ieee754.cc 101 KB
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// The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// The original source code covered by the above license above has been
// modified significantly by Google Inc.
// Copyright 2016 the V8 project authors. All rights reserved.

#include "src/base/ieee754.h"

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#include <cmath>
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#include <limits>

#include "src/base/build_config.h"
#include "src/base/macros.h"
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#include "src/base/overflowing-math.h"
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namespace v8 {
namespace base {
namespace ieee754 {

namespace {

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/* 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.
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 * TODO(jkummerow): This is undefined behavior. Use bit_cast instead.
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 */

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#if V8_TARGET_LITTLE_ENDIAN
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union ieee_double_shape_type {
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  double value;
  struct {
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    uint32_t lsw;
    uint32_t msw;
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  } parts;
  struct {
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    uint64_t w;
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  } xparts;
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};
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#else
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union ieee_double_shape_type {
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  double value;
  struct {
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    uint32_t msw;
    uint32_t lsw;
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  } parts;
  struct {
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    uint64_t w;
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  } xparts;
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};
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#endif

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/* Get two 32 bit ints from a double.  */
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#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;        \
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  } while (false)
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/* 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;        \
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  } while (false)
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/* 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;        \
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  } while (false)
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/* 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;        \
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  } while (false)
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/* 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;             \
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  } while (false)
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/* 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;            \
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  } while (false)
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/* 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;            \
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  } while (false)
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/* 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;            \
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  } while (false)
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/* Support macro. */

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

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int32_t __ieee754_rem_pio2(double x, double* y) V8_WARN_UNUSED_RESULT;
double __kernel_cos(double x, double y) V8_WARN_UNUSED_RESULT;
int __kernel_rem_pio2(double* x, double* y, int e0, int nx, int prec,
                      const int32_t* ipio2) V8_WARN_UNUSED_RESULT;
double __kernel_sin(double x, double y, int iy) V8_WARN_UNUSED_RESULT;
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/* __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;
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  uint32_t low;
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  z = 0;
  GET_HIGH_WORD(hx, x); /* high word of x */
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  ix = hx & 0x7FFFFFFF;
  if (ix <= 0x3FE921FB) { /* |x| ~<= pi/4 , no need for reduction */
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    y[0] = x;
    y[1] = 0;
    return 0;
  }
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  if (ix < 0x4002D97C) { /* |x| < 3pi/4, special case with n=+-1 */
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    if (hx > 0) {
      z = x - pio2_1;
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      if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */
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        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;
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      if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */
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        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;
    }
  }
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  if (ix <= 0x413921FB) { /* |x| ~<= 2^19*(pi/2), medium size */
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    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 {
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      uint32_t high;
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      j = ix >> 20;
      y[0] = r - w;
      GET_HIGH_WORD(high, y[0]);
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      i = j - ((high >> 20) & 0x7FF);
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      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]);
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        i = j - ((high >> 20) & 0x7FF);
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        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
   */
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  if (ix >= 0x7FF00000) { /* x is inf or NaN */
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    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; */
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  SET_HIGH_WORD(z, ix - static_cast<int32_t>(static_cast<uint32_t>(e0) << 20));
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  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.
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 *      2. if x < 2^-27 (hx<0x3E400000 0), return 1 with inexact if x!=0.
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 *      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 */

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  double a, iz, z, r, qx;
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  int32_t ix;
  GET_HIGH_WORD(ix, x);
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  ix &= 0x7FFFFFFF;                           /* ix = |x|'s high word*/
  if (ix < 0x3E400000) {                      /* if x < 2**27 */
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    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 {
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    if (ix > 0x3FE90000) { /* x > 0.78125 */
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      qx = 0.28125;
    } else {
      INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
    }
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    iz = 0.5 * z - qx;
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    a = one - qx;
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    return a - (iz - (z * r - x * y));
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  }
}

/* __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 {
590
        iq[i] = 0xFFFFFF - j;
591 592 593 594 595
      }
    }
    if (q0 > 0) { /* rare case: chance is 1 in 12 */
      switch (q0) {
        case 1:
596
          iq[jz - 1] &= 0x7FFFFF;
597 598
          break;
        case 2:
599
          iq[jz - 1] &= 0x3FFFFF;
600 601 602 603 604 605 606 607 608 609 610 611 612 613
          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 */
614
      for (k = 1; jk >= k && iq[jk - k] == 0; k++) {
615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710
        /* 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.
711
 *      2. if x < 2^-27 (hx<0x3E400000 0), return x with inexact if x!=0.
712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742
 *      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);
743 744
  ix &= 0x7FFFFFFF;      /* high word of x */
  if (ix < 0x3E400000) { /* |x| < 2**-27 */
745 746 747 748 749 750 751 752 753 754 755 756
    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);
  }
}

757 758 759 760 761 762 763 764 765
/* __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.
766
 *      2. if x < 2^-28 (hx<0x3E300000 0), return x with inexact if x!=0.
767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817
 *      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 */
818 819
  ix = hx & 0x7FFFFFFF;             /* high word of |x| */
  if (ix < 0x3E300000) {            /* x < 2**-28 */
820
    if (static_cast<int>(x) == 0) { /* generate inexact */
821
      uint32_t low;
822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893
      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
}

894 895
}  // namespace

896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938
/* acos(x)
 * Method :
 *      acos(x)  = pi/2 - asin(x)
 *      acos(-x) = pi/2 + asin(x)
 * For |x|<=0.5
 *      acos(x) = pi/2 - (x + x*x^2*R(x^2))     (see asin.c)
 * For x>0.5
 *      acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
 *              = 2asin(sqrt((1-x)/2))
 *              = 2s + 2s*z*R(z)        ...z=(1-x)/2, s=sqrt(z)
 *              = 2f + (2c + 2s*z*R(z))
 *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
 *     for f so that f+c ~ sqrt(z).
 * For x<-0.5
 *      acos(x) = pi - 2asin(sqrt((1-|x|)/2))
 *              = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
 *
 * Special cases:
 *      if x is NaN, return x itself;
 *      if |x|>1, return NaN with invalid signal.
 *
 * Function needed: sqrt
 */
double acos(double x) {
  static const double
      one = 1.00000000000000000000e+00,     /* 0x3FF00000, 0x00000000 */
      pi = 3.14159265358979311600e+00,      /* 0x400921FB, 0x54442D18 */
      pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
      pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
      pS0 = 1.66666666666666657415e-01,     /* 0x3FC55555, 0x55555555 */
      pS1 = -3.25565818622400915405e-01,    /* 0xBFD4D612, 0x03EB6F7D */
      pS2 = 2.01212532134862925881e-01,     /* 0x3FC9C155, 0x0E884455 */
      pS3 = -4.00555345006794114027e-02,    /* 0xBFA48228, 0xB5688F3B */
      pS4 = 7.91534994289814532176e-04,     /* 0x3F49EFE0, 0x7501B288 */
      pS5 = 3.47933107596021167570e-05,     /* 0x3F023DE1, 0x0DFDF709 */
      qS1 = -2.40339491173441421878e+00,    /* 0xC0033A27, 0x1C8A2D4B */
      qS2 = 2.02094576023350569471e+00,     /* 0x40002AE5, 0x9C598AC8 */
      qS3 = -6.88283971605453293030e-01,    /* 0xBFE6066C, 0x1B8D0159 */
      qS4 = 7.70381505559019352791e-02;     /* 0x3FB3B8C5, 0xB12E9282 */

