xref: /external/v8/src/base/ieee754.cc

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

#include "src/base/ieee754.h"

#include <cmath>
#include <limits>

#include "src/base/build_config.h"
#include "src/base/macros.h"

namespace v8 {
namespace base {
namespace ieee754 {

namespace {

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

#if V8_CC_MSVC

#pragma warning(disable : 4723)

#endif

/*
 * The original fdlibm code used statements like:
 *  n0 = ((*(int*)&one)>>29)^1;   * index of high word *
 *  ix0 = *(n0+(int*)&x);     * high word of x *
 *  ix1 = *((1-n0)+(int*)&x);   * low word of x *
 * to dig two 32 bit words out of the 64 bit IEEE floating point
 * value.  That is non-ANSI, and, moreover, the gcc instruction
 * scheduler gets it wrong.  We instead use the following macros.
 * Unlike the original code, we determine the endianness at compile
 * time, not at run time; I don't see much benefit to selecting
 * endianness at run time.
 */

/*
 * A union which permits us to convert between a double and two 32 bit
 * ints.
 */

#if V8_TARGET_LITTLE_ENDIAN

typedef union {
  double value;
  struct {
    uint32_t lsw;
    uint32_t msw;
  } parts;
  struct {
    uint64_t w;
  } xparts;
} ieee_double_shape_type;

#else

typedef union {
  double value;
  struct {
    uint32_t msw;
    uint32_t lsw;
  } parts;
  struct {
    uint64_t w;
  } xparts;
} ieee_double_shape_type;

#endif

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

#define EXTRACT_WORDS(ix0, ix1, d) \
  do {                             \
    ieee_double_shape_type ew_u;   \
    ew_u.value = (d);              \
    (ix0) = ew_u.parts.msw;        \
    (ix1) = ew_u.parts.lsw;        \
  } while (0)

/* Get a 64-bit int from a double. */
#define EXTRACT_WORD64(ix, d)    \
  do {                           \
    ieee_double_shape_type ew_u; \
    ew_u.value = (d);            \
    (ix) = ew_u.xparts.w;        \
  } while (0)

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

#define GET_HIGH_WORD(i, d)      \
  do {                           \
    ieee_double_shape_type gh_u; \
    gh_u.value = (d);            \
    (i) = gh_u.parts.msw;        \
  } while (0)

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

#define GET_LOW_WORD(i, d)       \
  do {                           \
    ieee_double_shape_type gl_u; \
    gl_u.value = (d);            \
    (i) = gl_u.parts.lsw;        \
  } while (0)

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

#define INSERT_WORDS(d, ix0, ix1) \
  do {                            \
    ieee_double_shape_type iw_u;  \
    iw_u.parts.msw = (ix0);       \
    iw_u.parts.lsw = (ix1);       \
    (d) = iw_u.value;             \
  } while (0)

/* Set a double from a 64-bit int. */
#define INSERT_WORD64(d, ix)     \
  do {                           \
    ieee_double_shape_type iw_u; \
    iw_u.xparts.w = (ix);        \
    (d) = iw_u.value;            \
  } while (0)

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

#define SET_HIGH_WORD(d, v)      \
  do {                           \
    ieee_double_shape_type sh_u; \
    sh_u.value = (d);            \
    sh_u.parts.msw = (v);        \
    (d) = sh_u.value;            \
  } while (0)

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

#define SET_LOW_WORD(d, v)       \
  do {                           \
    ieee_double_shape_type sl_u; \
    sl_u.value = (d);            \
    sl_u.parts.lsw = (v);        \
    (d) = sl_u.value;            \
  } while (0)

/* Support macro. */

#define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))

int32_t __ieee754_rem_pio2(double x, double *y) WARN_UNUSED_RESULT;
double __kernel_cos(double x, double y) WARN_UNUSED_RESULT;
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
                      const int32_t *ipio2) WARN_UNUSED_RESULT;
double __kernel_sin(double x, double y, int iy) WARN_UNUSED_RESULT;

/* __ieee754_rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __kernel_rem_pio2()
 */
int32_t __ieee754_rem_pio2(double x, double *y) {
  /*
   * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
   */
  static const int32_t two_over_pi[] = {
      0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C,
      0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649,
      0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44,
      0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B,
      0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D,
      0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
      0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330,
      0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08,
      0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
      0x73A8C9, 0x60E27B, 0xC08C6B,
  };

  static const int32_t npio2_hw[] = {
      0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
      0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
      0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
      0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
      0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
      0x404858EB, 0x404921FB,
  };

  /*
   * invpio2:  53 bits of 2/pi
   * pio2_1:   first  33 bit of pi/2
   * pio2_1t:  pi/2 - pio2_1
   * pio2_2:   second 33 bit of pi/2
   * pio2_2t:  pi/2 - (pio2_1+pio2_2)
   * pio2_3:   third  33 bit of pi/2
   * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
   */

  static const double
      zero = 0.00000000000000000000e+00,    /* 0x00000000, 0x00000000 */
      half = 5.00000000000000000000e-01,    /* 0x3FE00000, 0x00000000 */
      two24 = 1.67772160000000000000e+07,   /* 0x41700000, 0x00000000 */
      invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
      pio2_1 = 1.57079632673412561417e+00,  /* 0x3FF921FB, 0x54400000 */
      pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
      pio2_2 = 6.07710050630396597660e-11,  /* 0x3DD0B461, 0x1A600000 */
      pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
      pio2_3 = 2.02226624871116645580e-21,  /* 0x3BA3198A, 0x2E000000 */
      pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

  double z, w, t, r, fn;
  double tx[3];
  int32_t e0, i, j, nx, n, ix, hx;
  uint32_t low;

  z = 0;
  GET_HIGH_WORD(hx, x); /* high word of x */
  ix = hx & 0x7fffffff;
  if (ix <= 0x3fe921fb) { /* |x| ~<= pi/4 , no need for reduction */
    y[0] = x;
    y[1] = 0;
    return 0;
  }
  if (ix < 0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
    if (hx > 0) {
      z = x - pio2_1;
      if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
        y[0] = z - pio2_1t;
        y[1] = (z - y[0]) - pio2_1t;
      } else { /* near pi/2, use 33+33+53 bit pi */
        z -= pio2_2;
        y[0] = z - pio2_2t;
        y[1] = (z - y[0]) - pio2_2t;
      }
      return 1;
    } else { /* negative x */
      z = x + pio2_1;
      if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
        y[0] = z + pio2_1t;
        y[1] = (z - y[0]) + pio2_1t;
      } else { /* near pi/2, use 33+33+53 bit pi */
        z += pio2_2;
        y[0] = z + pio2_2t;
        y[1] = (z - y[0]) + pio2_2t;
      }
      return -1;
    }
  }
  if (ix <= 0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
    t = fabs(x);
    n = static_cast<int32_t>(t * invpio2 + half);
    fn = static_cast<double>(n);
    r = t - fn * pio2_1;
    w = fn * pio2_1t; /* 1st round good to 85 bit */
    if (n < 32 && ix != npio2_hw[n - 1]) {
      y[0] = r - w; /* quick check no cancellation */
    } else {
      uint32_t high;
      j = ix >> 20;
      y[0] = r - w;
      GET_HIGH_WORD(high, y[0]);
      i = j - ((high >> 20) & 0x7ff);
      if (i > 16) { /* 2nd iteration needed, good to 118 */
        t = r;
        w = fn * pio2_2;
        r = t - w;
        w = fn * pio2_2t - ((t - r) - w);
        y[0] = r - w;
        GET_HIGH_WORD(high, y[0]);
        i = j - ((high >> 20) & 0x7ff);
        if (i > 49) { /* 3rd iteration need, 151 bits acc */
          t = r;      /* will cover all possible cases */
          w = fn * pio2_3;
          r = t - w;
          w = fn * pio2_3t - ((t - r) - w);
          y[0] = r - w;
        }
      }
    }
    y[1] = (r - y[0]) - w;
    if (hx < 0) {
      y[0] = -y[0];
      y[1] = -y[1];
      return -n;
    } else {
      return n;
    }
  }
  /*
   * all other (large) arguments
   */
  if (ix >= 0x7ff00000) { /* x is inf or NaN */
    y[0] = y[1] = x - x;
    return 0;
  }
  /* set z = scalbn(|x|,ilogb(x)-23) */
  GET_LOW_WORD(low, x);
  SET_LOW_WORD(z, low);
  e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
  SET_HIGH_WORD(z, ix - static_cast<int32_t>(e0 << 20));
  for (i = 0; i < 2; i++) {
    tx[i] = static_cast<double>(static_cast<int32_t>(z));
    z = (z - tx[i]) * two24;
  }
  tx[2] = z;
  nx = 3;
  while (tx[nx - 1] == zero) nx--; /* skip zero term */
  n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
  if (hx < 0) {
    y[0] = -y[0];
    y[1] = -y[1];
    return -n;
  }
  return n;
}

