i];
if (carry == 0) {
if (j != 0) {
carry = 1;
iq[i] = 0x1000000 - j;
}
} else {
iq[i] = 0xFFFFFF - j;
}
}
if (q0 > 0) { /* rare case: chance is 1 in 12 */
switch (q0) {
case 1:
iq[jz - 1] &= 0x7FFFFF;
break;
case 2:
iq[jz - 1] &= 0x3FFFFF;
break;
}
}
if (ih == 2) {
z = one - z;
if (carry != 0) z -= scalbn(one, q0);
}
}
/* check if recomputation is needed */
if (z == zero) {
j = 0;
for (i = jz - 1; i >= jk; i--) j |= iq[i];
if (j == 0) { /* need recomputation */
for (k = 1; jk >= k && iq[jk - k] == 0; k++) {
/* k = no. of terms needed */
}
for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
f[jx + i] = ipio2[jv + i];
for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if (z == 0.0) {
jz -= 1;
q0 -= 24;
while (iq[jz] == 0) {
jz--;
q0 -= 24;
}
} else { /* break z into 24-bit if necessary */
z = scalbn(z, -q0);
if (z >= two24) {
fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
iq[jz] = z - two24 * fw;
jz += 1;
q0 += 24;
iq[jz] = fw;
} else {
iq[jz] = z;
}
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn(one, q0);
for (i = jz; i >= 0; i--) {
q[i] = fw * iq[i];
fw *= twon24;
}
/* compute PIo2[0,...,jp]*q[jz,...,0] */
for (i = jz; i >= 0; i--) {
for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) fw += PIo2[k] * q[i + k];
fq[jz - i] = fw;
}
/* compress fq[] into y[] */
switch (prec) {
case 0:
fw = 0.0;
for (i = jz; i >= 0; i--) fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i = jz; i >= 0; i--) fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
fw = fq[0] - fw;
for (i = 1; i <= jz; i++) fw += fq[i];
y[1] = (ih == 0) ? fw : -fw;
break;
case 3: /* painful */
for (i = jz; i > 0; i--) {
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (i = jz; i > 1; i--) {
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (fw = 0.0, i = jz; i >= 2; i--) fw += fq[i];
if (ih == 0) {
y[0] = fq[0];
y[1] = fq[1];
y[2] = fw;
} else {
y[0] = -fq[0];
y[1] = -fq[1];
y[2] = -fw;
}
}
return n & 7;
}
/* __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.
* Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
*
* Algorithm
* 1. Since sin(-x) = -sin(x), we need only to consider positive x.
* 2. if x < 2^-27 (hx<0x3E400000 0), return x with inexact if x!=0.
* 3. sin(x) is approximated by a polynomial of degree 13 on
* [0,pi/4]
* 3 13
* sin(x) ~ x + S1*x + ... + S6*x
* where
*
* |sin(x) 2 4 6 8 10 12 | -58
* |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
* | x |
*
* 4. sin(x+y) = sin(x) + sin'(x')*y
* ~ 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))
*/
V8_INLINE double __kernel_sin(double x, double y, int iy) {
static const double
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
double z, r, v;
int32_t ix;
GET_HIGH_WORD(ix, x);
ix &= 0x7FFFFFFF; /* high word of x */
if (ix < 0x3E400000) { /* |x| < 2**-27 */
if (static_cast<int>(x) == 0) return x;
} /* generate inexact */
z = x * x;
v = z * x;
r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
if (iy == 0) {
return x + v * (S1 + z * r);
} else {
return x - ((z * (half * y - v * r) - y) - v * S1);
}
}
/* __kernel_tan( x, y, k )
* 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 tan (if k=1) or
* -1/tan (if k= -1) is returned.
*
* Algorithm
* 1. Since tan(-x) = -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. 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
*
* |tan(x) 2 4 26 | -59.2
* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
* | x |
*
* Note: tan(x+y) = tan(x) + tan'(x)*y
* ~ tan(x) + (1+x*x)*y
* Therefore, for better accuracy in computing 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) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
double __kernel_tan(double x, double y, int iy) {
static const double xxx[] = {
3.33333333333334091986e-01, /* 3FD55555, 55555563 */
1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
};
#define one xxx[13]
#define pio4 xxx[14]
#define pio4lo xxx[15]
#define T xxx
double z, r, v, w, s;
int32_t ix, hx;
GET_HIGH_WORD(hx, x); /* high word of x */
ix = hx & 0x7FFFFFFF; /* high word of |x| */
if (ix < 0x3E300000) { /* x < 2**-28 */
if (static_cast<int>(x) == 0) { /* generate inexact */
uint32_t low;
GET_LOW_WORD(low, x);
if (((ix | low) | (iy + 1)) == 0) {
return one / fabs(x);
} else {
if (iy == 1) {
return x;
} else { /* compute -1 / (x+y) carefully */
double a, t;
z = w = x + y;
SET_LOW_WORD(z, 0);
v = y - (z - x);
t = a = -one / w;
SET_LOW_WORD(t, 0);
s = one + t * z;
return t + a * (s + t * v);
}
}
}
}
if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
if (hx < 0) {
x = -x;
y = -y;
}
z = pio4 - x;
w = pio4lo - y;
x = z + w;
y = 0.0;
}
z = x * x;
w = z * z;
/*
* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
v = z *
(T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
s = z * x;
r = y + z * (s * (r + v) + y);
r += T[0] * s;
w = x + r;
if (ix >= 0x3FE59428) {
v = iy;
return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
}
if (iy == 1) {
return w;
} else {
/*
* if allow error up to 2 ulp, simply return
* -1.0 / (x+r) here
*/
/* compute -1.0 / (x+r) accurately */
double a, t;
z = w;
SET_LOW_WORD(z, 0);
v = r - (z - x); /* z+v = r+x */
t = a = -1.0 / w; /* a = -1.0/w */
SET_LOW_WORD(t, 0);
s = 1.0 + t * z;
return t + a * (s + t * v);
}
#undef one
#undef pio4
#undef pio4lo
#undef T
}
} // namespace
/* acos(x)
* Method :
* acos(x) = pi/2 - asin(x)
* acos(-x) = pi/2 + asin(x)
* For |x|<=0.5
* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
* For x>0.5
* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
* = 2asin(sqrt((1-x)/2))
* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
* = 2f + (2c + 2s*z*R(z))
* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
* for f so that f+c ~ sqrt(z).