  double z, p, q, r, w, s, c, df;
  int32_t hx, ix;
  GET_HIGH_WORD(hx, x);
939 940
  ix = hx & 0x7FFFFFFF;
  if (ix >= 0x3FF00000) { /* |x| >= 1 */
941 942
    uint32_t lx;
    GET_LOW_WORD(lx, x);
943
    if (((ix - 0x3FF00000) | lx) == 0) { /* |x|==1 */
944 945 946 947 948
      if (hx > 0)
        return 0.0; /* acos(1) = 0  */
      else
        return pi + 2.0 * pio2_lo; /* acos(-1)= pi */
    }
949
    return std::numeric_limits<double>::signaling_NaN();  // acos(|x|>1) is NaN
950
  }
951 952
  if (ix < 0x3FE00000) {                            /* |x| < 0.5 */
    if (ix <= 0x3C600000) return pio2_hi + pio2_lo; /*if|x|<2**-57*/
953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000
    z = x * x;
    p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
    q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
    r = p / q;
    return pio2_hi - (x - (pio2_lo - x * r));
  } else if (hx < 0) { /* x < -0.5 */
    z = (one + x) * 0.5;
    p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
    q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
    s = sqrt(z);
    r = p / q;
    w = r * s - pio2_lo;
    return pi - 2.0 * (s + w);
  } else { /* x > 0.5 */
    z = (one - x) * 0.5;
    s = sqrt(z);
    df = s;
    SET_LOW_WORD(df, 0);
    c = (z - df * df) / (s + df);
    p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
    q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
    r = p / q;
    w = r * s + c;
    return 2.0 * (df + w);
  }
}

/* acosh(x)
 * Method :
 *      Based on
 *              acosh(x) = log [ x + sqrt(x*x-1) ]
 *      we have
 *              acosh(x) := log(x)+ln2, if x is large; else
 *              acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
 *              acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
 *
 * Special cases:
 *      acosh(x) is NaN with signal if x<1.
 *      acosh(NaN) is NaN without signal.
 */
double acosh(double x) {
  static const double
      one = 1.0,
      ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
  double t;
  int32_t hx;
  uint32_t lx;
  EXTRACT_WORDS(hx, lx, x);
1001
  if (hx < 0x3FF00000) { /* x < 1 */
1002
    return std::numeric_limits<double>::signaling_NaN();
1003 1004
  } else if (hx >= 0x41B00000) { /* x > 2**28 */
    if (hx >= 0x7FF00000) {      /* x is inf of NaN */
1005 1006 1007 1008
      return x + x;
    } else {
      return log(x) + ln2; /* acosh(huge)=log(2x) */
    }
1009
  } else if (((hx - 0x3FF00000) | lx) == 0) {
1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071
    return 0.0;                 /* acosh(1) = 0 */
  } else if (hx > 0x40000000) { /* 2**28 > x > 2 */
    t = x * x;
    return log(2.0 * x - one / (x + sqrt(t - one)));
  } else { /* 1<x<2 */
    t = x - one;
    return log1p(t + sqrt(2.0 * t + t * t));
  }
}

/* asin(x)
 * Method :
 *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
 *      we approximate asin(x) on [0,0.5] by
 *              asin(x) = x + x*x^2*R(x^2)
 *      where
 *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
 *      and its remez error is bounded by
 *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
 *
 *      For x in [0.5,1]
 *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
 *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
 *      then for x>0.98
 *              asin(x) = pi/2 - 2*(s+s*z*R(z))
 *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
 *      For x<=0.98, let pio4_hi = pio2_hi/2, then
 *              f = hi part of s;
 *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
 *      and
 *              asin(x) = pi/2 - 2*(s+s*z*R(z))
 *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
 *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
 *
 * Special cases:
 *      if x is NaN, return x itself;
 *      if |x|>1, return NaN with invalid signal.
 */
double asin(double x) {
  static const double
      one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
      huge = 1.000e+300,
      pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
      pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
      pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
                                            /* coefficient for R(x^2) */
      pS0 = 1.66666666666666657415e-01,     /* 0x3FC55555, 0x55555555 */
      pS1 = -3.25565818622400915405e-01,    /* 0xBFD4D612, 0x03EB6F7D */
      pS2 = 2.01212532134862925881e-01,     /* 0x3FC9C155, 0x0E884455 */
      pS3 = -4.00555345006794114027e-02,    /* 0xBFA48228, 0xB5688F3B */
      pS4 = 7.91534994289814532176e-04,     /* 0x3F49EFE0, 0x7501B288 */
      pS5 = 3.47933107596021167570e-05,     /* 0x3F023DE1, 0x0DFDF709 */
      qS1 = -2.40339491173441421878e+00,    /* 0xC0033A27, 0x1C8A2D4B */
      qS2 = 2.02094576023350569471e+00,     /* 0x40002AE5, 0x9C598AC8 */
      qS3 = -6.88283971605453293030e-01,    /* 0xBFE6066C, 0x1B8D0159 */
      qS4 = 7.70381505559019352791e-02;     /* 0x3FB3B8C5, 0xB12E9282 */

  double t, w, p, q, c, r, s;
  int32_t hx, ix;