/* __kernel_cos( x,  y )
 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 *
 * Algorithm
 *      1. Since cos(-x) = cos(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
 *      3. cos(x) is approximated by a polynomial of degree 14 on
 *         [0,pi/4]
 *                                       4            14
 *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
 *         where the remez error is
 *
 *      |              2     4     6     8     10    12     14 |     -58
 *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
 *      |                                                      |
 *
 *                     4     6     8     10    12     14
 *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
 *             cos(x) = 1 - x*x/2 + r
 *         since cos(x+y) ~ cos(x) - sin(x)*y
 *                        ~ cos(x) - x*y,
 *         a correction term is necessary in cos(x) and hence
 *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
 *         For better accuracy when x > 0.3, let qx = |x|/4 with
 *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
 *         Then
 *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
 *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
 *         magnitude of the latter is at least a quarter of x*x/2,
 *         thus, reducing the rounding error in the subtraction.
 */
V8_INLINE double __kernel_cos(double x, double y) {
  static const double
      one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
      C1 = 4.16666666666666019037e-02,  /* 0x3FA55555, 0x5555554C */
      C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
      C3 = 2.48015872894767294178e-05,  /* 0x3EFA01A0, 0x19CB1590 */
      C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
      C5 = 2.08757232129817482790e-09,  /* 0x3E21EE9E, 0xBDB4B1C4 */
      C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

  double a, iz, z, r, qx;
  int32_t ix;
  GET_HIGH_WORD(ix, x);
  ix &= 0x7fffffff;                           /* ix = |x|'s high word*/
  if (ix < 0x3e400000) {                      /* if x < 2**27 */
    if (static_cast<int>(x) == 0) return one; /* generate inexact */
  }
  z = x * x;
  r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
  if (ix < 0x3FD33333) { /* if |x| < 0.3 */
    return one - (0.5 * z - (z * r - x * y));
  } else {
    if (ix > 0x3fe90000) { /* x > 0.78125 */
      qx = 0.28125;
    } else {
      INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
    }
    iz = 0.5 * z - qx;
    a = one - qx;
    return a - (iz - (z * r - x * y));
  }
}

/* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
 * double x[],y[]; int e0,nx,prec; int ipio2[];
 *
 * __kernel_rem_pio2 return the last three digits of N with
 *              y = x - N*pi/2
 * so that |y| < pi/2.
 *
 * The method is to compute the integer (mod 8) and fraction parts of
 * (2/pi)*x without doing the full multiplication. In general we
 * skip the part of the product that are known to be a huge integer (
 * more accurately, = 0 mod 8 ). Thus the number of operations are
 * independent of the exponent of the input.
 *
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
 *
 * Input parameters:
 *      x[]     The input value (must be positive) is broken into nx
 *              pieces of 24-bit integers in double precision format.
 *              x[i] will be the i-th 24 bit of x. The scaled exponent
 *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
 *              match x's up to 24 bits.
 *
 *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
 *                      e0 = ilogb(z)-23
 *                      z  = scalbn(z,-e0)
 *              for i = 0,1,2
 *                      x[i] = floor(z)
 *                      z    = (z-x[i])*2**24
 *
 *
 *      y[]     output result in an array of double precision numbers.
 *              The dimension of y[] is:
 *                      24-bit  precision       1
 *                      53-bit  precision       2
 *                      64-bit  precision       2
 *                      113-bit precision       3
 *              The actual value is the sum of them. Thus for 113-bit
 *              precison, one may have to do something like:
 *
 *              long double t,w,r_head, r_tail;
 *              t = (long double)y[2] + (long double)y[1];
 *              w = (long double)y[0];
 *              r_head = t+w;
 *              r_tail = w - (r_head - t);
 *
 *      e0      The exponent of x[0]
 *
 *      nx      dimension of x[]
 *
 *      prec    an integer indicating the precision:
 *                      0       24  bits (single)
 *                      1       53  bits (double)
 *                      2       64  bits (extended)
 *                      3       113 bits (quad)
 *
 *      ipio2[]
 *              integer array, contains the (24*i)-th to (24*i+23)-th
 *              bit of 2/pi after binary point. The corresponding
 *              floating value is
 *
 *                      ipio2[i] * 2^(-24(i+1)).
 *
 * External function:
 *      double scalbn(), floor();
 *
 *
 * Here is the description of some local variables:
 *
 *      jk      jk+1 is the initial number of terms of ipio2[] needed
 *              in the computation. The recommended value is 2,3,4,
 *              6 for single, double, extended,and quad.
 *
 *      jz      local integer variable indicating the number of
 *              terms of ipio2[] used.
 *
 *      jx      nx - 1
 *
 *      jv      index for pointing to the suitable ipio2[] for the
 *              computation. In general, we want
 *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
 *              is an integer. Thus
 *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 *              Hence jv = max(0,(e0-3)/24).
 *
 *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 *
 *      q[]     double array with integral value, representing the
 *              24-bits chunk of the product of x and 2/pi.
 *
 *      q0      the corresponding exponent of q[0]. Note that the
 *              exponent for q[i] would be q0-24*i.
 *
 *      PIo2[]  double precision array, obtained by cutting pi/2
 *              into 24 bits chunks.
 *
 *      f[]     ipio2[] in floating point
 *
 *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 *
 *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 *
 *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 *              it also indicates the *sign* of the result.
 *
 */
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
                      const int32_t *ipio2) {
  /* Constants:
   * The hexadecimal values are the intended ones for the following
   * constants. The decimal values may be used, provided that the
   * compiler will convert from decimal to binary accurately enough
   * to produce the hexadecimal values shown.
   */
  static const int init_jk[] = {2, 3, 4, 6}; /* initial value for jk */

  static const double PIo2[] = {
      1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
      7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
      5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
      3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
      1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
      1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
      2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
      2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
  };

  static const double
      zero = 0.0,
      one = 1.0,
      two24 = 1.67772160000000000000e+07,  /* 0x41700000, 0x00000000 */
      twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

  int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
  double z, fw, f[20], fq[20], q[20];

  /* initialize jk*/
  jk = init_jk[prec];
  jp = jk;