* For x<-0.5
* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
* Function needed: sqrt
*/
double acos(double x) {
static const double
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
double z, p, q, r, w, s, c, df;
int32_t hx, ix;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7FFFFFFF;
if (ix >= 0x3FF00000) { /* |x| >= 1 */
uint32_t lx;
GET_LOW_WORD(lx, x);
if (((ix - 0x3FF00000) | lx) == 0) { /* |x|==1 */
if (hx > 0)
return 0.0; /* acos(1) = 0 */
else
return pi + 2.0 * pio2_lo; /* acos(-1)= pi */
}
return std::numeric_limits<double>::signaling_NaN(); // acos(|x|>1) is NaN
}
if (ix < 0x3FE00000) { /* |x| < 0.5 */
if (ix <= 0x3C600000) return pio2_hi + pio2_lo; /*if|x|<2**-57*/
z = x * x;
p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
r = p / q;
return pio2_hi - (x - (pio2_lo - x * r));
} else if (hx < 0) { /* x < -0.5 */
z = (one + x) * 0.5;
p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
s = sqrt(z);
r = p / q;
w = r * s - pio2_lo;
return pi - 2.0 * (s + w);
} else { /* x > 0.5 */
z = (one - x) * 0.5;
s = sqrt(z);
df = s;
SET_LOW_WORD(df, 0);
c = (z - df * df) / (s + df);
p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
r = p / q;
w = r * s + c;
return 2.0 * (df + w);
}
}
/* acosh(x)
* Method :
* Based on
* acosh(x) = log [ x + sqrt(x*x-1) ]
* we have
* acosh(x) := log(x)+ln2, if x is large; else
* acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
* acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
*
* Special cases:
* acosh(x) is NaN with signal if x<1.
* acosh(NaN) is NaN without signal.
*/
double acosh(double x) {
static const double
one = 1.0,
ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
double t;
int32_t hx;
uint32_t lx;
EXTRACT_WORDS(hx, lx, x);
if (hx < 0x3FF00000) { /* x < 1 */
return std::numeric_limits<double>::signaling_NaN();
} else if (hx >= 0x41B00000) { /* x > 2**28 */
if (hx >= 0x7FF00000) { /* x is inf of NaN */
return x + x;
} else {
return log(x) + ln2; /* acosh(huge)=log(2x) */
}
} else if (((hx - 0x3FF00000) | lx) == 0) {
return 0.0; /* acosh(1) = 0 */
} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
t = x * x;
return log(2.0 * x - one / (x + sqrt(t - one)));
} else { /* 1<x<2 */
t = x - one;
return log1p(t + sqrt(2.0 * t + t * t));
}
}
/* asin(x)
* Method :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
* where
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
* and its remez error is bounded by
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*/
double asin(double x) {
static const double
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
huge = 1.000e+300,
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
/* coefficient for R(x^2) */
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
double t, w, p, q, c, r, s;
int32_t hx, ix;
t = 0;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7FFFFFFF;
if (ix >= 0x3FF00000) { /* |x|>= 1 */
uint32_t lx;
GET_LOW_WORD(lx, x);
if (((ix - 0x3FF00000) | lx) == 0) { /* asin(1)=+-pi/2 with inexact */
return x * pio2_hi + x * pio2_lo;
}
return std::numeric_limits<double>::signaling_NaN(); // asin(|x|>1) is NaN
} else if (ix < 0x3FE00000) { /* |x|<0.5 */
if (ix < 0x3E400000) { /* if |x| < 2**-27 */
if (huge + x > one) return x; /* return x with inexact if x!=0*/
} else {
t = x * x;
}
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
w = p / q;
return x + x * w;
}
/* 1> |x|>= 0.5 */
w = one - fabs(x);
t = w * 0.5;
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
s = sqrt(t);
if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
w = p / q;
t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
} else {
w = s;
SET_LOW_WORD(w, 0);
c = (t - w * w) / (s + w);
r = p / q;
p = 2.0 * s * r - (pio2_lo - 2.0 * c);
q = pio4_hi - 2.0 * w;
t = pio4_hi - (p - q);
}
if (hx > 0)
return t;
else
return -t;
}
/* asinh(x)
* Method :
* Based on
* asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
* we have
* asinh(x) := x if 1+x*x=1,
* := sign(x)*(log(x)+ln2)) for large |x|, else
* := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
* := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
*/
double asinh(double x) {
static const double
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
huge = 1.00000000000000000000e+300;
double t, w;
int32_t hx, ix;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7FFFFFFF;
if (ix >= 0x7FF00000) return x + x; /* x is inf or NaN */
if (ix < 0x3E300000) { /* |x|<2**-28 */
if (huge + x > one) return x; /* return x inexact except 0 */
}
if (ix > 0x41B00000) { /* |x| > 2**28 */
w = log(fabs(x)) + ln2;
} else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
t = fabs(x);
w = log(2.0 * t + one / (sqrt(x * x + one) + t));
} else { /* 2.0 > |x| > 2**-28 */
t = x * x;
w = log1p(fabs(x) + t / (one + sqrt(one + t)));
}
if (hx > 0) {
return w;
} else {
return -w;
}
}
/* atan(x)
* Method
* 1. Reduce x to positive by atan(x) = -atan(-x).