  t = 0;
  GET_HIGH_WORD(hx, x);
1072 1073
  ix = hx & 0x7FFFFFFF;
  if (ix >= 0x3FF00000) { /* |x|>= 1 */
1074 1075
    uint32_t lx;
    GET_LOW_WORD(lx, x);
1076
    if (((ix - 0x3FF00000) | lx) == 0) { /* asin(1)=+-pi/2 with inexact */
1077
      return x * pio2_hi + x * pio2_lo;
1078 1079
    }
    return std::numeric_limits<double>::signaling_NaN();  // asin(|x|>1) is NaN
1080 1081
  } else if (ix < 0x3FE00000) {     /* |x|<0.5 */
    if (ix < 0x3E400000) {          /* if |x| < 2**-27 */
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      if (huge + x > one) return x; /* return x with inexact if x!=0*/
    } else {
      t = x * x;
    }
    p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
    q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
    w = p / q;
    return x + x * w;
  }
  /* 1> |x|>= 0.5 */
  w = one - fabs(x);
  t = w * 0.5;
  p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
  q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
  s = sqrt(t);
  if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
    w = p / q;
    t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
  } else {
    w = s;
    SET_LOW_WORD(w, 0);
    c = (t - w * w) / (s + w);
    r = p / q;
    p = 2.0 * s * r - (pio2_lo - 2.0 * c);
    q = pio4_hi - 2.0 * w;
    t = pio4_hi - (p - q);
  }
  if (hx > 0)
    return t;
  else
    return -t;
}
/* asinh(x)
 * Method :
 *      Based on
 *              asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
 *      we have
 *      asinh(x) := x  if  1+x*x=1,
 *               := sign(x)*(log(x)+ln2)) for large |x|, else
 *               := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
 *               := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
 */
double asinh(double x) {
  static const double
      one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
      ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
      huge = 1.00000000000000000000e+300;

  double t, w;
  int32_t hx, ix;
  GET_HIGH_WORD(hx, x);
1133 1134 1135
  ix = hx & 0x7FFFFFFF;
  if (ix >= 0x7FF00000) return x + x; /* x is inf or NaN */
  if (ix < 0x3E300000) {              /* |x|<2**-28 */
1136 1137
    if (huge + x > one) return x;     /* return x inexact except 0 */
  }
1138
  if (ix > 0x41B00000) { /* |x| > 2**28 */
1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153
    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;
  }
}

1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207
/* 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);
1208
  ix = hx & 0x7FFFFFFF;
1209
  if (ix >= 0x44100000) { /* if |x| >= 2^66 */
1210
    uint32_t low;
1211
    GET_LOW_WORD(low, x);
1212
    if (ix > 0x7FF00000 || (ix == 0x7FF00000 && (low != 0)))
1213 1214
      return x + x; /* NaN */
    if (hx > 0)
1215
      return atanhi[3] + *const_cast<volatile double*>(&atanlo[3]);
1216
    else
1217
      return -atanhi[3] - *const_cast<volatile double*>(&atanlo[3]);
1218
  }
1219 1220
  if (ix < 0x3FDC0000) {            /* |x| < 0.4375 */
    if (ix < 0x3E400000) {          /* |x| < 2^-27 */
1221 1222 1223 1224 1225
      if (huge + x > one) return x; /* raise inexact */
    }
    id = -1;
  } else {
    x = fabs(x);
1226 1227
    if (ix < 0x3FF30000) {   /* |x| < 1.1875 */
      if (ix < 0x3FE60000) { /* 7/16 <=|x|<11/16 */
1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296
        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;
1297
  uint32_t lx, ly;
1298 1299

  EXTRACT_WORDS(hx, lx, x);
1300
  ix = hx & 0x7FFFFFFF;
1301
  EXTRACT_WORDS(hy, ly, y);
1302
  iy = hy & 0x7FFFFFFF;
1303 1304
  if (((ix | ((lx | NegateWithWraparound<int32_t>(lx)) >> 31)) > 0x7FF00000) ||
      ((iy | ((ly | NegateWithWraparound<int32_t>(ly)) >> 31)) > 0x7FF00000)) {
1305 1306
    return x + y; /* x or y is NaN */
  }
1307 1308 1309
  if ((SubWithWraparound(hx, 0x3FF00000) | lx) == 0) {
    return atan(y); /* x=1.0 */
  }
1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327
  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 */
1328 1329
  if (ix == 0x7FF00000) {
    if (iy == 0x7FF00000) {
1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353
      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 */
1354
  if (iy == 0x7FF00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
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  /* 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(-,-) */
  }
}

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/* 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 */
1416 1417
  ix &= 0x7FFFFFFF;
  if (ix <= 0x3FE921FB) {
1418
    return __kernel_cos(x, z);
1419
  } else if (ix >= 0x7FF00000) {
1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437
    /* 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);
    }
  }
}

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/* 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 */
1505 1506 1507 1508 1509 1510
      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 */
1511 1512 1513 1514
      P1 = 1.66666666666666019037e-01,           /* 0x3FC55555, 0x5555553E */
      P2 = -2.77777777770155933842e-03,          /* 0xBF66C16C, 0x16BEBD93 */
      P3 = 6.61375632143793436117e-05,           /* 0x3F11566A, 0xAF25DE2C */
      P4 = -1.65339022054652515390e-06,          /* 0xBEBBBD41, 0xC5D26BF1 */
1515
      P5 = 4.13813679705723846039e-08,           /* 0x3E663769, 0x72BEA4D0 */
1516
      E = 2.718281828459045;                     /* 0x4005BF0A, 0x8B145769 */
1517 1518 1519 1520 1521 1522 1523 1524

  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;
1525
  uint32_t hx;
1526 1527 1528

  GET_HIGH_WORD(hx, x);
  xsb = (hx >> 31) & 1; /* sign bit of x */
1529
  hx &= 0x7FFFFFFF;     /* high word of |x| */
1530 1531 1532

  /* filter out non-finite argument */
  if (hx >= 0x40862E42) { /* if |x|>=709.78... */
1533
    if (hx >= 0x7FF00000) {
1534
      uint32_t lx;
1535
      GET_LOW_WORD(lx, x);
1536
      if (((hx & 0xFFFFF) | lx) != 0)
1537 1538 1539 1540 1541 1542 1543 1544 1545
        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 */
1546
  if (hx > 0x3FD62E42) {   /* if  |x| > 0.5 ln2 */
1547
    if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
1548 1549 1550 1551 1552
      /* 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;
1553 1554 1555 1556 1557 1558 1559 1560 1561 1562
      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);
1563
  } else if (hx < 0x3E300000) {         /* when |x|<2**-28 */
1564 1565 1566 1567 1568 1569 1570 1571
    if (huge + x > one) return one + x; /* trigger inexact */
  } else {
    k = 0;
  }

