LCOV - code coverage report
Current view: top level - src/base - ieee754.cc (source / functions) Hit Total Coverage
Test: app.info Lines: 721 755 95.5 %
Date: 2017-04-26 Functions: 24 24 100.0 %

          Line data    Source code
       1             : // The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
       2             : //
       3             : // ====================================================
       4             : // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
       5             : //
       6             : // Developed at SunSoft, a Sun Microsystems, Inc. business.
       7             : // Permission to use, copy, modify, and distribute this
       8             : // software is freely granted, provided that this notice
       9             : // is preserved.
      10             : // ====================================================
      11             : //
      12             : // The original source code covered by the above license above has been
      13             : // modified significantly by Google Inc.
      14             : // Copyright 2016 the V8 project authors. All rights reserved.
      15             : 
      16             : #include "src/base/ieee754.h"
      17             : 
      18             : #include <cmath>
      19             : #include <limits>
      20             : 
      21             : #include "src/base/build_config.h"
      22             : #include "src/base/macros.h"
      23             : 
      24             : namespace v8 {
      25             : namespace base {
      26             : namespace ieee754 {
      27             : 
      28             : namespace {
      29             : 
      30             : /* Disable "potential divide by 0" warning in Visual Studio compiler. */
      31             : 
      32             : #if V8_CC_MSVC
      33             : 
      34             : #pragma warning(disable : 4723)
      35             : 
      36             : #endif
      37             : 
      38             : /*
      39             :  * The original fdlibm code used statements like:
      40             :  *  n0 = ((*(int*)&one)>>29)^1;   * index of high word *
      41             :  *  ix0 = *(n0+(int*)&x);     * high word of x *
      42             :  *  ix1 = *((1-n0)+(int*)&x);   * low word of x *
      43             :  * to dig two 32 bit words out of the 64 bit IEEE floating point
      44             :  * value.  That is non-ANSI, and, moreover, the gcc instruction
      45             :  * scheduler gets it wrong.  We instead use the following macros.
      46             :  * Unlike the original code, we determine the endianness at compile
      47             :  * time, not at run time; I don't see much benefit to selecting
      48             :  * endianness at run time.
      49             :  */
      50             : 
      51             : /*
      52             :  * A union which permits us to convert between a double and two 32 bit
      53             :  * ints.
      54             :  */
      55             : 
      56             : #if V8_TARGET_LITTLE_ENDIAN
      57             : 
      58             : typedef union {
      59             :   double value;
      60             :   struct {
      61             :     uint32_t lsw;
      62             :     uint32_t msw;
      63             :   } parts;
      64             :   struct {
      65             :     uint64_t w;
      66             :   } xparts;
      67             : } ieee_double_shape_type;
      68             : 
      69             : #else
      70             : 
      71             : typedef union {
      72             :   double value;
      73             :   struct {
      74             :     uint32_t msw;
      75             :     uint32_t lsw;
      76             :   } parts;
      77             :   struct {
      78             :     uint64_t w;
      79             :   } xparts;
      80             : } ieee_double_shape_type;
      81             : 
      82             : #endif
      83             : 
      84             : /* Get two 32 bit ints from a double.  */
      85             : 
      86             : #define EXTRACT_WORDS(ix0, ix1, d) \
      87             :   do {                             \
      88             :     ieee_double_shape_type ew_u;   \
      89             :     ew_u.value = (d);              \
      90             :     (ix0) = ew_u.parts.msw;        \
      91             :     (ix1) = ew_u.parts.lsw;        \
      92             :   } while (0)
      93             : 
      94             : /* Get a 64-bit int from a double. */
      95             : #define EXTRACT_WORD64(ix, d)    \
      96             :   do {                           \
      97             :     ieee_double_shape_type ew_u; \
      98             :     ew_u.value = (d);            \
      99             :     (ix) = ew_u.xparts.w;        \
     100             :   } while (0)
     101             : 
     102             : /* Get the more significant 32 bit int from a double.  */
     103             : 
     104             : #define GET_HIGH_WORD(i, d)      \
     105             :   do {                           \
     106             :     ieee_double_shape_type gh_u; \
     107             :     gh_u.value = (d);            \
     108             :     (i) = gh_u.parts.msw;        \
     109             :   } while (0)
     110             : 
     111             : /* Get the less significant 32 bit int from a double.  */
     112             : 
     113             : #define GET_LOW_WORD(i, d)       \
     114             :   do {                           \