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
* is further reduced to one of the following intervals and the
* arctangent of t is evaluated by the corresponding formula:
*
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
double atan(double x) {
static const double atanhi[] = {
4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
};
static const double atanlo[] = {
2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
};
static const double aT[] = {
3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
-1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
-1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
-7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
-5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
-3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
};
static const double one = 1.0, huge = 1.0e300;
double w, s1, s2, z;
int32_t ix, hx, id;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7FFFFFFF;
if (ix >= 0x44100000) { /* if |x| >= 2^66 */
uint32_t low;
GET_LOW_WORD(low, x);
if (ix > 0x7FF00000 || (ix == 0x7FF00000 && (low != 0)))
return x + x; /* NaN */
if (hx > 0)
return atanhi[3] + *const_cast<volatile double*>(&atanlo[3]);
else
return -atanhi[3] - *const_cast<volatile double*>(&atanlo[3]);
}
if (ix < 0x3FDC0000) { /* |x| < 0.4375 */
if (ix < 0x3E400000) { /* |x| < 2^-27 */
if (huge + x > one) return x; /* raise inexact */
}
id = -1;
} else {
x = fabs(x);
if (ix < 0x3FF30000) { /* |x| < 1.1875 */
if (ix < 0x3FE60000) { /* 7/16 <=|x|<11/16 */
id = 0;
x = (2.0 * x - one) / (2.0 + x);
} else { /* 11/16<=|x|< 19/16 */
id = 1;
x = (x - one) / (x + one);
}
} else {
if (ix < 0x40038000) { /* |x| < 2.4375 */
id = 2;
x = (x - 1.5) / (one + 1.5 * x);
} else { /* 2.4375 <= |x| < 2^66 */
id = 3;
x = -1.0 / x;
}
}
}
/* end of argument reduction */
z = x * x;
w = z * z;
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
s1 = z * (aT[0] +
w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
if (id < 0) {
return x - x * (s1 + s2);
} else {
z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
return (hx < 0) ? -z : z;
}
}
/* atan2(y,x)
* Method :
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
* 2. Reduce x to positive by (if x and y are unexceptional):
* ARG (x+iy) = arctan(y/x) ... if x > 0,
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
*
* Special cases:
*
* ATAN2((anything), NaN ) is NaN;
* ATAN2(NAN , (anything) ) is NaN;
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
* ATAN2(+-INF,+INF ) is +-pi/4 ;
* ATAN2(+-INF,-INF ) is +-3pi/4;
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
double atan2(double y, double x) {
static volatile double tiny = 1.0e-300;
static const double
zero = 0.0,
pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */
static volatile double pi_lo =
1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
double z;
int32_t k, m, hx, hy, ix, iy;
uint32_t lx, ly;
EXTRACT_WORDS(hx, lx, x);
ix = hx & 0x7FFFFFFF;
EXTRACT_WORDS(hy, ly, y);
iy = hy & 0x7FFFFFFF;
if (((ix | ((lx | NegateWithWraparound<int32_t>(lx)) >> 31)) > 0x7FF00000) ||
((iy | ((ly | NegateWithWraparound<int32_t>(ly)) >> 31)) > 0x7FF00000)) {
return x + y; /* x or y is NaN */
}
if ((SubWithWraparound(hx, 0x3FF00000) | lx) == 0) {
return atan(y); /* x=1.0 */
}
m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */
/* when y = 0 */
if ((iy | ly) == 0) {
switch (m) {
case 0:
case 1:
return y; /* atan(+-0,+anything)=+-0 */
case 2:
return pi + tiny; /* atan(+0,-anything) = pi */
case 3:
return -pi - tiny; /* atan(-0,-anything) =-pi */
}
}
/* when x = 0 */
if ((ix | lx) == 0) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
/* when x is INF */
if (ix == 0x7FF00000) {
if (iy == 0x7FF00000) {
switch (m) {
case 0:
return pi_o_4 + tiny; /* atan(+INF,+INF) */
case 1:
return -pi_o_4 - tiny; /* atan(-INF,+INF) */
case 2:
return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/
case 3:
return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/
}
} else {
switch (m) {
case 0:
return zero; /* atan(+...,+INF) */
case 1:
return -zero; /* atan(-...,+INF) */
case 2:
return pi + tiny; /* atan(+...,-INF) */
case 3:
return -pi - tiny; /* atan(-...,-INF) */
}
}
}
/* when y is INF */
if (iy == 0x7FF00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
/* compute y/x */
k = (iy - ix) >> 20;
if (k > 60) { /* |y/x| > 2**60 */
z = pi_o_2 + 0.5 * pi_lo;
m &= 1;
} else if (hx < 0 && k < -60) {
z = 0.0; /* 0 > |y|/x > -2**-60 */
} else {
z = atan(fabs(y / x)); /* safe to do y/x */
}
switch (m) {
case 0:
return z; /* atan(+,+) */
case 1:
return -z; /* atan(-,+) */
case 2:
return pi - (z - pi_lo); /* atan(+,-) */
default: /* case 3 */
return (z - pi_lo) - pi; /* atan(-,-) */
}
}
/* cos(x)
* Return cosine function of x.
*
* kernel function:
* __kernel_sin ... sine function on [-pi/4,pi/4]
* __kernel_cos ... cosine function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
double cos(double x) {
double y[2], z = 0.0;
int32_t n, ix;
/* High word of x. */
GET_HIGH_WORD(ix, x);
/* |x| ~< pi/4 */
ix &= 0x7FFFFFFF;
if (ix <= 0x3FE921FB) {
return __kernel_cos(x, z);
} else if (ix >= 0x7FF00000) {
/* cos(Inf or NaN) is NaN */
return x - x;
} else {
/* argument reduction needed */
n = __ieee754_rem_pio2(x, y);
switch (n & 3) {
case 0:
return __kernel_cos(y[0], y[1]);
case 1:
return -__kernel_sin(y[0], y[1], 1);
case 2:
return -__kernel_cos(y[0], y[1]);
default:
return __kernel_sin(y[0], y[1], 1);
}
}
}
/* exp(x)
* Returns the exponential of x.
*
* Method
* 1. Argument reduction:
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2.
*
* Here r will be represented as r = hi-lo for better
* accuracy.
*
* 2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Remes algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
* (where z=r*r, and the values of P1 to P5 are listed below)
* and
* | 5 | -59
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + -------
* R - r
* r*R1(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - R1(r)
* where
* 2 4 10
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
* exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 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 exp(x) overflow
* if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
double exp(double x) {
static const double
one = 1.0,
halF[2] = {0.5, -0.5},
o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
u_threshold = -7.45133219101941108420e+02, /* 0xC0874910, 0xD52D3051 */
ln2HI[2] = {6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */
-6.93147180369123816490e-01}, /* 0xBFE62E42, 0xFEE00000 */
ln2LO[2] = {1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */
-1.90821492927058770002e-10}, /* 0xBDEA39EF, 0x35793C76 */
invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
E = 2.718281828459045; /* 0x4005BF0A, 0x8B145769 */
static volatile double
huge = 1.0e+300,
twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
two1023 = 8.988465674311579539e307; /* 0x1p1023 */
double y, hi = 0.0, lo = 0.0, c, t, twopk;
int32_t k = 0, xsb;
uint32_t hx;
GET_HIGH_WORD(hx, x);
xsb = (hx >> 31) & 1; /* sign bit of x */
hx &= 0x7FFFFFFF; /* high word of |x| */
/* filter out non-finite argument */
if (hx >= 0x40862E42) { /* if |x|>=709.78... */
if (hx >= 0x7FF00000) {
uint32_t lx;
GET_LOW_WORD(lx, x);
if (((hx & 0xFFFFF) | lx) != 0)
return x + x; /* NaN */
else
return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
}
if (x > o_threshold) return huge * huge; /* overflow */
if (x < u_threshold) return twom1000 * twom1000; /* underflow */
}
/* argument reduction */
if (hx > 0x3FD62E42) { /* if |x| > 0.5 ln2 */
if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
/* TODO(rtoy): We special case exp(1) here to return the correct
* value of E, as the computation below would get the last bit
* wrong. We should probably fix the algorithm instead.