  /* determine jx,jv,q0, note that 3>q0 */
  jx = nx - 1;
  jv = (e0 - 3) / 24;
  if (jv < 0) jv = 0;
  q0 = e0 - 24 * (jv + 1);

  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
  j = jv - jx;
  m = jx + jk;
  for (i = 0; i <= m; i++, j++) {
    f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]);
  }

  /* compute q[0],q[1],...q[jk] */
  for (i = 0; i <= jk; i++) {
    for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
    q[i] = fw;
  }

  jz = jk;
recompute:
  /* distill q[] into iq[] reversingly */
  for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
    fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
    iq[i] = static_cast<int32_t>(z - two24 * fw);
    z = q[j - 1] + fw;
  }

  /* compute n */
  z = scalbn(z, q0);           /* actual value of z */
  z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
  n = static_cast<int32_t>(z);
  z -= static_cast<double>(n);
  ih = 0;
  if (q0 > 0) { /* need iq[jz-1] to determine n */
    i = (iq[jz - 1] >> (24 - q0));
    n += i;
    iq[jz - 1] -= i << (24 - q0);
    ih = iq[jz - 1] >> (23 - q0);
  } else if (q0 == 0) {
    ih = iq[jz - 1] >> 23;
  } else if (z >= 0.5) {
    ih = 2;
  }

  if (ih > 0) { /* q > 0.5 */
    n += 1;
    carry = 0;
    for (i = 0; i < jz; i++) { /* compute 1-q */
      j = iq[i];
      if (carry == 0) {
        if (j != 0) {
          carry = 1;
          iq[i] = 0x1000000 - j;
        }
      } else {
        iq[i] = 0xffffff - j;
      }
    }
    if (q0 > 0) { /* rare case: chance is 1 in 12 */
      switch (q0) {
        case 1:
          iq[jz - 1] &= 0x7fffff;
          break;
        case 2:
          iq[jz - 1] &= 0x3fffff;
          break;
      }
    }
    if (ih == 2) {
      z = one - z;
      if (carry != 0) z -= scalbn(one, q0);
    }
  }

  /* check if recomputation is needed */
  if (z == zero) {
    j = 0;
    for (i = jz - 1; i >= jk; i--) j |= iq[i];
    if (j == 0) { /* need recomputation */
      for (k = 1; jk >= k && iq[jk - k] == 0; k++) {
        /* k = no. of terms needed */
      }

      for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
        f[jx + i] = ipio2[jv + i];
        for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
        q[i] = fw;
      }
      jz += k;
      goto recompute;
    }
  }

  /* chop off zero terms */
  if (z == 0.0) {
    jz -= 1;
    q0 -= 24;
    while (iq[jz] == 0) {
      jz--;
      q0 -= 24;
    }
  } else { /* break z into 24-bit if necessary */
    z = scalbn(z, -q0);
    if (z >= two24) {
      fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
      iq[jz] = z - two24 * fw;
      jz += 1;
      q0 += 24;
      iq[jz] = fw;
    } else {
      iq[jz] = z;
    }
  }

  /* convert integer "bit" chunk to floating-point value */
  fw = scalbn(one, q0);
  for (i = jz; i >= 0; i--) {
    q[i] = fw * iq[i];
    fw *= twon24;
  }

  /* compute PIo2[0,...,jp]*q[jz,...,0] */
  for (i = jz; i >= 0; i--) {
    for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) fw += PIo2[k] * q[i + k];
    fq[jz - i] = fw;
  }

  /* compress fq[] into y[] */
  switch (prec) {
    case 0:
      fw = 0.0;
      for (i = jz; i >= 0; i--) fw += fq[i];
      y[0] = (ih == 0) ? fw : -fw;
      break;
    case 1:
    case 2:
      fw = 0.0;
      for (i = jz; i >= 0; i--) fw += fq[i];
      y[0] = (ih == 0) ? fw : -fw;
      fw = fq[0] - fw;
      for (i = 1; i <= jz; i++) fw += fq[i];
      y[1] = (ih == 0) ? fw : -fw;
      break;
    case 3: /* painful */
      for (i = jz; i > 0; i--) {
        fw = fq[i - 1] + fq[i];
        fq[i] += fq[i - 1] - fw;
        fq[i - 1] = fw;
      }
      for (i = jz; i > 1; i--) {
        fw = fq[i - 1] + fq[i];
        fq[i] += fq[i - 1] - fw;
        fq[i - 1] = fw;
      }
      for (fw = 0.0, i = jz; i >= 2; i--) fw += fq[i];
      if (ih == 0) {
        y[0] = fq[0];
        y[1] = fq[1];
        y[2] = fw;
      } else {
        y[0] = -fq[0];
        y[1] = -fq[1];
        y[2] = -fw;
      }
  }
  return n & 7;
}

/* __kernel_sin( x, y, iy)
 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
 *
 * Algorithm
 *      1. Since sin(-x) = -sin(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
 *      3. sin(x) is approximated by a polynomial of degree 13 on
 *         [0,pi/4]
 *                               3            13
 *              sin(x) ~ x + S1*x + ... + S6*x
 *         where
 *
 *      |sin(x)         2     4     6     8     10     12  |     -58
 *      |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
 *      |  x                                               |
 *
 *      4. sin(x+y) = sin(x) + sin'(x')*y
 *                  ~ sin(x) + (1-x*x/2)*y
 *         For better accuracy, let
 *                   3      2      2      2      2
 *              r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
 *         then                   3    2
 *              sin(x) = x + (S1*x + (x *(r-y/2)+y))
 */
V8_INLINE double __kernel_sin(double x, double y, int iy) {
  static const double
      half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
      S1 = -1.66666666666666324348e-01,  /* 0xBFC55555, 0x55555549 */
      S2 = 8.33333333332248946124e-03,   /* 0x3F811111, 0x1110F8A6 */
      S3 = -1.98412698298579493134e-04,  /* 0xBF2A01A0, 0x19C161D5 */
      S4 = 2.75573137070700676789e-06,   /* 0x3EC71DE3, 0x57B1FE7D */
      S5 = -2.50507602534068634195e-08,  /* 0xBE5AE5E6, 0x8A2B9CEB */
      S6 = 1.58969099521155010221e-10;   /* 0x3DE5D93A, 0x5ACFD57C */

  double z, r, v;
  int32_t ix;
  GET_HIGH_WORD(ix, x);
  ix &= 0x7fffffff;      /* high word of x */
  if (ix < 0x3e400000) { /* |x| < 2**-27 */
    if (static_cast<int>(x) == 0) return x;
  } /* generate inexact */
  z = x * x;
  v = z * x;
  r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
  if (iy == 0) {
    return x + v * (S1 + z * r);
  } else {
    return x - ((z * (half * y - v * r) - y) - v * S1);
  }
}