     115             :     ieee_double_shape_type gl_u; \
     116             :     gl_u.value = (d);            \
     117             :     (i) = gl_u.parts.lsw;        \
     118             :   } while (0)
     119             : 
     120             : /* Set a double from two 32 bit ints.  */
     121             : 
     122             : #define INSERT_WORDS(d, ix0, ix1) \
     123             :   do {                            \
     124             :     ieee_double_shape_type iw_u;  \
     125             :     iw_u.parts.msw = (ix0);       \
     126             :     iw_u.parts.lsw = (ix1);       \
     127             :     (d) = iw_u.value;             \
     128             :   } while (0)
     129             : 
     130             : /* Set a double from a 64-bit int. */
     131             : #define INSERT_WORD64(d, ix)     \
     132             :   do {                           \
     133             :     ieee_double_shape_type iw_u; \
     134             :     iw_u.xparts.w = (ix);        \
     135             :     (d) = iw_u.value;            \
     136             :   } while (0)
     137             : 
     138             : /* Set the more significant 32 bits of a double from an int.  */
     139             : 
     140             : #define SET_HIGH_WORD(d, v)      \
     141             :   do {                           \
     142             :     ieee_double_shape_type sh_u; \
     143             :     sh_u.value = (d);            \
     144             :     sh_u.parts.msw = (v);        \
     145             :     (d) = sh_u.value;            \
     146             :   } while (0)
     147             : 
     148             : /* Set the less significant 32 bits of a double from an int.  */
     149             : 
     150             : #define SET_LOW_WORD(d, v)       \
     151             :   do {                           \
     152             :     ieee_double_shape_type sl_u; \
     153             :     sl_u.value = (d);            \
     154             :     sl_u.parts.lsw = (v);        \
     155             :     (d) = sl_u.value;            \
     156             :   } while (0)
     157             : 
     158             : /* Support macro. */
     159             : 
     160             : #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))
     161             : 
     162             : int32_t __ieee754_rem_pio2(double x, double *y) WARN_UNUSED_RESULT;
     163             : double __kernel_cos(double x, double y) WARN_UNUSED_RESULT;
     164             : int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
     165             :                       const int32_t *ipio2) WARN_UNUSED_RESULT;
     166             : double __kernel_sin(double x, double y, int iy) WARN_UNUSED_RESULT;
     167             : 
     168             : /* __ieee754_rem_pio2(x,y)
     169             :  *
     170             :  * return the remainder of x rem pi/2 in y[0]+y[1]
     171             :  * use __kernel_rem_pio2()
     172             :  */
     173     9333683 : int32_t __ieee754_rem_pio2(double x, double *y) {
     174             :   /*
     175             :    * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
     176             :    */
     177             :   static const int32_t two_over_pi[] = {
     178             :       0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C,
     179             :       0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649,
     180             :       0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44,
     181             :       0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B,
     182             :       0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D,
     183             :       0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
     184             :       0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330,
     185             :       0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08,
     186             :       0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
     187             :       0x73A8C9, 0x60E27B, 0xC08C6B,
     188             :   };
     189             : 
     190             :   static const int32_t npio2_hw[] = {
     191             :       0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
     192             :       0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
     193             :       0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
     194             :       0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
     195             :       0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
     196             :       0x404858EB, 0x404921FB,
     197             :   };
     198             : 
     199             :   /*
     200             :    * invpio2:  53 bits of 2/pi
     201             :    * pio2_1:   first  33 bit of pi/2
     202             :    * pio2_1t:  pi/2 - pio2_1
     203             :    * pio2_2:   second 33 bit of pi/2
     204             :    * pio2_2t:  pi/2 - (pio2_1+pio2_2)
     205             :    * pio2_3:   third  33 bit of pi/2
     206             :    * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
     207             :    */
     208             : 
     209             :   static const double
     210             :       zero = 0.00000000000000000000e+00,    /* 0x00000000, 0x00000000 */
     211             :       half = 5.00000000000000000000e-01,    /* 0x3FE00000, 0x00000000 */
     212             :       two24 = 1.67772160000000000000e+07,   /* 0x41700000, 0x00000000 */