*/
if (x == 1.0) return E;
hi = x - ln2HI[xsb];
lo = ln2LO[xsb];
k = 1 - xsb - xsb;
} else {
k = static_cast<int>(invln2 * x + halF[xsb]);
t = k;
hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
lo = t * ln2LO[0];
}
x = hi - lo;
} else if (hx < 0x3E300000) { /* when |x|<2**-28 */
if (huge + x > one) return one + x; /* trigger inexact */
} else {
k = 0;
}
/* x is now in primary range */
t = x * x;
if (k >= -1021) {
INSERT_WORDS(
twopk,
0x3FF00000 + static_cast<int32_t>(static_cast<uint32_t>(k) << 20), 0);
} else {
INSERT_WORDS(twopk, 0x3FF00000 + (static_cast<uint32_t>(k + 1000) << 20),
0);
}
c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
if (k == 0) {
return one - ((x * c) / (c - 2.0) - x);
} else {
y = one - ((lo - (x * c) / (2.0 - c)) - hi);
}
if (k >= -1021) {
if (k == 1024) return y * 2.0 * two1023;
return y * twopk;
} else {
return y * twopk * twom1000;
}
}
/*
* Method :
* 1.Reduced x to positive by atanh(-x) = -atanh(x)
* 2.For x>=0.5
* 1 2x x
* atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
* 2 1 - x 1 - x
*
* For x<0.5
* atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
*
* Special cases:
* atanh(x) is NaN if |x| > 1 with signal;
* atanh(NaN) is that NaN with no signal;
* atanh(+-1) is +-INF with signal.
*
*/
double atanh(double x) {
static const double one = 1.0, huge = 1e300;
static const double zero = 0.0;
double t;
int32_t hx, ix;
uint32_t lx;
EXTRACT_WORDS(hx, lx, x);
ix = hx & 0x7FFFFFFF;
if ((ix | ((lx | NegateWithWraparound<int32_t>(lx)) >> 31)) > 0x3FF00000) {
/* |x|>1 */
return std::numeric_limits<double>::signaling_NaN();
}
if (ix == 0x3FF00000) {
return x > 0 ? std::numeric_limits<double>::infinity()
: -std::numeric_limits<double>::infinity();
}
if (ix < 0x3E300000 && (huge + x) > zero) return x; /* x<2**-28 */
SET_HIGH_WORD(x, ix);
if (ix < 0x3FE00000) { /* x < 0.5 */
t = x + x;
t = 0.5 * log1p(t + t * x / (one - x));
} else {
t = 0.5 * log1p((x + x) / (one - x));
}
if (hx >= 0)
return t;
else
return -t;
}
/* log(x)
* Return the logrithm of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*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
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+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:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(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:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
double log(double x) {
static const double /* -- */
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
static const double zero = 0.0;
double hfsq, f, s, z, R, w, t1, t2, dk;
int32_t k, hx, i, j;
uint32_t lx;
EXTRACT_WORDS(hx, lx, x);
k = 0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx & 0x7FFFFFFF) | lx) == 0) {
return -std::numeric_limits<double>::infinity(); /* log(+-0)=-inf */
}
if (hx < 0) {
return std::numeric_limits<double>::signaling_NaN(); /* log(-#) = NaN */
}
k -= 54;
x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx, x);
}
if (hx >= 0x7FF00000) return x + x;
k += (hx >> 20) - 1023;
hx &= 0x000FFFFF;
i = (hx + 0x95F64) & 0x100000;
SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */
k += (i >> 20);
f = x - 1.0;
if ((0x000FFFFF & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
if (f == zero) {
if (k == 0) {
return zero;
} else {
dk = static_cast<double>(k);
return dk * ln2_hi + dk * ln2_lo;
}
}
R = f * f * (0.5 - 0.33333333333333333 * f);
if (k == 0) {
return f - R;
} else {
dk = static_cast<double>(k);
return dk * ln2_hi - ((R - dk * ln2_lo) - f);
}
}
s = f / (2.0 + f);
dk = static_cast<double>(k);
z = s * s;
i = hx - 0x6147A;
w = z * z;
j = 0x6B851 - hx;
t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
i |= j;
R = t2 + t1;
if (i > 0) {
hfsq = 0.5 * f * f;
if (k == 0)
return f - (hfsq - s * (hfsq + R));
else
return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
} else {
if (k == 0)
return f - s * (f - R);
else
return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
}
}
/* double log1p(double x)
*
* 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:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*
* 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.
*/
double log1p(double x) {
static const double /* -- */
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
static const double zero = 0.0;
double hfsq, f, c, s, z, R, u;
int32_t k, hx, hu, ax;
GET_HIGH_WORD(hx, x);
ax = hx & 0x7FFFFFFF;
k = 1;
if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
if (ax >= 0x3FF00000) { /* x <= -1.0 */
if (x == -1.0)
return -std::numeric_limits<double>::infinity(); /* log1p(-1)=+inf */
else
return std::numeric_limits<double>::signaling_NaN(); // log1p(x<-1)=NaN
}
if (ax < 0x3E200000) { /* |x| < 2**-29 */
if (two54 + x > zero /* raise inexact */
&& ax < 0x3C900000) /* |x| < 2**-54 */
return x;
else
return x - x * x * 0.5;
}
if (hx > 0 || hx <= static_cast<int32_t>(0xBFD2BEC4)) {
k = 0;
f = x;
hu = 1;
} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
}
if (hx >= 0x7FF00000) return x + x;
if (k != 0) {
if (hx < 0x43400000) {
u = 1.0 + x;
GET_HIGH_WORD(hu, u);
k = (hu >> 20) - 1023;
c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
c /= u;
} else {
u = x;
GET_HIGH_WORD(hu, u);
k = (hu >> 20) - 1023;
c = 0;
}
hu &= 0x000FFFFF;
/*
* The approximation to sqrt(2) used in thresholds is not
* critical. However, the ones used above must give less
* strict bounds than the one here so that the k==0 case is
* never reached from here, since here we have committed to
* using the correction term but don't use it if k==0.
*/
if (hu < 0x6A09E) { /* u ~< sqrt(2) */
SET_HIGH_WORD(u, hu | 0x3FF00000); /* normalize u */
} else {
k += 1;
SET_HIGH_WORD(u, hu | 0x3FE00000); /* normalize u/2 */
hu = (0x00100000 - hu) >> 2;
}
f = u - 1.0;
}
hfsq = 0.5 * f * f;
if (hu == 0) { /* |f| < 2**-20 */
if (f == zero) {
if (k == 0) {
return zero;
} else {
c += k * ln2_lo;
return k * ln2_hi + c;
}
}
R = hfsq * (1.0 - 0.66666666666666666 * f);
if (k == 0)
return f - R;
else
return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
}
s = f / (2.0 + f);
z = s * s;
R = z * (Lp1 +
z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
if (k == 0)
return f - (hfsq - s * (hfsq + R));
else
return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
}
/*
* k_log1p(f):
* Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
*
* The following describes the overall strategy for computing
* logarithms in base e. The argument reduction and adding the final
* term of the polynomial are done by the caller for increased accuracy
* when different bases are used.