/* __kernel_tan( x, y, k )
 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input k indicates whether tan (if k=1) or
 * -1/tan (if k= -1) is returned.
 *
 * Algorithm
 *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
 *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
 *      3. tan(x) is approximated by a odd polynomial of degree 27 on
 *         [0,0.67434]
 *                               3             27
 *              tan(x) ~ x + T1*x + ... + T13*x
 *         where
 *
 *              |tan(x)         2     4            26   |     -59.2
 *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
 *              |  x                                    |
 *
 *         Note: tan(x+y) = tan(x) + tan'(x)*y
 *                        ~ tan(x) + (1+x*x)*y
 *         Therefore, for better accuracy in computing tan(x+y), let
 *                   3      2      2       2       2
 *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
 *         then
 *                                  3    2
 *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
 *
 *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
 *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
 *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
 */
double __kernel_tan(double x, double y, int iy) {
  static const double xxx[] = {
      3.33333333333334091986e-01,             /* 3FD55555, 55555563 */
      1.33333333333201242699e-01,             /* 3FC11111, 1110FE7A */
      5.39682539762260521377e-02,             /* 3FABA1BA, 1BB341FE */
      2.18694882948595424599e-02,             /* 3F9664F4, 8406D637 */
      8.86323982359930005737e-03,             /* 3F8226E3, E96E8493 */
      3.59207910759131235356e-03,             /* 3F6D6D22, C9560328 */
      1.45620945432529025516e-03,             /* 3F57DBC8, FEE08315 */
      5.88041240820264096874e-04,             /* 3F4344D8, F2F26501 */
      2.46463134818469906812e-04,             /* 3F3026F7, 1A8D1068 */
      7.81794442939557092300e-05,             /* 3F147E88, A03792A6 */
      7.14072491382608190305e-05,             /* 3F12B80F, 32F0A7E9 */
      -1.85586374855275456654e-05,            /* BEF375CB, DB605373 */
      2.59073051863633712884e-05,             /* 3EFB2A70, 74BF7AD4 */
      /* one */ 1.00000000000000000000e+00,   /* 3FF00000, 00000000 */
      /* pio4 */ 7.85398163397448278999e-01,  /* 3FE921FB, 54442D18 */
      /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
  };
#define one xxx[13]
#define pio4 xxx[14]
#define pio4lo xxx[15]
#define T xxx

  double z, r, v, w, s;
  int32_t ix, hx;

  GET_HIGH_WORD(hx, x);             /* high word of x */
  ix = hx & 0x7fffffff;             /* high word of |x| */
  if (ix < 0x3e300000) {            /* x < 2**-28 */
    if (static_cast<int>(x) == 0) { /* generate inexact */
      uint32_t low;
      GET_LOW_WORD(low, x);
      if (((ix | low) | (iy + 1)) == 0) {
        return one / fabs(x);
      } else {
        if (iy == 1) {
          return x;
        } else { /* compute -1 / (x+y) carefully */
          double a, t;

          z = w = x + y;
          SET_LOW_WORD(z, 0);
          v = y - (z - x);
          t = a = -one / w;
          SET_LOW_WORD(t, 0);
          s = one + t * z;
          return t + a * (s + t * v);
        }
      }
    }
  }
  if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
    if (hx < 0) {
      x = -x;
      y = -y;
    }
    z = pio4 - x;
    w = pio4lo - y;
    x = z + w;
    y = 0.0;
  }
  z = x * x;
  w = z * z;
  /*
   * Break x^5*(T[1]+x^2*T[2]+...) into
   * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
   * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
   */
  r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
  v = z *
      (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
  s = z * x;
  r = y + z * (s * (r + v) + y);
  r += T[0] * s;
  w = x + r;
  if (ix >= 0x3FE59428) {
    v = iy;
    return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
  }
  if (iy == 1) {
    return w;
  } else {
    /*
     * if allow error up to 2 ulp, simply return
     * -1.0 / (x+r) here
     */
    /* compute -1.0 / (x+r) accurately */
    double a, t;
    z = w;
    SET_LOW_WORD(z, 0);
    v = r - (z - x);  /* z+v = r+x */
    t = a = -1.0 / w; /* a = -1.0/w */
    SET_LOW_WORD(t, 0);
    s = 1.0 + t * z;
    return t + a * (s + t * v);
  }

#undef one
#undef pio4
#undef pio4lo
#undef T
}

}  // namespace

/* atan(x)
 * Method
 *   1. Reduce x to positive by atan(x) = -atan(-x).
 *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
 *      is further reduced to one of the following intervals and the
 *      arctangent of t is evaluated by the corresponding formula:
 *
 *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
 *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
 *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
 *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
 *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
double atan(double x) {
  static const double atanhi[] = {
      4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
      7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
      9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
      1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
  };

  static const double atanlo[] = {
      2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
      3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
      1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
      6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
  };

  static const double aT[] = {
      3.33333333333329318027e-01,  /* 0x3FD55555, 0x5555550D */
      -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
      1.42857142725034663711e-01,  /* 0x3FC24924, 0x920083FF */
      -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
      9.09088713343650656196e-02,  /* 0x3FB745CD, 0xC54C206E */
      -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
      6.66107313738753120669e-02,  /* 0x3FB10D66, 0xA0D03D51 */
      -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
      4.97687799461593236017e-02,  /* 0x3FA97B4B, 0x24760DEB */
      -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
      1.62858201153657823623e-02,  /* 0x3F90AD3A, 0xE322DA11 */
  };

  static const double one = 1.0, huge = 1.0e300;

  double w, s1, s2, z;
  int32_t ix, hx, id;

  GET_HIGH_WORD(hx, x);
  ix = hx & 0x7fffffff;
  if (ix >= 0x44100000) { /* if |x| >= 2^66 */
    uint32_t low;
    GET_LOW_WORD(low, x);
    if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0)))
      return x + x; /* NaN */
    if (hx > 0)
      return atanhi[3] + *(volatile double *)&atanlo[3];
    else
      return -atanhi[3] - *(volatile double *)&atanlo[3];
  }
  if (ix < 0x3fdc0000) {            /* |x| < 0.4375 */
    if (ix < 0x3e400000) {          /* |x| < 2^-27 */
      if (huge + x > one) return x; /* raise inexact */
    }
    id = -1;
  } else {
    x = fabs(x);
    if (ix < 0x3ff30000) {   /* |x| < 1.1875 */
      if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
        id = 0;
        x = (2.0 * x - one) / (2.0 + x);
      } else { /* 11/16<=|x|< 19/16 */
        id = 1;
        x = (x - one) / (x + one);
      }
    } else {
      if (ix < 0x40038000) { /* |x| < 2.4375 */
        id = 2;
        x = (x - 1.5) / (one + 1.5 * x);
      } else { /* 2.4375 <= |x| < 2^66 */
        id = 3;
        x = -1.0 / x;
      }
    }
  }
  /* end of argument reduction */
  z = x * x;
  w = z * z;
  /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
  s1 = z * (aT[0] +
            w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
  s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
  if (id < 0) {
    return x - x * (s1 + s2);
  } else {
    z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
    return (hx < 0) ? -z : z;
  }
}

/* atan2(y,x)
 * Method :
 *  1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
 *  2. Reduce x to positive by (if x and y are unexceptional):
 *    ARG (x+iy) = arctan(y/x)       ... if x > 0,
 *    ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
 *
 * Special cases:
 *
 *  ATAN2((anything), NaN ) is NaN;
 *  ATAN2(NAN , (anything) ) is NaN;
 *  ATAN2(+-0, +(anything but NaN)) is +-0  ;
 *  ATAN2(+-0, -(anything but NaN)) is +-pi ;
 *  ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
 *  ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
 *  ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
 *  ATAN2(+-INF,+INF ) is +-pi/4 ;
 *  ATAN2(+-INF,-INF ) is +-3pi/4;
 *  ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
double atan2(double y, double x) {
  static volatile double tiny = 1.0e-300;
  static const double
      zero = 0.0,
      pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
      pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
      pi = 3.1415926535897931160E+00;     /* 0x400921FB, 0x54442D18 */
  static volatile double pi_lo =
      1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */

  double z;
  int32_t k, m, hx, hy, ix, iy;
  uint32_t lx, ly;