  /* x is now in primary range */
  t = x * x;
  if (k >= -1021) {
1572 1573 1574
    INSERT_WORDS(
        twopk,
        0x3FF00000 + static_cast<int32_t>(static_cast<uint32_t>(k) << 20), 0);
1575
  } else {
1576 1577
    INSERT_WORDS(twopk, 0x3FF00000 + (static_cast<uint32_t>(k + 1000) << 20),
                 0);
1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592
  }
  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;
  }
}

1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615
/*
 * 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;
1616
  uint32_t lx;
1617
  EXTRACT_WORDS(hx, lx, x);
1618
  ix = hx & 0x7FFFFFFF;
1619 1620 1621 1622 1623 1624 1625 1626
  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();
  }
1627
  if (ix < 0x3E300000 && (huge + x) > zero) return x; /* x<2**-28 */
1628
  SET_HIGH_WORD(x, ix);
1629
  if (ix < 0x3FE00000) { /* x < 0.5 */
1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640
    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;
}

1641 1642 1643 1644 1645
/* log(x)
 * Return the logrithm of x
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
1646
 *     x = 2^k * (1+f),
1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691
 *     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) {
1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707
  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;
1708
  uint32_t lx;
1709

1710 1711 1712
  EXTRACT_WORDS(hx, lx, x);

  k = 0;
1713
  if (hx < 0x00100000) { /* x < 2**-1022  */
1714 1715 1716 1717 1718 1719
    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 */
    }
1720
    k -= 54;
1721 1722
    x *= two54; /* subnormal number, scale up x */
    GET_HIGH_WORD(hx, x);
1723
  }
1724
  if (hx >= 0x7FF00000) return x + x;
1725
  k += (hx >> 20) - 1023;
1726 1727 1728
  hx &= 0x000FFFFF;
  i = (hx + 0x95F64) & 0x100000;
  SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */
1729 1730
  k += (i >> 20);
  f = x - 1.0;
1731
  if ((0x000FFFFF & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
1732
    if (f == zero) {
1733
      if (k == 0) {
1734
        return zero;
1735 1736
      } else {
        dk = static_cast<double>(k);
1737
        return dk * ln2_hi + dk * ln2_lo;
1738 1739
      }
    }
1740
    R = f * f * (0.5 - 0.33333333333333333 * f);
1741
    if (k == 0) {
1742
      return f - R;
1743 1744
    } else {
      dk = static_cast<double>(k);
1745
      return dk * ln2_hi - ((R - dk * ln2_lo) - f);
1746 1747 1748 1749 1750
    }
  }
  s = f / (2.0 + f);
  dk = static_cast<double>(k);
  z = s * s;
1751
  i = hx - 0x6147A;
1752
  w = z * z;
1753
  j = 0x6B851 - hx;
1754 1755
  t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
  t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
1756
  i |= j;
1757
  R = t2 + t1;
1758 1759
  if (i > 0) {
    hfsq = 0.5 * f * f;
1760 1761 1762 1763
    if (k == 0)
      return f - (hfsq - s * (hfsq + R));
    else
      return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
1764
  } else {
1765 1766 1767 1768
    if (k == 0)
      return f - s * (f - R);
    else
      return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
1769 1770 1771
  }
}

1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854
/* double log1p(double x)
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *      1+x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *  may not be representable exactly. In that case, a correction
 *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *  and add back the correction term c/u.
 *  (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log1p(f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
 *    (the values of Lp1 to Lp7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log1p(f) = f - (hfsq - s*(hfsq+R)).
 *
 *  3. Finally, log1p(x) = k*ln2 + log1p(f).
 *           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *  log1p(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 *
 * Note: Assuming log() return accurate answer, the following
 *   algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *    u = 1+x;
 *    if(u==1.0) return x ; else
 *         return log(u)*(x/(u-1.0));
 *
 *   See HP-15C Advanced Functions Handbook, p.193.
 */
double log1p(double x) {
  static const double                      /* -- */
      ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
      ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
      two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
      Lp1 = 6.666666666666735130e-01,      /* 3FE55555 55555593 */
      Lp2 = 3.999999999940941908e-01,      /* 3FD99999 9997FA04 */
      Lp3 = 2.857142874366239149e-01,      /* 3FD24924 94229359 */
      Lp4 = 2.222219843214978396e-01,      /* 3FCC71C5 1D8E78AF */
      Lp5 = 1.818357216161805012e-01,      /* 3FC74664 96CB03DE */
      Lp6 = 1.531383769920937332e-01,      /* 3FC39A09 D078C69F */
      Lp7 = 1.479819860511658591e-01;      /* 3FC2F112 DF3E5244 */

  static const double zero = 0.0;

  double hfsq, f, c, s, z, R, u;
  int32_t k, hx, hu, ax;

  GET_HIGH_WORD(hx, x);
1855
  ax = hx & 0x7FFFFFFF;
1856 1857 1858

  k = 1;
  if (hx < 0x3FDA827A) {    /* 1+x < sqrt(2)+ */
1859
    if (ax >= 0x3FF00000) { /* x <= -1.0 */
1860
      if (x == -1.0)
1861
        return -std::numeric_limits<double>::infinity(); /* log1p(-1)=+inf */
1862
      else
1863
        return std::numeric_limits<double>::signaling_NaN();  // log1p(x<-1)=NaN
1864
    }
1865
    if (ax < 0x3E200000) {    /* |x| < 2**-29 */
1866
      if (two54 + x > zero    /* raise inexact */
1867
          && ax < 0x3C900000) /* |x| < 2**-54 */
1868 1869 1870 1871
        return x;
      else
        return x - x * x * 0.5;
    }
1872
    if (hx > 0 || hx <= static_cast<int32_t>(0xBFD2BEC4)) {
1873 1874 1875 1876 1877
      k = 0;
      f = x;
      hu = 1;
    } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
  }
1878
  if (hx >= 0x7FF00000) return x + x;
1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891
  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;
    }
1892
    hu &= 0x000FFFFF;
1893 1894 1895 1896 1897 1898 1899
    /*
     * 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.
     */
1900 1901
    if (hu < 0x6A09E) {                  /* u ~< sqrt(2) */
      SET_HIGH_WORD(u, hu | 0x3FF00000); /* normalize u */
1902 1903
    } else {
      k += 1;
1904
      SET_HIGH_WORD(u, hu | 0x3FE00000); /* normalize u/2 */
1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934
      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);
}