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*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
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+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:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(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:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
static const double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
/*
* We always inline k_log1p(), since doing so produces a
* substantial performance improvement (~40% on amd64).
*/
static inline double k_log1p(double f) {
double hfsq, s, z, R, w, t1, t2;
s = f / (2.0 + f);
z = s * s;
w = z * z;
t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
R = t2 + t1;
hfsq = 0.5 * f * f;
return s * (hfsq + R);
}
/*
* Return the base 2 logarithm of x. See e_log.c and k_log.h for most
* comments.
*
* This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
* then does the combining and scaling steps
* log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
* in not-quite-routine extra precision.
*/
double log2(double x) {
static const double
two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
ivln2hi = 1.44269504072144627571e+00, /* 0x3FF71547, 0x65200000 */
ivln2lo = 1.67517131648865118353e-10; /* 0x3DE705FC, 0x2EEFA200 */
double f, hfsq, hi, lo, r, val_hi, val_lo, w, y;
int32_t i, k, hx;
uint32_t lx;
EXTRACT_WORDS(hx, lx, x);
k = 0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx & 0x7FFFFFFF) | lx) == 0) {
return -std::numeric_limits<double>::infinity(); /* log(+-0)=-inf */
}
if (hx < 0) {
return std::numeric_limits<double>::signaling_NaN(); /* log(-#) = NaN */
}
k -= 54;
x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx, x);
}
if (hx >= 0x7FF00000) return x + x;
if (hx == 0x3FF00000 && lx == 0) return 0.0; /* log(1) = +0 */
k += (hx >> 20) - 1023;
hx &= 0x000FFFFF;
i = (hx + 0x95F64) & 0x100000;
SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */
k += (i >> 20);
y = static_cast<double>(k);
f = x - 1.0;
hfsq = 0.5 * f * f;
r = k_log1p(f);
/*
* f-hfsq must (for args near 1) be evaluated in extra precision
* to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
* This is fairly efficient since f-hfsq only depends on f, so can
* be evaluated in parallel with R. Not combining hfsq with R also
* keeps R small (though not as small as a true `lo' term would be),
* so that extra precision is not needed for terms involving R.
*
* Compiler bugs involving extra precision used to break Dekker's
* theorem for spitting f-hfsq as hi+lo, unless double_t was used
* or the multi-precision calculations were avoided when double_t
* has extra precision. These problems are now automatically
* avoided as a side effect of the optimization of combining the
* Dekker splitting step with the clear-low-bits step.
*
* y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
* precision to avoid a very large cancellation when x is very near
* these values. Unlike the above cancellations, this problem is
* specific to base 2. It is strange that adding +-1 is so much
* harder than adding +-ln2 or +-log10_2.
*
* This uses Dekker's theorem to normalize y+val_hi, so the
* compiler bugs are back in some configurations, sigh. And I
* don't want to used double_t to avoid them, since that gives a
* pessimization and the support for avoiding the pessimization
* is not yet available.
*
* The multi-precision calculations for the multiplications are
* routine.
*/
hi = f - hfsq;
SET_LOW_WORD(hi, 0);
lo = (f - hi) - hfsq + r;
val_hi = hi * ivln2hi;
val_lo = (lo + hi) * ivln2lo + lo * ivln2hi;
/* spadd(val_hi, val_lo, y), except for not using double_t: */
w = y + val_hi;
val_lo += (y - w) + val_hi;
val_hi = w;
return val_lo + val_hi;
}
/*
* Return the base 10 logarithm of x
*
* Method :
* Let log10_2hi = leading 40 bits of log10(2) and
* log10_2lo = log10(2) - log10_2hi,
* ivln10 = 1/log(10) rounded.
* Then
* n = ilogb(x),
* if(n<0) n = n+1;
* x = scalbn(x,-n);
* log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
*
* Note 1:
* To guarantee log10(10**n)=n, where 10**n is normal, the rounding
* mode must set to Round-to-Nearest.
* Note 2:
* [1/log(10)] rounded to 53 bits has error .198 ulps;
* log10 is monotonic at all binary break points.
*
* Special cases:
* log10(x) is NaN if x < 0;
* log10(+INF) is +INF; log10(0) is -INF;
* log10(NaN) is that NaN;
* log10(10**N) = N for N=0,1,...,22.
*/
double log10(double x) {
static const double
two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
ivln10 = 4.34294481903251816668e-01,
log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
double y;
int32_t i, k, hx;
uint32_t lx;
EXTRACT_WORDS(hx, lx, x);
k = 0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx & 0x7FFFFFFF) | lx) == 0) {
return -std::numeric_limits<double>::infinity(); /* log(+-0)=-inf */
}
if (hx < 0) {
return std::numeric_limits<double>::quiet_NaN(); /* log(-#) = NaN */
}
k -= 54;
x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx, x);
GET_LOW_WORD(lx, x);
}
if (hx >= 0x7FF00000) return x + x;
if (hx == 0x3FF00000 && lx == 0) return 0.0; /* log(1) = +0 */
k += (hx >> 20) - 1023;
i = (k & 0x80000000) >> 31;
hx = (hx & 0x000FFFFF) | ((0x3FF - i) << 20);
y = k + i;
SET_HIGH_WORD(x, hx);
SET_LOW_WORD(x, lx);
double z = y * log10_2lo + ivln10 * log(x);
return z + y * log10_2hi;
}
/* expm1(x)
* 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 Reme 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;
* z = r*r,
* 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
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
double expm1(double x) {
static const double
one = 1.0,
tiny = 1.0e-300,
o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
ln2_hi = 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */
ln2_lo = 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */
invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */
/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs =
x*x/2: */
Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
static volatile double huge = 1.0e+300;
double y, hi, lo, c, t, e, hxs, hfx, r1, twopk;
int32_t k, xsb;
uint32_t hx;
GET_HIGH_WORD(hx, x);
xsb = hx & 0x80000000; /* sign bit of 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) {
uint32_t low;
GET_LOW_WORD(low, x);
if (((hx & 0xFFFFF) | low) != 0)
return x + x; /* NaN */
else
return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
}
if (x > o_threshold) return huge * huge; /* overflow */
}
if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */
if (x + tiny < 0.0) /* raise inexact */
return tiny - one; /* 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);
t = k;
hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
lo = t * ln2_lo;
}
x = hi - lo;
c = (hi - x) - lo;
} else if (hx < 0x3C900000) { /* when |x|<2**-54, return x */
t = huge + x; /* return x with inexact flags when x!=0 */
return x - (t - (huge + x));
} else {
k = 0;
}
/* x is now in primary range */
hfx = 0.5 * x;
hxs = x * hfx;
r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
t = 3.0 - r1 * hfx;
e = hxs * ((r1 - t) / (6.0 - x * t));
if (k == 0) {
return x - (x * e - hxs); /* c is 0 */
} else {
INSERT_WORDS(
twopk,
0x3FF00000 + static_cast<int32_t>(static_cast<uint32_t>(k) << 20),
0); /* 2^k */
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 -2.0 * (e - (x + 0.5));
else
return one + 2.0 * (x - e);
}
if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
y = one - (e - x);
// TODO(mvstanton): is this replacement for the hex float
// sufficient?
