../deps/v8/src/base/ieee754.cc

Bug Summary

File:	out/../deps/v8/src/base/ieee754.cc
Warning:	line 492, column 52 The right operand of '*' is a garbage value
Annotated Source Code

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1// The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
2//
3// ====================================================
4// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5//
6// Developed at SunSoft, a Sun Microsystems, Inc. business.
7// Permission to use, copy, modify, and distribute this
8// software is freely granted, provided that this notice
9// is preserved.
10// ====================================================
11//
12// The original source code covered by the above license above has been
13// modified significantly by Google Inc.
14// Copyright 2016 the V8 project authors. All rights reserved.

16#include "src/base/ieee754.h"

18#include <cmath>
19#include <limits>

21#include "src/base/build_config.h"
22#include "src/base/macros.h"
23#include "src/base/overflowing-math.h"

25namespace v8 {
26namespace base {
27namespace ieee754 {

29namespace {

31/* Disable "potential divide by 0" warning in Visual Studio compiler. */

33#if V8_CC_MSVC

35#pragma warning(disable : 4723)

37#endif

39/*
* 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.
*/

52/* Get two 32 bit ints from a double.  */

54#define EXTRACT_WORDS(ix0, ix1, d)         \
do {                                     \
  uint64_t bits = bit_cast<uint64_t>(d); \
  (ix0) = bits >> 32;                    \
  (ix1) = bits & 0xFFFFFFFFu;            \
} while (false)

61/* Get the more significant 32 bit int from a double.  */

63#define GET_HIGH_WORD(i, d)                \
do {                                     \
  uint64_t bits = bit_cast<uint64_t>(d); \
  (i) = bits >> 32;                      \
} while (false)

69/* Get the less significant 32 bit int from a double.  */

71#define GET_LOW_WORD(i, d)                 \
do {                                     \
  uint64_t bits = bit_cast<uint64_t>(d); \
  (i) = bits & 0xFFFFFFFFu;              \
} while (false)

77/* Set a double from two 32 bit ints.  */

79#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)

87/* Set the more significant 32 bits of a double from an int.  */

89#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)

97/* Set the less significant 32 bits of a double from an int.  */

99#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)

107int32_t __ieee754_rem_pio2(double x, double* y) V8_WARN_UNUSED_RESULT__attribute__((warn_unused_result));
108double __kernel_cos(double x, double y) V8_WARN_UNUSED_RESULT__attribute__((warn_unused_result));
109int __kernel_rem_pio2(double* x, double* y, int e0, int nx, int prec,
                    const int32_t* ipio2) V8_WARN_UNUSED_RESULT__attribute__((warn_unused_result));
111double __kernel_sin(double x, double y, int iy) V8_WARN_UNUSED_RESULT__attribute__((warn_unused_result));

113/* __ieee754_rem_pio2(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __kernel_rem_pio2()
*/
118int32_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;
270}

272/* __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.
*/
305V8_INLINEinline __attribute__((always_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));
}
336}

338/* __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.
*
*/
443int __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;
1
Assuming 'jv' is >= 0→
2
←
Taking false branch→
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++) {
3
←
Assuming 'i' is > 'm'→
4
←
Loop condition is false. Execution continues on line 491→
  f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]);
}

/* compute q[0],q[1],...q[jk] */
for (i = 0; i <= jk; i++) {
5
←
Assuming 'i' is <= 'jk'→
6
←
Loop condition is true.  Entering loop body→
13
←
Loop condition is true.  Entering loop body→
  for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
7
←
Assuming 'j' is <= 'jx'→
8
←
Loop condition is true.  Entering loop body→
9
←
Assuming 'j' is <= 'jx'→
10
←
Loop condition is true.  Entering loop body→
11
←
Assuming 'j' is > 'jx'→
12
←
Loop condition is false. Execution continues on line 493→
14
←
Loop condition is true.  Entering loop body→
15
←
The right operand of '*' is a garbage value
  q[i] = fw;
}

jz = jk;
497recompute:
/* 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;
644}

646/* __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))
*/
673V8_INLINEinline __attribute__((always_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);
}
698}

700/* __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)))
*/
733double __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 */
};
752#define one xxx[13]
753#define pio4 xxx[14]
754#define pio4lo xxx[15]
755#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);
}

831#undef one
832#undef pio4
833#undef pio4lo
834#undef T
835}

837}  // namespace

839/* 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
*/
862double 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);
}
921}

923/* 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.
*/
936double 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));
}
961}

963/* 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.
*/
991double 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;
1056}
1057/* 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)))
*/
1067double 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;
}
1095}

1097/* 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.
*/
1116double 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;
}
1200}

1202/* 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.
*/
1228double 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(-,-) */
}
1319}

1321/* 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
*/
1351double 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);
  }
}
1379}

1381/* 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.
*/
1443double 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;
}
1534}

1536/*
* 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.
*
*/
1553double 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;
1582}

1584/* 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.
*/
1634double 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);
}
1713}

1715/* 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.
*/
1779double 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);
1876}

1878/*
* 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.
*/

1935static 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 */

1943/*
* We always inline k_log1p(), since doing so produces a
* substantial performance improvement (~40% on amd64).
*/
1947static 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);
1958}

1960/*
* 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.
*/
1969double 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;
2047}

2049/*
* 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.
*/
2075double 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;
2113}

2115/* 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.
*/
2209double 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;
2327}

2329double 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);
2411}

2413/* 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
*/
2443double 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]);
  }
}
2471}

2473/* 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
*/
2502double 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));
}
2522}

2524/*
* 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.
*/
2546double 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;
2587}

2589/*
* 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.
*/

2637double 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;
2898}

2900/*
* 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.
*/
2919double 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;
2950}

2952/* 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.
*/
2975double 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;
3009}

3011#undef EXTRACT_WORDS
3012#undef GET_HIGH_WORD
3013#undef GET_LOW_WORD
3014#undef INSERT_WORDS
3015#undef SET_HIGH_WORD
3016#undef SET_LOW_WORD

3018}  // namespace ieee754
3019}  // namespace base
3020}  // namespace v8