1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990
/*
 * 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.
 */
1991

1992 1993 1994 1995 1996 1997 1998
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 */
1999

2000 2001 2002 2003 2004 2005
/*
 * 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;
2006

2007 2008 2009 2010 2011 2012 2013 2014 2015
  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);
}
2016

2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028
/*
 * 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 */
2029 2030
      ivln2hi = 1.44269504072144627571e+00, /* 0x3FF71547, 0x65200000 */
      ivln2lo = 1.67517131648865118353e-10; /* 0x3DE705FC, 0x2EEFA200 */
2031

2032 2033
  double f, hfsq, hi, lo, r, val_hi, val_lo, w, y;
  int32_t i, k, hx;
2034
  uint32_t lx;
2035

2036
  EXTRACT_WORDS(hx, lx, x);
2037

2038 2039
  k = 0;
  if (hx < 0x00100000) { /* x < 2**-1022  */
2040 2041 2042 2043 2044 2045
    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 */
    }
2046 2047 2048
    k -= 54;
    x *= two54; /* subnormal number, scale up x */
    GET_HIGH_WORD(hx, x);
2049
  }
2050
  if (hx >= 0x7FF00000) return x + x;
2051
  if (hx == 0x3FF00000 && lx == 0) return 0.0; /* log(1) = +0 */
2052
  k += (hx >> 20) - 1023;
2053 2054 2055
  hx &= 0x000FFFFF;
  i = (hx + 0x95F64) & 0x100000;
  SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */
2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103
  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;
2104 2105
}

2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131
/*
 * 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.
 */
2132 2133 2134 2135 2136 2137 2138 2139 2140
double log10(double x) {
  static const double
      two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
      ivln10 = 4.34294481903251816668e-01,
      log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
      log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */

  double y;
  int32_t i, k, hx;
2141
  uint32_t lx;
2142 2143 2144 2145 2146

  EXTRACT_WORDS(hx, lx, x);

  k = 0;
  if (hx < 0x00100000) { /* x < 2**-1022  */
2147 2148 2149 2150 2151 2152
    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 */
    }
2153 2154 2155 2156 2157
    k -= 54;
    x *= two54; /* subnormal number, scale up x */
    GET_HIGH_WORD(hx, x);
    GET_LOW_WORD(lx, x);
  }
2158
  if (hx >= 0x7FF00000) return x + x;
2159
  if (hx == 0x3FF00000 && lx == 0) return 0.0; /* log(1) = +0 */
2160 2161 2162
  k += (hx >> 20) - 1023;

  i = (k & 0x80000000) >> 31;
2163
  hx = (hx & 0x000FFFFF) | ((0x3FF - i) << 20);
2164 2165 2166 2167 2168 2169 2170 2171
  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;
}

2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270
/* 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 */
2271 2272 2273
      ln2_hi = 6.93147180369123816490e-01,      /* 0x3FE62E42, 0xFEE00000 */
      ln2_lo = 1.90821492927058770002e-10,      /* 0x3DEA39EF, 0x35793C76 */
      invln2 = 1.44269504088896338700e+00,      /* 0x3FF71547, 0x652B82FE */
2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285
      /* 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;
2286
  uint32_t hx;
2287 2288 2289

  GET_HIGH_WORD(hx, x);
  xsb = hx & 0x80000000; /* sign bit of x */
2290
  hx &= 0x7FFFFFFF;      /* high word of |x| */
2291 2292 2293 2294

  /* filter out huge and non-finite argument */
  if (hx >= 0x4043687A) {   /* if |x|>=56*ln2 */
    if (hx >= 0x40862E42) { /* if |x|>=709.78... */
2295
      if (hx >= 0x7FF00000) {
2296
        uint32_t low;
2297
        GET_LOW_WORD(low, x);
2298
        if (((hx & 0xFFFFF) | low) != 0)
2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311
          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 */
2312
  if (hx > 0x3FD62E42) {   /* if  |x| > 0.5 ln2 */
2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330
    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;
2331
  } else if (hx < 0x3C900000) { /* when |x|<2**-54, return x */
2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343 2344 2345 2346
    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 {
2347 2348 2349 2350
    INSERT_WORDS(
        twopk,
        0x3FF00000 + static_cast<int32_t>(static_cast<uint32_t>(k) << 20),
        0); /* 2^k */
2351 2352 2353 2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370 2371 2372
    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) {
2373
      SET_HIGH_WORD(t, 0x3FF00000 - (0x200000 >> k)); /* t=1-2^-k */
2374 2375 2376
      y = t - (e - x);
      y = y * twopk;
    } else {
2377
      SET_HIGH_WORD(t, ((0x3FF - k) << 20)); /* 2^-k */
2378 2379 2380 2381 2382 2383 2384 2385 2386
      y = x - (e + t);
      y += one;
      y = y * twopk;
    }
  }
  return y;
}

double cbrt(double x) {
2387
  static const uint32_t
2388 2389 2390 2391
      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]). */
2392 2393 2394 2395 2396
  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 */
2397 2398 2399 2400 2401 2402 2403

  int32_t hx;
  union {
    double value;
    uint64_t bits;
  } u;
  double r, s, t = 0.0, w;
2404 2405
  uint32_t sign;
  uint32_t high, low;
2406 2407 2408 2409

  EXTRACT_WORDS(hx, low, x);
  sign = hx & 0x80000000; /* sign= sign(x) */
  hx ^= sign;
2410
  if (hx >= 0x7FF00000) return (x + x); /* cbrt(NaN,INF) is itself */
2411 2412 2413 2414 2415 2416 2417 2418 2419 2420

  /*
   * 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,
2421
   * ordinary integer division of the value in bits magically gives
2422 2423 2424 2425 2426 2427 2428 2429 2430 2431
   * 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);
2432
    INSERT_WORDS(t, sign | ((high & 0x7FFFFFFF) / 3 + B2), 0);
2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460
  } 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;
2461
  u.bits = (u.bits + 0x80000000) & 0xFFFFFFFFC0000000ULL;
2462 2463 2464 2465 2466 2467 2468 2469 2470 2471 2472 2473
  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);
}

2474 2475 2476 2477 2478 2479 2480 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496 2497 2498 2499 2500 2501 2502 2503 2504 2505 2506 2507 2508 2509 2510 2511
/* 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 */
2512 2513
  ix &= 0x7FFFFFFF;
  if (ix <= 0x3FE921FB) {
2514
    return __kernel_sin(x, z, 0);
2515
  } else if (ix >= 0x7FF00000) {
2516 2517 2518 2519 2520 2521 2522 2523 2524 2525 2526 2527 2528 2529 2530 2531 2532 2533
    /* 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]);
    }
  }
}