// if (k == 1024) y = y*2.0*0x1p1023;
if (k == 1024)
y = y * 2.0 * 8.98846567431158e+307;
else
y = y * twopk;
return y - one;
}
t = one;
if (k < 20) {
SET_HIGH_WORD(t, 0x3FF00000 - (0x200000 >> k)); /* t=1-2^-k */
y = t - (e - x);
y = y * twopk;
} else {
SET_HIGH_WORD(t, ((0x3FF - k) << 20)); /* 2^-k */
y = x - (e + t);
y += one;
y = y * twopk;
}
}
return y;
}
double cbrt(double x) {
static const uint32_t
B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
static const double P0 = 1.87595182427177009643, /* 0x3FFE03E6, 0x0F61E692 */
P1 = -1.88497979543377169875, /* 0xBFFE28E0, 0x92F02420 */
P2 = 1.621429720105354466140, /* 0x3FF9F160, 0x4A49D6C2 */
P3 = -0.758397934778766047437, /* 0xBFE844CB, 0xBEE751D9 */
P4 = 0.145996192886612446982; /* 0x3FC2B000, 0xD4E4EDD7 */
int32_t hx;
double r, s, t = 0.0, w;
uint32_t sign;
uint32_t high, low;
EXTRACT_WORDS(hx, low, x);
sign = hx & 0x80000000; /* sign= sign(x) */
hx ^= sign;
if (hx >= 0x7FF00000) return (x + x); /* cbrt(NaN,INF) is itself */
/*
* Rough cbrt to 5 bits:
* cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
* where e is integral and >= 0, m is real and in [0, 1), and "/" and
* "%" are integer division and modulus with rounding towards minus
* infinity. The RHS is always >= the LHS and has a maximum relative
* error of about 1 in 16. Adding a bias of -0.03306235651 to the
* (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
* floating point representation, for finite positive normal values,
* ordinary integer division of the value in bits magically gives
* almost exactly the RHS of the above provided we first subtract the
* exponent bias (1023 for doubles) and later add it back. We do the
* subtraction virtually to keep e >= 0 so that ordinary integer
* division rounds towards minus infinity; this is also efficient.
*/
if (hx < 0x00100000) { /* zero or subnormal? */
if ((hx | low) == 0) return (x); /* cbrt(0) is itself */
SET_HIGH_WORD(t, 0x43500000); /* set t= 2**54 */
t *= x;
GET_HIGH_WORD(high, t);
INSERT_WORDS(t, sign | ((high & 0x7FFFFFFF) / 3 + B2), 0);
} else {
INSERT_WORDS(t, sign | (hx / 3 + B1), 0);
}
/*
* New cbrt to 23 bits:
* cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
* where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
* to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
* has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
* gives us bounds for r = t**3/x.
*
* Try to optimize for parallel evaluation as in k_tanf.c.
*/
r = (t * t) * (t / x);
t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));
/*
* Round t away from zero to 23 bits (sloppily except for ensuring that
* the result is larger in magnitude than cbrt(x) but not much more than
* 2 23-bit ulps larger). With rounding towards zero, the error bound
* would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
* in the rounded t, the infinite-precision error in the Newton
* approximation barely affects third digit in the final error
* 0.667; the error in the rounded t can be up to about 3 23-bit ulps
* before the final error is larger than 0.667 ulps.
*/
uint64_t bits = bit_cast<uint64_t>(t);
bits = (bits + 0x80000000) & 0xFFFFFFFFC0000000ULL;
t = bit_cast<double>(bits);
/* one step Newton iteration to 53 bits with error < 0.667 ulps */
s = t * t; /* t*t is exact */
r = x / s; /* error <= 0.5 ulps; |r| < |t| */
w = t + t; /* t+t is exact */
r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */
return (t);
}
/* sin(x)
* Return sine function of x.
*
* kernel function:
* __kernel_sin ... sine function on [-pi/4,pi/4]
* __kernel_cos ... cose function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
double sin(double x) {
double y[2], z = 0.0;
int32_t n, ix;
/* High word of x. */
GET_HIGH_WORD(ix, x);
/* |x| ~< pi/4 */
ix &= 0x7FFFFFFF;
if (ix <= 0x3FE921FB) {
return __kernel_sin(x, z, 0);
} else if (ix >= 0x7FF00000) {
/* sin(Inf or NaN) is NaN */
return x - x;
} else {
/* argument reduction needed */
n = __ieee754_rem_pio2(x, y);
switch (n & 3) {
case 0:
return __kernel_sin(y[0], y[1], 1);
case 1:
return __kernel_cos(y[0], y[1]);
case 2:
return -__kernel_sin(y[0], y[1], 1);
default:
return -__kernel_cos(y[0], y[1]);
}
}
}
/* tan(x)
* Return tangent function of x.
*
* kernel function:
* __kernel_tan ... tangent function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
double tan(double x) {
double y[2], z = 0.0;
int32_t n, ix;
/* High word of x. */
GET_HIGH_WORD(ix, x);
/* |x| ~< pi/4 */
ix &= 0x7FFFFFFF;
if (ix <= 0x3FE921FB) {
return __kernel_tan(x, z, 1);
} else if (ix >= 0x7FF00000) {
/* tan(Inf or NaN) is NaN */
return x - x; /* NaN */
} else {
/* argument reduction needed */
n = __ieee754_rem_pio2(x, y);
/* 1 -> n even, -1 -> n odd */
return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1));
}
}
/*
* 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.
*/
double cosh(double x) {
static const double KCOSH_OVERFLOW = 710.4758600739439;
static const double one = 1.0, half = 0.5;
static volatile double huge = 1.0e+300;
int32_t ix;
/* High word of |x|. */
GET_HIGH_WORD(ix, x);
ix &= 0x7FFFFFFF;
// |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
if (ix < 0x3FD62E43) {
double t = expm1(fabs(x));
double w = one + t;
// For |x| < 2^-55, cosh(x) = 1
if (ix < 0x3C800000) return w;
return one + (t * t) / (w + w);
}
// |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
if (ix < 0x40360000) {
double t = exp(fabs(x));
return half * t + half / t;
}
// |x| in [22, log(maxdouble)], return half*exp(|x|)
if (ix < 0x40862E42) return half * exp(fabs(x));
// |x| in [log(maxdouble), overflowthreshold]
if (fabs(x) <= KCOSH_OVERFLOW) {
double w = exp(half * fabs(x));
double t = half * w;
return t * w;
}
/* x is INF or NaN */
if (ix >= 0x7FF00000) return x * x;
// |x| > overflowthreshold.
return huge * huge;
}
/*
* ES2019 Draft 2019-01-02 12.6.4
* Math.pow & Exponentiation Operator
*
* Return X raised to the Yth power
*
* Method:
* Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53-24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
* arithmetic, where |y'|<=0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. (anything) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. +-1 ** +-INF is NAN
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
* 15. +INF ** (+anything except 0,NAN) is +INF
* 16. +INF ** (-anything except 0,NAN) is +0
* 17. -INF ** (anything) = -0 ** (-anything)
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular,
* pow(integer, integer) always returns the correct integer provided it is
* representable.