  EXTRACT_WORDS(hx, lx, x);
  ix = hx & 0x7fffffff;
  EXTRACT_WORDS(hy, ly, y);
  iy = hy & 0x7fffffff;
  if (((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x7ff00000) ||
      ((iy | ((ly | -static_cast<int32_t>(ly)) >> 31)) > 0x7ff00000)) {
    return x + y; /* x or y is NaN */
  }
  if (((hx - 0x3ff00000) | lx) == 0) return atan(y); /* x=1.0 */
  m = ((hy >> 31) & 1) | ((hx >> 30) & 2);           /* 2*sign(x)+sign(y) */

  /* when y = 0 */
  if ((iy | ly) == 0) {
    switch (m) {
      case 0:
      case 1:
        return y; /* atan(+-0,+anything)=+-0 */
      case 2:
        return pi + tiny; /* atan(+0,-anything) = pi */
      case 3:
        return -pi - tiny; /* atan(-0,-anything) =-pi */
    }
  }
  /* when x = 0 */
  if ((ix | lx) == 0) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;

  /* when x is INF */
  if (ix == 0x7ff00000) {
    if (iy == 0x7ff00000) {
      switch (m) {
        case 0:
          return pi_o_4 + tiny; /* atan(+INF,+INF) */
        case 1:
          return -pi_o_4 - tiny; /* atan(-INF,+INF) */
        case 2:
          return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/
        case 3:
          return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/
      }
    } else {
      switch (m) {
        case 0:
          return zero; /* atan(+...,+INF) */
        case 1:
          return -zero; /* atan(-...,+INF) */
        case 2:
          return pi + tiny; /* atan(+...,-INF) */
        case 3:
          return -pi - tiny; /* atan(-...,-INF) */
      }
    }
  }
  /* when y is INF */
  if (iy == 0x7ff00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;

  /* compute y/x */
  k = (iy - ix) >> 20;
  if (k > 60) { /* |y/x| >  2**60 */
    z = pi_o_2 + 0.5 * pi_lo;
    m &= 1;
  } else if (hx < 0 && k < -60) {
    z = 0.0; /* 0 > |y|/x > -2**-60 */
  } else {
    z = atan(fabs(y / x)); /* safe to do y/x */
  }
  switch (m) {
    case 0:
      return z; /* atan(+,+) */
    case 1:
      return -z; /* atan(-,+) */
    case 2:
      return pi - (z - pi_lo); /* atan(+,-) */
    default:                   /* case 3 */
      return (z - pi_lo) - pi; /* atan(-,-) */
  }
}

/* cos(x)
 * Return cosine function of x.
 *
 * kernel function:
 *      __kernel_sin            ... sine function on [-pi/4,pi/4]
 *      __kernel_cos            ... cosine function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 *
 * Method.
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */
double cos(double x) {
  double y[2], z = 0.0;
  int32_t n, ix;

  /* High word of x. */
  GET_HIGH_WORD(ix, x);

  /* |x| ~< pi/4 */
  ix &= 0x7fffffff;
  if (ix <= 0x3fe921fb) {
    return __kernel_cos(x, z);
  } else if (ix >= 0x7ff00000) {
    /* cos(Inf or NaN) is NaN */
    return x - x;
  } else {
    /* argument reduction needed */
    n = __ieee754_rem_pio2(x, y);
    switch (n & 3) {
      case 0:
        return __kernel_cos(y[0], y[1]);
      case 1:
        return -__kernel_sin(y[0], y[1], 1);
      case 2:
        return -__kernel_cos(y[0], y[1]);
      default:
        return __kernel_sin(y[0], y[1], 1);
    }
  }
}

/* exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *      Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 *
 *      Here r will be represented as r = hi-lo for better
 *      accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Write
 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Remes algorithm on [0,0.34658] to generate
 *      a polynomial of degree 5 to approximate R. The maximum error
 *      of this polynomial approximation is bounded by 2**-59. In
 *      other words,
 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *      (where z=r*r, and the values of P1 to P5 are listed below)
 *      and
 *          |                  5          |     -59
 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 *          |                             |
 *      The computation of exp(r) thus becomes
 *                             2*r
 *              exp(r) = 1 + -------
 *                            R - r
 *                                 r*R1(r)
 *                     = 1 + r + ----------- (for better accuracy)
 *                                2 - R1(r)
 *      where
 *                               2       4             10
 *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *
 *   3. Scale back to obtain exp(x):
 *      From step 1, we have
 *         exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *      exp(INF) is INF, exp(NaN) is NaN;
 *      exp(-INF) is 0, and
 *      for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Misc. info.
 *      For IEEE double
 *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
double exp(double x) {
  static const double
      one = 1.0,
      halF[2] = {0.5, -0.5},
      o_threshold = 7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
      u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
      ln2HI[2] = {6.93147180369123816490e-01,    /* 0x3fe62e42, 0xfee00000 */
                  -6.93147180369123816490e-01},  /* 0xbfe62e42, 0xfee00000 */
      ln2LO[2] = {1.90821492927058770002e-10,    /* 0x3dea39ef, 0x35793c76 */
                  -1.90821492927058770002e-10},  /* 0xbdea39ef, 0x35793c76 */
      invln2 = 1.44269504088896338700e+00,       /* 0x3ff71547, 0x652b82fe */
      P1 = 1.66666666666666019037e-01,           /* 0x3FC55555, 0x5555553E */
      P2 = -2.77777777770155933842e-03,          /* 0xBF66C16C, 0x16BEBD93 */
      P3 = 6.61375632143793436117e-05,           /* 0x3F11566A, 0xAF25DE2C */
      P4 = -1.65339022054652515390e-06,          /* 0xBEBBBD41, 0xC5D26BF1 */
      P5 = 4.13813679705723846039e-08,           /* 0x3E663769, 0x72BEA4D0 */
      E = 2.718281828459045;                     /* 0x4005bf0a, 0x8b145769 */

  static volatile double
      huge = 1.0e+300,
      twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
      two1023 = 8.988465674311579539e307;     /* 0x1p1023 */

  double y, hi = 0.0, lo = 0.0, c, t, twopk;
  int32_t k = 0, xsb;
  uint32_t hx;

  GET_HIGH_WORD(hx, x);
  xsb = (hx >> 31) & 1; /* sign bit of x */
  hx &= 0x7fffffff;     /* high word of |x| */

  /* filter out non-finite argument */
  if (hx >= 0x40862E42) { /* if |x|>=709.78... */
    if (hx >= 0x7ff00000) {
      uint32_t lx;
      GET_LOW_WORD(lx, x);
      if (((hx & 0xfffff) | lx) != 0)
        return x + x; /* NaN */
      else
        return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
    }
    if (x > o_threshold) return huge * huge;         /* overflow */
    if (x < u_threshold) return twom1000 * twom1000; /* underflow */
  }

  /* argument reduction */
  if (hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
    if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
      /* TODO(rtoy): We special case exp(1) here to return the correct
       * value of E, as the computation below would get the last bit
       * wrong. We should probably fix the algorithm instead.
       */
      if (x == 1.0) return E;
      hi = x - ln2HI[xsb];
      lo = ln2LO[xsb];
      k = 1 - xsb - xsb;
    } else {
      k = static_cast<int>(invln2 * x + halF[xsb]);
      t = k;
      hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
      lo = t * ln2LO[0];
    }
    STRICT_ASSIGN(double, 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 + (k << 20), 0);
  } else {
    INSERT_WORDS(twopk, 0x3ff00000 + ((k + 1000) << 20), 0);
  }
  c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
  if (k == 0) {
    return one - ((x * c) / (c - 2.0) - x);
  } else {
    y = one - ((lo - (x * c) / (2.0 - c)) - hi);
  }
  if (k >= -1021) {
    if (k == 1024) return y * 2.0 * two1023;
    return y * twopk;
  } else {
    return y * twopk * twom1000;
  }
}