2534 2535 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 2568 2569 2570
/* 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 */
2571 2572
  ix &= 0x7FFFFFFF;
  if (ix <= 0x3FE921FB) {
2573
    return __kernel_tan(x, z, 1);
2574
  } else if (ix >= 0x7FF00000) {
2575 2576 2577 2578 2579 2580 2581 2582 2583 2584
    /* 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));
  }
}

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/*
 * ES6 draft 09-27-13, section 20.2.2.12.
 * Math.cosh
 * Method :
 * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
 *      1. Replace x by |x| (cosh(x) = cosh(-x)).
 *      2.
 *                                                      [ exp(x) - 1 ]^2
 *          0        <= x <= ln2/2  :  cosh(x) := 1 + -------------------
 *                                                         2*exp(x)
 *
 *                                                 exp(x) + 1/exp(x)
 *          ln2/2    <= x <= 22     :  cosh(x) := -------------------
 *                                                        2
 *          22       <= x <= lnovft :  cosh(x) := exp(x)/2
 *          lnovft   <= x <= ln2ovft:  cosh(x) := exp(x/2)/2 * exp(x/2)
 *          ln2ovft  <  x           :  cosh(x) := huge*huge (overflow)
 *
 * Special cases:
 *      cosh(x) is |x| if x is +INF, -INF, or NaN.
 *      only cosh(0)=1 is exact for finite x.
 */
double cosh(double x) {
  static const double KCOSH_OVERFLOW = 710.4758600739439;
  static const double one = 1.0, half = 0.5;
  static volatile double huge = 1.0e+300;

  int32_t ix;

  /* High word of |x|. */
  GET_HIGH_WORD(ix, x);
2616
  ix &= 0x7FFFFFFF;
2617 2618

  // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
2619
  if (ix < 0x3FD62E43) {
2620 2621 2622
    double t = expm1(fabs(x));
    double w = one + t;
    // For |x| < 2^-55, cosh(x) = 1
2623
    if (ix < 0x3C800000) return w;
2624 2625 2626 2627 2628 2629 2630 2631 2632 2633
    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|)
2634
  if (ix < 0x40862E42) return half * exp(fabs(x));
2635 2636 2637 2638 2639 2640 2641 2642 2643

  // |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 */
2644
  if (ix >= 0x7FF00000) return x * x;
2645 2646 2647 2648 2649

  // |x| > overflowthreshold.
  return huge * huge;
}

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/*
 * ES2019 Draft 2019-01-02 12.6.4
 * Math.pow & Exponentiation Operator
 *
 * Return X raised to the Yth power
 *
 * Method:
 *     Let x =  2   * (1+f)
 *     1. Compute and return log2(x) in two pieces:
 *        log2(x) = w1 + w2,
 *        where w1 has 53-24 = 29 bit trailing zeros.
 *     2. Perform y*log2(x) = n+y' by simulating muti-precision
 *        arithmetic, where |y'|<=0.5.
 *     3. Return x**y = 2**n*exp(y'*log2)
 *
 * Special cases:
 *     1.  (anything) ** 0  is 1
 *     2.  (anything) ** 1  is itself
 *     3.  (anything) ** NAN is NAN
 *     4.  NAN ** (anything except 0) is NAN
 *     5.  +-(|x| > 1) **  +INF is +INF
 *     6.  +-(|x| > 1) **  -INF is +0
 *     7.  +-(|x| < 1) **  +INF is +0
 *     8.  +-(|x| < 1) **  -INF is +INF
 *     9.  +-1         ** +-INF is NAN
 *     10. +0 ** (+anything except 0, NAN)               is +0
 *     11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 *     12. +0 ** (-anything except 0, NAN)               is +INF
 *     13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 *     14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 *     15. +INF ** (+anything except 0,NAN) is +INF
 *     16. +INF ** (-anything except 0,NAN) is +0
 *     17. -INF ** (anything)  = -0 ** (-anything)
 *     18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 *     19. (-anything except 0 and inf) ** (non-integer) is NAN
 *
 * Accuracy:
 *      pow(x,y) returns x**y nearly rounded. In particular,
 *      pow(integer, integer) always returns the correct integer provided it is
 *      representable.
 *
 * Constants:
 *     The hexadecimal values are the intended ones for the following
 *     constants. The decimal values may be used, provided that the
 *     compiler will convert from decimal to binary accurately enough
 *     to produce the hexadecimal values shown.
 */

double pow(double x, double y) {
  static const double
      bp[] = {1.0, 1.5},
      dp_h[] = {0.0, 5.84962487220764160156e-01},  // 0x3FE2B803, 0x40000000
      dp_l[] = {0.0, 1.35003920212974897128e-08},  // 0x3E4CFDEB, 0x43CFD006
      zero = 0.0, one = 1.0, two = 2.0,
      two53 = 9007199254740992.0,  // 0x43400000, 0x00000000
      huge = 1.0e300, tiny = 1.0e-300,
      // poly coefs for (3/2)*(log(x)-2s-2/3*s**3
      L1 = 5.99999999999994648725e-01,      // 0x3FE33333, 0x33333303
      L2 = 4.28571428578550184252e-01,      // 0x3FDB6DB6, 0xDB6FABFF
      L3 = 3.33333329818377432918e-01,      // 0x3FD55555, 0x518F264D
      L4 = 2.72728123808534006489e-01,      // 0x3FD17460, 0xA91D4101
      L5 = 2.30660745775561754067e-01,      // 0x3FCD864A, 0x93C9DB65
      L6 = 2.06975017800338417784e-01,      // 0x3FCA7E28, 0x4A454EEF
      P1 = 1.66666666666666019037e-01,      // 0x3FC55555, 0x5555553E
      P2 = -2.77777777770155933842e-03,     // 0xBF66C16C, 0x16BEBD93
      P3 = 6.61375632143793436117e-05,      // 0x3F11566A, 0xAF25DE2C
      P4 = -1.65339022054652515390e-06,     // 0xBEBBBD41, 0xC5D26BF1
      P5 = 4.13813679705723846039e-08,      // 0x3E663769, 0x72BEA4D0
      lg2 = 6.93147180559945286227e-01,     // 0x3FE62E42, 0xFEFA39EF
      lg2_h = 6.93147182464599609375e-01,   // 0x3FE62E43, 0x00000000
      lg2_l = -1.90465429995776804525e-09,  // 0xBE205C61, 0x0CA86C39
      ovt = 8.0085662595372944372e-0017,    // -(1024-log2(ovfl+.5ulp))
      cp = 9.61796693925975554329e-01,      // 0x3FEEC709, 0xDC3A03FD =2/(3ln2)
      cp_h = 9.61796700954437255859e-01,    // 0x3FEEC709, 0xE0000000 =(float)cp
      cp_l = -7.02846165095275826516e-09,   // 0xBE3E2FE0, 0x145B01F5 =tail cp_h
      ivln2 = 1.44269504088896338700e+00,   // 0x3FF71547, 0x652B82FE =1/ln2
      ivln2_h =
          1.44269502162933349609e+00,  // 0x3FF71547, 0x60000000 =24b 1/ln2
      ivln2_l =
          1.92596299112661746887e-08;  // 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail