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
double pow(double x, double y) {
static const double
bp[] = {1.0, 1.5},
dp_h[] = {0.0, 5.84962487220764160156e-01}, // 0x3FE2B803, 0x40000000
dp_l[] = {0.0, 1.35003920212974897128e-08}, // 0x3E4CFDEB, 0x43CFD006
zero = 0.0, one = 1.0, two = 2.0,
two53 = 9007199254740992.0, // 0x43400000, 0x00000000
huge = 1.0e300, tiny = 1.0e-300,
// poly coefs for (3/2)*(log(x)-2s-2/3*s**3
L1 = 5.99999999999994648725e-01, // 0x3FE33333, 0x33333303
L2 = 4.28571428578550184252e-01, // 0x3FDB6DB6, 0xDB6FABFF
L3 = 3.33333329818377432918e-01, // 0x3FD55555, 0x518F264D
L4 = 2.72728123808534006489e-01, // 0x3FD17460, 0xA91D4101
L5 = 2.30660745775561754067e-01, // 0x3FCD864A, 0x93C9DB65
L6 = 2.06975017800338417784e-01, // 0x3FCA7E28, 0x4A454EEF
P1 = 1.66666666666666019037e-01, // 0x3FC55555, 0x5555553E
P2 = -2.77777777770155933842e-03, // 0xBF66C16C, 0x16BEBD93
P3 = 6.61375632143793436117e-05, // 0x3F11566A, 0xAF25DE2C
P4 = -1.65339022054652515390e-06, // 0xBEBBBD41, 0xC5D26BF1
P5 = 4.13813679705723846039e-08, // 0x3E663769, 0x72BEA4D0
lg2 = 6.93147180559945286227e-01, // 0x3FE62E42, 0xFEFA39EF
lg2_h = 6.93147182464599609375e-01, // 0x3FE62E43, 0x00000000
lg2_l = -1.90465429995776804525e-09, // 0xBE205C61, 0x0CA86C39
ovt = 8.0085662595372944372e-0017, // -(1024-log2(ovfl+.5ulp))
cp = 9.61796693925975554329e-01, // 0x3FEEC709, 0xDC3A03FD =2/(3ln2)
cp_h = 9.61796700954437255859e-01, // 0x3FEEC709, 0xE0000000 =(float)cp
cp_l = -7.02846165095275826516e-09, // 0xBE3E2FE0, 0x145B01F5 =tail cp_h
ivln2 = 1.44269504088896338700e+00, // 0x3FF71547, 0x652B82FE =1/ln2
ivln2_h =
1.44269502162933349609e+00, // 0x3FF71547, 0x60000000 =24b 1/ln2
ivln2_l =
1.92596299112661746887e-08; // 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail
double z, ax, z_h, z_l, p_h, p_l;
double y1, t1, t2, r, s, t, u, v, w;
int i, j, k, yisint, n;
int hx, hy, ix, iy;
unsigned lx, ly;
EXTRACT_WORDS(hx, lx, x);
EXTRACT_WORDS(hy, ly, y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* y==zero: x**0 = 1 */
if ((iy | ly) == 0) return one;
/* +-NaN return x+y */
if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 ||
((iy == 0x7ff00000) && (ly != 0))) {
return x + y;
}
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if (hx < 0) {
if (iy >= 0x43400000) {
yisint = 2; /* even integer y */
} else if (iy >= 0x3ff00000) {
k = (iy >> 20) - 0x3ff; /* exponent */
if (k > 20) {
j = ly >> (52 - k);
if ((j << (52 - k)) == static_cast<int>(ly)) yisint = 2 - (j & 1);
} else if (ly == 0) {
j = iy >> (20 - k);
if ((j << (20 - k)) == iy) yisint = 2 - (j & 1);
}
}
}
/* special value of y */
if (ly == 0) {
if (iy == 0x7ff00000) { /* y is +-inf */
if (((ix - 0x3ff00000) | lx) == 0) {
return y - y; /* inf**+-1 is NaN */
} else if (ix >= 0x3ff00000) { /* (|x|>1)**+-inf = inf,0 */
return (hy >= 0) ? y : zero;
} else { /* (|x|<1)**-,+inf = inf,0 */
return (hy < 0) ? -y : zero;
}
}
if (iy == 0x3ff00000) { /* y is +-1 */
if (hy < 0) {
return base::Divide(one, x);
} else {
return x;
}
}
if (hy == 0x40000000) return x * x; /* y is 2 */
if (hy == 0x3fe00000) { /* y is 0.5 */
if (hx >= 0) { /* x >= +0 */
return sqrt(x);
}
}
}
ax = fabs(x);
/* special value of x */
if (lx == 0) {
if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
z = ax; /*x is +-0,+-inf,+-1*/
if (hy < 0) z = base::Divide(one, z); /* z = (1/|x|) */
if (hx < 0) {
if (((ix - 0x3ff00000) | yisint) == 0) {
/* (-1)**non-int is NaN */
z = std::numeric_limits<double>::signaling_NaN();
} else if (yisint == 1) {
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
}
return z;
}
}
n = (hx >> 31) + 1;
/* (x<0)**(non-int) is NaN */
if ((n | yisint) == 0) {
return std::numeric_limits<double>::signaling_NaN();
}
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
if ((n | (yisint - 1)) == 0) s = -one; /* (-ve)**(odd int) */
/* |y| is huge */
if (iy > 0x41e00000) { /* if |y| > 2**31 */
if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
if (ix <= 0x3fefffff) return (hy < 0) ? huge * huge : tiny * tiny;
if (ix >= 0x3ff00000) return (hy > 0) ? huge * huge : tiny * tiny;
}
/* over/underflow if x is not close to one */
if (ix < 0x3fefffff) return (hy < 0) ? s * huge * huge : s * tiny * tiny;
if (ix > 0x3ff00000) return (hy > 0) ? s * huge * huge : s * tiny * tiny;
/* now |1-x| is tiny <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
t = ax - one; /* t has 20 trailing zeros */
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
v = t * ivln2_l - w * ivln2;
t1 = u + v;
SET_LOW_WORD(t1, 0);
t2 = v - (t1 - u);
} else {
double ss, s2, s_h, s_l, t_h, t_l;
n = 0;
/* take care subnormal number */
if (ix < 0x00100000) {
ax *= two53;
n -= 53;
GET_HIGH_WORD(ix, ax);
}
n += ((ix) >> 20) - 0x3ff;
j = ix & 0x000fffff;
/* determine interval */
ix = j | 0x3ff00000; /* normalize ix */
if (j <= 0x3988E) {