/*
 * Method :
 *    1.Reduced x to positive by atanh(-x) = -atanh(x)
 *    2.For x>=0.5
 *              1              2x                          x
 *  atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
 *              2             1 - x                      1 - x
 *
 *   For x<0.5
 *  atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
 *
 * Special cases:
 *  atanh(x) is NaN if |x| > 1 with signal;
 *  atanh(NaN) is that NaN with no signal;
 *  atanh(+-1) is +-INF with signal.
 *
 */
double atanh(double x) {
  static const double one = 1.0, huge = 1e300;
  static const double zero = 0.0;

  double t;
  int32_t hx, ix;
  uint32_t lx;
  EXTRACT_WORDS(hx, lx, x);
  ix = hx & 0x7fffffff;
  if ((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x3ff00000) /* |x|>1 */
    return (x - x) / (x - x);
  if (ix == 0x3ff00000) return x / zero;
  if (ix < 0x3e300000 && (huge + x) > zero) return x; /* x<2**-28 */
  SET_HIGH_WORD(x, ix);
  if (ix < 0x3fe00000) { /* x < 0.5 */
    t = x + x;
    t = 0.5 * log1p(t + t * x / (one - x));
  } else {
    t = 0.5 * log1p((x + x) / (one - x));
  }
  if (hx >= 0)
    return t;
  else
    return -t;
}

/* log(x)
 * Return the logrithm of x
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *     x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *    (the values of Lg1 to Lg7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log(1+f) = f - s*(f - R)  (if f is not too large)
 *    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
 *
 *  3. Finally,  log(x) = k*ln2 + log(1+f).
 *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log(x) is NaN with signal if x < 0 (including -INF) ;
 *  log(+INF) is +INF; log(0) is -INF with signal;
 *  log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
double log(double x) {
  static const double                      /* -- */
      ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
      ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
      two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
      Lg1 = 6.666666666666735130e-01,      /* 3FE55555 55555593 */
      Lg2 = 3.999999999940941908e-01,      /* 3FD99999 9997FA04 */
      Lg3 = 2.857142874366239149e-01,      /* 3FD24924 94229359 */
      Lg4 = 2.222219843214978396e-01,      /* 3FCC71C5 1D8E78AF */
      Lg5 = 1.818357216161805012e-01,      /* 3FC74664 96CB03DE */
      Lg6 = 1.531383769920937332e-01,      /* 3FC39A09 D078C69F */
      Lg7 = 1.479819860511658591e-01;      /* 3FC2F112 DF3E5244 */

  static const double zero = 0.0;
  static volatile double vzero = 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 -two54 / vzero;           /* log(+-0)=-inf */
    if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
    k -= 54;
    x *= two54; /* subnormal number, scale up x */
    GET_HIGH_WORD(hx, x);
  }
  if (hx >= 0x7ff00000) return x + x;
  k += (hx >> 20) - 1023;
  hx &= 0x000fffff;
  i = (hx + 0x95f64) & 0x100000;
  SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
  k += (i >> 20);
  f = x - 1.0;
  if ((0x000fffff & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
    if (f == zero) {
      if (k == 0) {
        return zero;
      } else {
        dk = static_cast<double>(k);
        return dk * ln2_hi + dk * ln2_lo;
      }
    }
    R = f * f * (0.5 - 0.33333333333333333 * f);
    if (k == 0) {
      return f - R;
    } else {
      dk = static_cast<double>(k);
      return dk * ln2_hi - ((R - dk * ln2_lo) - f);
    }
  }
  s = f / (2.0 + f);
  dk = static_cast<double>(k);
  z = s * s;
  i = hx - 0x6147a;
  w = z * z;
  j = 0x6b851 - hx;
  t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
  t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
  i |= j;
  R = t2 + t1;
  if (i > 0) {
    hfsq = 0.5 * f * f;
    if (k == 0)
      return f - (hfsq - s * (hfsq + R));
    else
      return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
  } else {
    if (k == 0)
      return f - s * (f - R);
    else
      return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
  }
}

/* double log1p(double x)
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *      1+x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *  may not be representable exactly. In that case, a correction
 *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *  and add back the correction term c/u.
 *  (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log1p(f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
 *    (the values of Lp1 to Lp7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log1p(f) = f - (hfsq - s*(hfsq+R)).
 *
 *  3. Finally, log1p(x) = k*ln2 + log1p(f).
 *           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *  log1p(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 *
 * Note: Assuming log() return accurate answer, the following
 *   algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *    u = 1+x;
 *    if(u==1.0) return x ; else
 *         return log(u)*(x/(u-1.0));
 *
 *   See HP-15C Advanced Functions Handbook, p.193.
 */
double log1p(double x) {
  static const double                      /* -- */
      ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
      ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
      two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
      Lp1 = 6.666666666666735130e-01,      /* 3FE55555 55555593 */
      Lp2 = 3.999999999940941908e-01,      /* 3FD99999 9997FA04 */
      Lp3 = 2.857142874366239149e-01,      /* 3FD24924 94229359 */
      Lp4 = 2.222219843214978396e-01,      /* 3FCC71C5 1D8E78AF */
      Lp5 = 1.818357216161805012e-01,      /* 3FC74664 96CB03DE */
      Lp6 = 1.531383769920937332e-01,      /* 3FC39A09 D078C69F */
      Lp7 = 1.479819860511658591e-01;      /* 3FC2F112 DF3E5244 */

  static const double zero = 0.0;
  static volatile double vzero = 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 -two54 / vzero; /* log1p(-1)=+inf */
      else
        return (x - x) / (x - x); /* 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) {
      STRICT_ASSIGN(double, u, 1.0 + x);
      GET_HIGH_WORD(hu, u);
      k = (hu >> 20) - 1023;
      c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
      c /= u;
    } else {
      u = x;
      GET_HIGH_WORD(hu, u);
      k = (hu >> 20) - 1023;
      c = 0;
    }
    hu &= 0x000fffff;
    /*
     * The approximation to sqrt(2) used in thresholds is not
     * critical.  However, the ones used above must give less
     * strict bounds than the one here so that the k==0 case is
     * never reached from here, since here we have committed to
     * using the correction term but don't use it if k==0.
     */
    if (hu < 0x6a09e) {                  /* u ~< sqrt(2) */
      SET_HIGH_WORD(u, hu | 0x3ff00000); /* normalize u */
    } else {
      k += 1;
      SET_HIGH_WORD(u, hu | 0x3fe00000); /* normalize u/2 */
      hu = (0x00100000 - hu) >> 2;
    }
    f = u - 1.0;
  }
  hfsq = 0.5 * f * f;
  if (hu == 0) { /* |f| < 2**-20 */
    if (f == zero) {
      if (k == 0) {
        return zero;
      } else {
        c += k * ln2_lo;
        return k * ln2_hi + c;
      }
    }
    R = hfsq * (1.0 - 0.66666666666666666 * f);
    if (k == 0)
      return f - R;
    else
      return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
  }
  s = f / (2.0 + f);
  z = s * s;
  R = z * (Lp1 +
           z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
  if (k == 0)
    return f - (hfsq - s * (hfsq + R));
  else
    return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
}