  double z, ax, z_h, z_l, p_h, p_l;
  double y1, t1, t2, r, s, t, u, v, w;
  int i, j, k, yisint, n;
  int hx, hy, ix, iy;
  unsigned lx, ly;

  EXTRACT_WORDS(hx, lx, x);
  EXTRACT_WORDS(hy, ly, y);
  ix = hx & 0x7fffffff;
  iy = hy & 0x7fffffff;

  /* y==zero: x**0 = 1 */
  if ((iy | ly) == 0) return one;

  /* +-NaN return x+y */
  if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 ||
      ((iy == 0x7ff00000) && (ly != 0))) {
    return x + y;
  }

  /* determine if y is an odd int when x < 0
   * yisint = 0 ... y is not an integer
   * yisint = 1 ... y is an odd int
   * yisint = 2 ... y is an even int
   */
  yisint = 0;
  if (hx < 0) {
    if (iy >= 0x43400000) {
      yisint = 2; /* even integer y */
    } else if (iy >= 0x3ff00000) {
      k = (iy >> 20) - 0x3ff; /* exponent */
      if (k > 20) {
        j = ly >> (52 - k);
        if ((j << (52 - k)) == static_cast<int>(ly)) yisint = 2 - (j & 1);
      } else if (ly == 0) {
        j = iy >> (20 - k);
        if ((j << (20 - k)) == iy) yisint = 2 - (j & 1);
      }
    }
  }

  /* special value of y */
  if (ly == 0) {
    if (iy == 0x7ff00000) { /* y is +-inf */
      if (((ix - 0x3ff00000) | lx) == 0) {
        return y - y;                /* inf**+-1 is NaN */
      } else if (ix >= 0x3ff00000) { /* (|x|>1)**+-inf = inf,0 */
        return (hy >= 0) ? y : zero;
      } else { /* (|x|<1)**-,+inf = inf,0 */
        return (hy < 0) ? -y : zero;
      }
    }
    if (iy == 0x3ff00000) { /* y is  +-1 */
      if (hy < 0) {
        return base::Divide(one, x);
      } else {
        return x;
      }
    }
    if (hy == 0x40000000) return x * x; /* y is  2 */
    if (hy == 0x3fe00000) {             /* y is  0.5 */
      if (hx >= 0) {                    /* x >= +0 */
        return sqrt(x);
      }
    }
  }

  ax = fabs(x);
  /* special value of x */
  if (lx == 0) {
    if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
      z = ax;                         /*x is +-0,+-inf,+-1*/
      if (hy < 0) z = base::Divide(one, z); /* z = (1/|x|) */
      if (hx < 0) {
        if (((ix - 0x3ff00000) | yisint) == 0) {
          /* (-1)**non-int is NaN */
          z = std::numeric_limits<double>::signaling_NaN();
        } else if (yisint == 1) {
          z = -z; /* (x<0)**odd = -(|x|**odd) */
        }
      }
      return z;
    }
  }

  n = (hx >> 31) + 1;

  /* (x<0)**(non-int) is NaN */
  if ((n | yisint) == 0) {
    return std::numeric_limits<double>::signaling_NaN();
  }

  s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  if ((n | (yisint - 1)) == 0) s = -one; /* (-ve)**(odd int) */

  /* |y| is huge */
  if (iy > 0x41e00000) {   /* if |y| > 2**31 */
    if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
      if (ix <= 0x3fefffff) return (hy < 0) ? huge * huge : tiny * tiny;
      if (ix >= 0x3ff00000) return (hy > 0) ? huge * huge : tiny * tiny;
    }
    /* over/underflow if x is not close to one */
    if (ix < 0x3fefffff) return (hy < 0) ? s * huge * huge : s * tiny * tiny;
    if (ix > 0x3ff00000) return (hy > 0) ? s * huge * huge : s * tiny * tiny;
    /* now |1-x| is tiny <= 2**-20, suffice to compute
       log(x) by x-x^2/2+x^3/3-x^4/4 */
    t = ax - one; /* t has 20 trailing zeros */
    w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
    u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
    v = t * ivln2_l - w * ivln2;
    t1 = u + v;
    SET_LOW_WORD(t1, 0);
    t2 = v - (t1 - u);
  } else {
    double ss, s2, s_h, s_l, t_h, t_l;
    n = 0;
    /* take care subnormal number */
    if (ix < 0x00100000) {
      ax *= two53;
      n -= 53;
      GET_HIGH_WORD(ix, ax);
    }
    n += ((ix) >> 20) - 0x3ff;
    j = ix & 0x000fffff;
    /* determine interval */
    ix = j | 0x3ff00000; /* normalize ix */
    if (j <= 0x3988E) {
      k = 0; /* |x|<sqrt(3/2) */
    } else if (j < 0xBB67A) {
      k = 1; /* |x|<sqrt(3)   */
    } else {
      k = 0;
      n += 1;
      ix -= 0x00100000;
    }
    SET_HIGH_WORD(ax, ix);