k = 0; /* |x|<sqrt(3/2) */
} else if (j < 0xBB67A) {
k = 1; /* |x|<sqrt(3) */
} else {
k = 0;
n += 1;
ix -= 0x00100000;
}
SET_HIGH_WORD(ax, ix);
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = base::Divide(one, ax + bp[k]);
ss = u * v;
s_h = ss;
SET_LOW_WORD(s_h, 0);
/* t_h=ax+bp[k] High */
t_h = zero;
SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
t_l = ax - (t_h - bp[k]);
s_l = v * ((u - s_h * t_h) - s_h * t_l);
/* compute log(ax) */
s2 = ss * ss;
r = s2 * s2 *
(L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
r += s_l * (s_h + ss);
s2 = s_h * s_h;
t_h = 3.0 + s2 + r;
SET_LOW_WORD(t_h, 0);
t_l = r - ((t_h - 3.0) - s2);
/* u+v = ss*(1+...) */
u = s_h * t_h;
v = s_l * t_h + t_l * ss;
/* 2/(3log2)*(ss+...) */
p_h = u + v;
SET_LOW_WORD(p_h, 0);
p_l = v - (p_h - u);
z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = cp_l * p_h + p_l * cp + dp_l[k];
/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = static_cast<double>(n);
t1 = (((z_h + z_l) + dp_h[k]) + t);
SET_LOW_WORD(t1, 0);
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
}
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
y1 = y;
SET_LOW_WORD(y1, 0);
p_l = (y - y1) * t1 + y * t2;
p_h = y1 * t1;
z = p_l + p_h;
EXTRACT_WORDS(j, i, z);
if (j >= 0x40900000) { /* z >= 1024 */
if (((j - 0x40900000) | i) != 0) { /* if z > 1024 */
return s * huge * huge; /* overflow */
} else {
if (p_l + ovt > z - p_h) return s * huge * huge; /* overflow */
}
} else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */
if (((j - 0xc090cc00) | i) != 0) { /* z < -1075 */
return s * tiny * tiny; /* underflow */
} else {
if (p_l <= z - p_h) return s * tiny * tiny; /* underflow */
}
}
/*
* compute 2**(p_h+p_l)
*/
i = j & 0x7fffffff;
k = (i >> 20) - 0x3ff;
n = 0;
if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
n = j + (0x00100000 >> (k + 1));
k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
t = zero;
SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
if (j < 0) n = -n;
p_h -= t;
}
t = p_l + p_h;
SET_LOW_WORD(t, 0);
u = t * lg2_h;
v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
z = u + v;
w = v - (z - u);
t = z * z;
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
r = base::Divide(z * t1, (t1 - two) - (w + z * w));
z = one - (r - z);
GET_HIGH_WORD(j, z);
j += static_cast<int>(static_cast<uint32_t>(n) << 20);
if ((j >> 20) <= 0) {
z = scalbn(z, n); /* subnormal output */
} else {
int tmp;
GET_HIGH_WORD(tmp, z);
SET_HIGH_WORD(z, tmp + static_cast<int>(static_cast<uint32_t>(n) << 20));
}
return s * z;
}
/*
* 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.
*/
double sinh(double x) {
static const double KSINH_OVERFLOW = 710.4758600739439,
TWO_M28 =
3.725290298461914e-9, // 2^-28, empty lower half
LOG_MAXD = 709.7822265625; // 0x40862E42 00000000, empty lower half
static const double shuge = 1.0e307;
double h = (x < 0) ? -0.5 : 0.5;
// |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
double ax = fabs(x);
if (ax < 22) {
// For |x| < 2^-28, sinh(x) = x
if (ax < TWO_M28) return x;
double t = expm1(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 * exp(ax);
// |x| in [log(maxdouble), overflowthreshold]
// overflowthreshold = 710.4758600739426
if (ax <= KSINH_OVERFLOW) {
double w = exp(0.5 * ax);
double t = h * w;
return t * w;
}
// |x| > overflowthreshold or is NaN.
// Return Infinity of the appropriate sign or NaN.
return x * shuge;
}
/* Tanh(x)
* Return the Hyperbolic Tangent of x
*
* Method :
* x -x
* e - e
* 0. tanh(x) is defined to be -----------
* x -x
* e + e
* 1. reduce x to non-negative by tanh(-x) = -tanh(x).
* 2. 0 <= x < 2**-28 : tanh(x) := x with inexact if x != 0
* -t
* 2**-28 <= x < 1 : tanh(x) := -----; t = expm1(-2x)
* t + 2
* 2
* 1 <= x < 22 : tanh(x) := 1 - -----; t = expm1(2x)
* t + 2
* 22 <= x <= INF : tanh(x) := 1.
*
* Special cases:
* tanh(NaN) is NaN;
* only tanh(0)=0 is exact for finite argument.
*/
double tanh(double x) {
static const volatile double tiny = 1.0e-300;
static const double one = 1.0, two = 2.0, huge = 1.0e300;
double t, z;
int32_t jx, ix;
GET_HIGH_WORD(jx, x);
ix = jx & 0x7FFFFFFF;
/* x is INF or NaN */
if (ix >= 0x7FF00000) {
if (jx >= 0)
return one / x + one; /* tanh(+-inf)=+-1 */
else
return one / x - one; /* tanh(NaN) = NaN */
}
/* |x| < 22 */
if (ix < 0x40360000) { /* |x|<22 */
if (ix < 0x3E300000) { /* |x|<2**-28 */
if (huge + x > one) return x; /* tanh(tiny) = tiny with inexact */
}
if (ix >= 0x3FF00000) { /* |x|>=1 */
t = expm1(two * fabs(x));
z = one - two / (t + two);
} else {
t = expm1(-two * fabs(x));
z = -t / (t + two);
}
/* |x| >= 22, return +-1 */
} else {
z = one - tiny; /* raise inexact flag */
}
return (jx >= 0) ? z : -z;
}
#undef EXTRACT_WORDS
#undef GET_HIGH_WORD
#undef GET_LOW_WORD
#undef INSERT_WORDS
#undef SET_HIGH_WORD
#undef SET_LOW_WORD
} // namespace ieee754
} // namespace base
} // namespace v8