/*
 * k_log1p(f):
 * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
 *
 * The following describes the overall strategy for computing
 * logarithms in base e.  The argument reduction and adding the final
 * term of the polynomial are done by the caller for increased accuracy
 * when different bases are used.
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *         x = 2^k * (1+f),
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *            = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *            = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *      a polynomial of degree 14 to approximate R The maximum error
 *      of this polynomial approximation is bounded by 2**-58.45. In
 *      other words,
 *          2      4      6      8      10      12      14
 *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *      (the values of Lg1 to Lg7 are listed in the program)
 *      and
 *          |      2          14          |     -58.45
 *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *          |                             |
 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *      In order to guarantee error in log below 1ulp, we compute log
 *      by
 *          log(1+f) = f - s*(f - R)            (if f is not too large)
 *          log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
 *
 *   3. Finally,  log(x) = k*ln2 + log(1+f).
 *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *      Here ln2 is split into two floating point number:
 *          ln2_hi + ln2_lo,
 *      where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *      log(x) is NaN with signal if x < 0 (including -INF) ;
 *      log(+INF) is +INF; log(0) is -INF with signal;
 *      log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

static const double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
    Lg2 = 3.999999999940941908e-01,                 /* 3FD99999 9997FA04 */
    Lg3 = 2.857142874366239149e-01,                 /* 3FD24924 94229359 */
    Lg4 = 2.222219843214978396e-01,                 /* 3FCC71C5 1D8E78AF */
    Lg5 = 1.818357216161805012e-01,                 /* 3FC74664 96CB03DE */
    Lg6 = 1.531383769920937332e-01,                 /* 3FC39A09 D078C69F */
    Lg7 = 1.479819860511658591e-01;                 /* 3FC2F112 DF3E5244 */

/*
 * We always inline k_log1p(), since doing so produces a
 * substantial performance improvement (~40% on amd64).
 */
static inline double k_log1p(double f) {
  double hfsq, s, z, R, w, t1, t2;

  s = f / (2.0 + f);
  z = s * s;
  w = z * z;
  t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
  t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
  R = t2 + t1;
  hfsq = 0.5 * f * f;
  return s * (hfsq + R);
}

/*
 * Return the base 2 logarithm of x.  See e_log.c and k_log.h for most
 * comments.
 *
 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
 * then does the combining and scaling steps
 *    log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
 * in not-quite-routine extra precision.
 */
double log2(double x) {
  static const double
      two54 = 1.80143985094819840000e+16,   /* 0x43500000, 0x00000000 */
      ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
      ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */

  static const double zero = 0.0;
  static volatile double vzero = 0.0;

  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 -two54 / vzero;           /* log(+-0)=-inf */
    if (hx < 0) return (x - x) / zero; /* 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 zero; /* log(1) = +0 */
  k += (hx >> 20) - 1023;
  hx &= 0x000fffff;
  i = (hx + 0x95f64) & 0x100000;
  SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
  k += (i >> 20);
  y = static_cast<double>(k);
  f = x - 1.0;
  hfsq = 0.5 * f * f;
  r = k_log1p(f);

  /*
   * f-hfsq must (for args near 1) be evaluated in extra precision
   * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
   * This is fairly efficient since f-hfsq only depends on f, so can
   * be evaluated in parallel with R.  Not combining hfsq with R also
   * keeps R small (though not as small as a true `lo' term would be),
   * so that extra precision is not needed for terms involving R.
   *
   * Compiler bugs involving extra precision used to break Dekker's
   * theorem for spitting f-hfsq as hi+lo, unless double_t was used
   * or the multi-precision calculations were avoided when double_t
   * has extra precision.  These problems are now automatically
   * avoided as a side effect of the optimization of combining the
   * Dekker splitting step with the clear-low-bits step.
   *
   * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
   * precision to avoid a very large cancellation when x is very near
   * these values.  Unlike the above cancellations, this problem is
   * specific to base 2.  It is strange that adding +-1 is so much
   * harder than adding +-ln2 or +-log10_2.
   *
   * This uses Dekker's theorem to normalize y+val_hi, so the
   * compiler bugs are back in some configurations, sigh.  And I
   * don't want to used double_t to avoid them, since that gives a
   * pessimization and the support for avoiding the pessimization
   * is not yet available.
   *
   * The multi-precision calculations for the multiplications are
   * routine.
   */
  hi = f - hfsq;
  SET_LOW_WORD(hi, 0);
  lo = (f - hi) - hfsq + r;
  val_hi = hi * ivln2hi;
  val_lo = (lo + hi) * ivln2lo + lo * ivln2hi;

  /* spadd(val_hi, val_lo, y), except for not using double_t: */
  w = y + val_hi;
  val_lo += (y - w) + val_hi;
  val_hi = w;

  return val_lo + val_hi;
}

/*
 * Return the base 10 logarithm of x
 *
 * Method :
 *      Let log10_2hi = leading 40 bits of log10(2) and
 *          log10_2lo = log10(2) - log10_2hi,
 *          ivln10   = 1/log(10) rounded.
 *      Then
 *              n = ilogb(x),
 *              if(n<0)  n = n+1;
 *              x = scalbn(x,-n);
 *              log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
 *
 *  Note 1:
 *     To guarantee log10(10**n)=n, where 10**n is normal, the rounding
 *     mode must set to Round-to-Nearest.
 *  Note 2:
 *      [1/log(10)] rounded to 53 bits has error .198 ulps;
 *      log10 is monotonic at all binary break points.
 *
 *  Special cases:
 *      log10(x) is NaN if x < 0;
 *      log10(+INF) is +INF; log10(0) is -INF;
 *      log10(NaN) is that NaN;
 *      log10(10**N) = N  for N=0,1,...,22.
 */
double log10(double x) {
  static const double
      two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
      ivln10 = 4.34294481903251816668e-01,
      log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
      log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */

  static const double zero = 0.0;
  static volatile double vzero = 0.0;

  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 -two54 / vzero;           /* log(+-0)=-inf */
    if (hx < 0) return (x - x) / zero; /* 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 zero; /* log(1) = +0 */
  k += (hx >> 20) - 1023;

  i = (k & 0x80000000) >> 31;
  hx = (hx & 0x000fffff) | ((0x3ff - i) << 20);
  y = k + i;
  SET_HIGH_WORD(x, hx);
  SET_LOW_WORD(x, lx);