    /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
    u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
    v = base::Divide(one, ax + bp[k]);
    ss = u * v;
    s_h = ss;
    SET_LOW_WORD(s_h, 0);
    /* t_h=ax+bp[k] High */
    t_h = zero;
    SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
    t_l = ax - (t_h - bp[k]);
    s_l = v * ((u - s_h * t_h) - s_h * t_l);
    /* compute log(ax) */
    s2 = ss * ss;
    r = s2 * s2 *
        (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
    r += s_l * (s_h + ss);
    s2 = s_h * s_h;
    t_h = 3.0 + s2 + r;
    SET_LOW_WORD(t_h, 0);
    t_l = r - ((t_h - 3.0) - s2);
    /* u+v = ss*(1+...) */
    u = s_h * t_h;
    v = s_l * t_h + t_l * ss;
    /* 2/(3log2)*(ss+...) */
    p_h = u + v;
    SET_LOW_WORD(p_h, 0);
    p_l = v - (p_h - u);
    z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
    z_l = cp_l * p_h + p_l * cp + dp_l[k];
    /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
    t = static_cast<double>(n);
    t1 = (((z_h + z_l) + dp_h[k]) + t);
    SET_LOW_WORD(t1, 0);
    t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
  }

  /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
  y1 = y;
  SET_LOW_WORD(y1, 0);
  p_l = (y - y1) * t1 + y * t2;
  p_h = y1 * t1;
  z = p_l + p_h;
  EXTRACT_WORDS(j, i, z);
  if (j >= 0x40900000) {               /* z >= 1024 */
    if (((j - 0x40900000) | i) != 0) { /* if z > 1024 */
      return s * huge * huge;          /* overflow */
    } else {
      if (p_l + ovt > z - p_h) return s * huge * huge; /* overflow */
    }
  } else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */
    if (((j - 0xc090cc00) | i) != 0) {         /* z < -1075 */
      return s * tiny * tiny;                  /* underflow */
    } else {
      if (p_l <= z - p_h) return s * tiny * tiny; /* underflow */
    }
  }
  /*
   * compute 2**(p_h+p_l)
   */
  i = j & 0x7fffffff;
  k = (i >> 20) - 0x3ff;
  n = 0;
  if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
    n = j + (0x00100000 >> (k + 1));
    k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
    t = zero;
    SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
    n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
    if (j < 0) n = -n;
    p_h -= t;
  }
  t = p_l + p_h;
  SET_LOW_WORD(t, 0);
  u = t * lg2_h;
  v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
  z = u + v;
  w = v - (z - u);
  t = z * z;
  t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
  r = base::Divide(z * t1, (t1 - two) - (w + z * w));
  z = one - (r - z);
  GET_HIGH_WORD(j, z);
  j += static_cast<int>(static_cast<uint32_t>(n) << 20);
  if ((j >> 20) <= 0) {
    z = scalbn(z, n); /* subnormal output */
  } else {
    int tmp;
    GET_HIGH_WORD(tmp, z);
    SET_HIGH_WORD(z, tmp + static_cast<int>(static_cast<uint32_t>(n) << 20));
  }
  return s * z;
}

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/*
 * ES6 draft 09-27-13, section 20.2.2.30.
 * Math.sinh
 * Method :
 * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
 *      1. Replace x by |x| (sinh(-x) = -sinh(x)).
 *      2.
 *                                                  E + E/(E+1)
 *          0        <= x <= 22     :  sinh(x) := --------------, E=expm1(x)
 *                                                      2
 *
 *          22       <= x <= lnovft :  sinh(x) := exp(x)/2
 *          lnovft   <= x <= ln2ovft:  sinh(x) := exp(x/2)/2 * exp(x/2)
 *          ln2ovft  <  x           :  sinh(x) := x*shuge (overflow)
 *
 * Special cases:
 *      sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
 *      only sinh(0)=0 is exact for finite x.
 */
double sinh(double x) {
  static const double KSINH_OVERFLOW = 710.4758600739439,
                      TWO_M28 =
                          3.725290298461914e-9,  // 2^-28, empty lower half
2984
      LOG_MAXD = 709.7822265625;  // 0x40862E42 00000000, empty lower half
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  static const double shuge = 1.0e307;

  double h = (x < 0) ? -0.5 : 0.5;
  // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
  double ax = fabs(x);
  if (ax < 22) {
    // For |x| < 2^-28, sinh(x) = x
    if (ax < TWO_M28) return x;
    double t = expm1(ax);
    if (ax < 1) {
      return h * (2 * t - t * t / (t + 1));
    }
    return h * (t + t / (t + 1));
  }
  // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
  if (ax < LOG_MAXD) return h * exp(ax);
  // |x| in [log(maxdouble), overflowthreshold]
  // overflowthreshold = 710.4758600739426
  if (ax <= KSINH_OVERFLOW) {
    double w = exp(0.5 * ax);
    double t = h * w;
    return t * w;
  }
  // |x| > overflowthreshold or is NaN.
  // Return Infinity of the appropriate sign or NaN.
  return x * shuge;
}

/* Tanh(x)
 * Return the Hyperbolic Tangent of x
 *
 * Method :
 *                                 x    -x
 *                                e  - e
 *  0. tanh(x) is defined to be -----------
 *                                 x    -x
 *                                e  + e
 *  1. reduce x to non-negative by tanh(-x) = -tanh(x).
 *  2.  0      <= x <  2**-28 : tanh(x) := x with inexact if x != 0
 *                                          -t
 *      2**-28 <= x <  1      : tanh(x) := -----; t = expm1(-2x)
 *                                         t + 2
 *                                               2
 *      1      <= x <  22     : tanh(x) := 1 - -----; t = expm1(2x)
 *                                             t + 2
 *      22     <= x <= INF    : tanh(x) := 1.
 *
 * Special cases:
 *      tanh(NaN) is NaN;
 *      only tanh(0)=0 is exact for finite argument.
 */
double tanh(double x) {
  static const volatile double tiny = 1.0e-300;
  static const double one = 1.0, two = 2.0, huge = 1.0e300;
  double t, z;
  int32_t jx, ix;

  GET_HIGH_WORD(jx, x);
3043
  ix = jx & 0x7FFFFFFF;
3044 3045

  /* x is INF or NaN */
3046
  if (ix >= 0x7FF00000) {
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    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 */
3055
    if (ix < 0x3E300000) {          /* |x|<2**-28 */
3056 3057
      if (huge + x > one) return x; /* tanh(tiny) = tiny with inexact */
    }
3058
    if (ix >= 0x3FF00000) { /* |x|>=1  */
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      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;
}

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#undef EXTRACT_WORDS
#undef EXTRACT_WORD64
#undef GET_HIGH_WORD
#undef GET_LOW_WORD
#undef INSERT_WORDS
#undef INSERT_WORD64
#undef SET_HIGH_WORD
#undef SET_LOW_WORD
#undef STRICT_ASSIGN

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