  double z = y * log10_2lo + ivln10 * log(x);
  return z + y * log10_2hi;
}

/* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *   1. Argument reduction:
 *  Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *      Here a correction term c will be computed to compensate
 *  the error in r when rounded to a floating-point number.
 *
 *   2. Approximating expm1(r) by a special rational function on
 *  the interval [0,0.34658]:
 *  Since
 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *  we define R1(r*r) by
 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *  That is,
 *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *         = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *         = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *      We use a special Reme algorithm on [0,0.347] to generate
 *   a polynomial of degree 5 in r*r to approximate R1. The
 *  maximum error of this polynomial approximation is bounded
 *  by 2**-61. In other words,
 *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *  where   Q1  =  -1.6666666666666567384E-2,
 *     Q2  =   3.9682539681370365873E-4,
 *     Q3  =  -9.9206344733435987357E-6,
 *     Q4  =   2.5051361420808517002E-7,
 *     Q5  =  -6.2843505682382617102E-9;
 *    z   =  r*r,
 *  with error bounded by
 *      |                  5           |     -61
 *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *      |                              |
 *
 *  expm1(r) = exp(r)-1 is then computed by the following
 *   specific way which minimize the accumulation rounding error:
 *             2     3
 *            r     r    [ 3 - (R1 + R1*r/2)  ]
 *        expm1(r) = r + --- + --- * [--------------------]
 *                  2     2    [ 6 - r*(3 - R1*r/2) ]
 *
 *  To compensate the error in the argument reduction, we use
 *    expm1(r+c) = expm1(r) + c + expm1(r)*c
 *         ~ expm1(r) + c + r*c
 *  Thus c+r*c will be added in as the correction terms for
 *  expm1(r+c). Now rearrange the term to avoid optimization
 *   screw up:
 *            (      2                                    2 )
 *            ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                             )
 *
 *       = r - E
 *   3. Scale back to obtain expm1(x):
 *  From step 1, we have
 *     expm1(x) = either 2^k*[expm1(r)+1] - 1
 *        = or     2^k*[expm1(r) + (1-2^-k)]
 *   4. Implementation notes:
 *  (A). To save one multiplication, we scale the coefficient Qi
 *       to Qi*2^i, and replace z by (x^2)/2.
 *  (B). To achieve maximum accuracy, we compute expm1(x) by
 *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *    (ii)  if k=0, return r-E
 *    (iii) if k=-1, return 0.5*(r-E)-0.5
 *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *                  else       return  1.0+2.0*(r-E);
 *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *    (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *  expm1(INF) is INF, expm1(NaN) is NaN;
 *  expm1(-INF) is -1, and
 *  for finite argument, only expm1(0)=0 is exact.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Misc. info.
 *  For IEEE double
 *      if x >  7.09782712893383973096e+02 then expm1(x) overflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
double expm1(double x) {
  static const double
      one = 1.0,
      tiny = 1.0e-300,
      o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
      ln2_hi = 6.93147180369123816490e-01,      /* 0x3fe62e42, 0xfee00000 */
      ln2_lo = 1.90821492927058770002e-10,      /* 0x3dea39ef, 0x35793c76 */
      invln2 = 1.44269504088896338700e+00,      /* 0x3ff71547, 0x652b82fe */
      /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs =
         x*x/2: */
      Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
      Q2 = 1.58730158725481460165e-03,  /* 3F5A01A0 19FE5585 */
      Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
      Q4 = 4.00821782732936239552e-06,  /* 3ED0CFCA 86E65239 */
      Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

  static volatile double huge = 1.0e+300;

  double y, hi, lo, c, t, e, hxs, hfx, r1, twopk;
  int32_t k, xsb;
  uint32_t hx;

  GET_HIGH_WORD(hx, x);
  xsb = hx & 0x80000000; /* sign bit of x */
  hx &= 0x7fffffff;      /* high word of |x| */

  /* filter out huge and non-finite argument */
  if (hx >= 0x4043687A) {   /* if |x|>=56*ln2 */
    if (hx >= 0x40862E42) { /* if |x|>=709.78... */
      if (hx >= 0x7ff00000) {
        uint32_t low;
        GET_LOW_WORD(low, x);
        if (((hx & 0xfffff) | low) != 0)
          return x + x; /* NaN */
        else
          return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
      }
      if (x > o_threshold) return huge * huge; /* overflow */
    }
    if (xsb != 0) {        /* x < -56*ln2, return -1.0 with inexact */
      if (x + tiny < 0.0)  /* raise inexact */
        return tiny - one; /* return -1 */
    }
  }

  /* argument reduction */
  if (hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
    if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
      if (xsb == 0) {
        hi = x - ln2_hi;
        lo = ln2_lo;
        k = 1;
      } else {
        hi = x + ln2_hi;
        lo = -ln2_lo;
        k = -1;
      }
    } else {
      k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
      t = k;
      hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
      lo = t * ln2_lo;
    }
    STRICT_ASSIGN(double, 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 + (k << 20), 0); /* 2^k */
    e = (x * (e - c) - c);
    e -= hxs;
    if (k == -1) return 0.5 * (x - e) - 0.5;
    if (k == 1) {
      if (x < -0.25)
        return -2.0 * (e - (x + 0.5));
      else
        return one + 2.0 * (x - e);
    }
    if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
      y = one - (e - x);
      // TODO(mvstanton): is this replacement for the hex float
      // sufficient?
      // if (k == 1024) y = y*2.0*0x1p1023;
      if (k == 1024)
        y = y * 2.0 * 8.98846567431158e+307;
      else
        y = y * twopk;
      return y - one;
    }
    t = one;
    if (k < 20) {
      SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
      y = t - (e - x);
      y = y * twopk;
    } else {
      SET_HIGH_WORD(t, ((0x3ff - k) << 20)); /* 2^-k */
      y = x - (e + t);
      y += one;
      y = y * twopk;
    }
  }
  return y;
}

double cbrt(double x) {
  static const uint32_t
      B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
      B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */

  /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
  static const double P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
      P1 = -1.88497979543377169875,                /* 0xbffe28e0, 0x92f02420 */
      P2 = 1.621429720105354466140,                /* 0x3ff9f160, 0x4a49d6c2 */
      P3 = -0.758397934778766047437,               /* 0xbfe844cb, 0xbee751d9 */
      P4 = 0.145996192886612446982;                /* 0x3fc2b000, 0xd4e4edd7 */

  int32_t hx;
  union {
    double value;
    uint64_t bits;
  } u;
  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 divison 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.
   */
  u.value = t;
  u.bits = (u.bits + 0x80000000) & 0xffffffffc0000000ULL;
  t = u.value;

  /* one step Newton iteration to 53 bits with error < 0.667 ulps */
  s = t * t;             /* t*t is exact */
  r = x / s;             /* error <= 0.5 ulps; |r| < |t| */
  w = t + t;             /* t+t is exact */
  r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
  t = t + t * r;         /* error <= 0.5 + 0.5/3 + epsilon */

  return (t);
}

/* sin(x)
 * Return sine function of x.
 *
 * kernel function:
 *      __kernel_sin            ... sine function on [-pi/4,pi/4]
 *      __kernel_cos            ... cose function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 *
 * Method.
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */
double sin(double x) {
  double y[2], z = 0.0;
  int32_t n, ix;

  /* High word of x. */
  GET_HIGH_WORD(ix, x);

  /* |x| ~< pi/4 */
  ix &= 0x7fffffff;
  if (ix <= 0x3fe921fb) {
    return __kernel_sin(x, z, 0);
  } else if (ix >= 0x7ff00000) {
    /* sin(Inf or NaN) is NaN */
    return x - x;
  } else {
    /* argument reduction needed */
    n = __ieee754_rem_pio2(x, y);
    switch (n & 3) {
      case 0:
        return __kernel_sin(y[0], y[1], 1);
      case 1:
        return __kernel_cos(y[0], y[1]);
      case 2:
        return -__kernel_sin(y[0], y[1], 1);
      default:
        return -__kernel_cos(y[0], y[1]);
    }
  }
}

/* tan(x)
 * Return tangent function of x.
 *
 * kernel function:
 *      __kernel_tan            ... tangent function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 *
 * Method.
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */
double tan(double x) {
  double y[2], z = 0.0;
  int32_t n, ix;

  /* High word of x. */
  GET_HIGH_WORD(ix, x);

  /* |x| ~< pi/4 */
  ix &= 0x7fffffff;
  if (ix <= 0x3fe921fb) {
    return __kernel_tan(x, z, 1);
  } else if (ix >= 0x7ff00000) {
    /* tan(Inf or NaN) is NaN */
    return x - x; /* NaN */
  } else {
    /* argument reduction needed */
    n = __ieee754_rem_pio2(x, y);
    /* 1 -> n even, -1 -> n odd */
    return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1));
  }
}

}  // namespace ieee754
}  // namespace base
}  // namespace v8