     213             :       invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
     214             :       pio2_1 = 1.57079632673412561417e+00,  /* 0x3FF921FB, 0x54400000 */
     215             :       pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
     216             :       pio2_2 = 6.07710050630396597660e-11,  /* 0x3DD0B461, 0x1A600000 */
     217             :       pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
     218             :       pio2_3 = 2.02226624871116645580e-21,  /* 0x3BA3198A, 0x2E000000 */
     219             :       pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
     220             : 
     221             :   double z, w, t, r, fn;
     222             :   double tx[3];
     223             :   int32_t e0, i, j, nx, n, ix, hx;
     224             :   uint32_t low;
     225             : 
     226             :   z = 0;
     227     9333683 :   GET_HIGH_WORD(hx, x); /* high word of x */
     228     9333683 :   ix = hx & 0x7fffffff;
     229     9333683 :   if (ix <= 0x3fe921fb) { /* |x| ~<= pi/4 , no need for reduction */
     230           0 :     y[0] = x;
     231           0 :     y[1] = 0;
     232           0 :     return 0;
     233             :   }
     234     9333683 :   if (ix < 0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
     235     6001594 :     if (hx > 0) {
     236     5996415 :       z = x - pio2_1;
     237     5996415 :       if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
     238     5996189 :         y[0] = z - pio2_1t;
     239     5996189 :         y[1] = (z - y[0]) - pio2_1t;
     240             :       } else { /* near pi/2, use 33+33+53 bit pi */
     241         226 :         z -= pio2_2;
     242         226 :         y[0] = z - pio2_2t;
     243         226 :         y[1] = (z - y[0]) - pio2_2t;
     244             :       }
     245             :       return 1;
     246             :     } else { /* negative x */
     247        5179 :       z = x + pio2_1;
     248        5179 :       if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
     249        5082 :         y[0] = z + pio2_1t;
     250        5082 :         y[1] = (z - y[0]) + pio2_1t;
     251             :       } else { /* near pi/2, use 33+33+53 bit pi */
     252          97 :         z += pio2_2;
     253          97 :         y[0] = z + pio2_2t;
     254          97 :         y[1] = (z - y[0]) + pio2_2t;
     255             :       }
     256             :       return -1;
     257             :     }
     258             :   }
     259     3332089 :   if (ix <= 0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
     260     3293146 :     t = fabs(x);
     261     3293146 :     n = static_cast<int32_t>(t * invpio2 + half);
     262     3293146 :     fn = static_cast<double>(n);
     263     3293146 :     r = t - fn * pio2_1;
     264     3293146 :     w = fn * pio2_1t; /* 1st round good to 85 bit */
     265     3293146 :     if (n < 32 && ix != npio2_hw[n - 1]) {
     266       82159 :       y[0] = r - w; /* quick check no cancellation */
     267             :     } else {
     268             :       uint32_t high;
     269     3210987 :       j = ix >> 20;
     270     3210987 :       y[0] = r - w;
     271     3210987 :       GET_HIGH_WORD(high, y[0]);
     272     3210987 :       i = j - ((high >> 20) & 0x7ff);
     273     3210987 :       if (i > 16) { /* 2nd iteration needed, good to 118 */
     274             :         t = r;
     275     1895991 :         w = fn * pio2_2;
     276     1895991 :         r = t - w;
     277     1895991 :         w = fn * pio2_2t - ((t - r) - w);
     278     1895991 :         y[0] = r - w;
     279     1895991 :         GET_HIGH_WORD(high, y[0]);
     280     1895991 :         i = j - ((high >> 20) & 0x7ff);
     281     1895991 :         if (i > 49) { /* 3rd iteration need, 151 bits acc */
     282             :           t = r;      /* will cover all possible cases */
     283         288 :           w = fn * pio2_3;
     284         288 :           r = t - w;
     285         288 :           w = fn * pio2_3t - ((t - r) - w);
     286         288 :           y[0] = r - w;
     287             :         }
     288             :       }
     289             :     }
     290     3293146 :     y[1] = (r - y[0]) - w;
     291     3293146 :     if (hx < 0) {
     292       11490 :       y[0] = -y[0];
     293       11490 :       y[1] = -y[1];
     294       11490 :       return -n;
     295             :     } else {
     296             :       return n;
     297             :     }
     298             :   }
     299             :   /*
     300             :    * all other (large) arguments
     301             :    */
     302       38943 :   if (ix >= 0x7ff00000) { /* x is inf or NaN */
     303           0 :     y[0] = y[1] = x - x;
     304           0 :     return 0;
     305             :   }
     306             :   /* set z = scalbn(|x|,ilogb(x)-23) */
     307       38943 :   GET_LOW_WORD(low, x);
     308       38943 :   SET_LOW_WORD(z, low);
     309       38943 :   e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
     310       38943 :   SET_HIGH_WORD(z, ix - static_cast<int32_t>(e0 << 20));
     311      116829 :   for (i = 0; i < 2; i++) {
     312       77886 :     tx[i] = static_cast<double>(static_cast<int32_t>(z));
     313       77886 :     z = (z - tx[i]) * two24;
     314             :   }
     315       38943 :   tx[2] = z;
     316             :   nx = 3;
     317       38943 :   while (tx[nx - 1] == zero) nx--; /* skip zero term */
     318       38943 :   n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
     319       38943 :   if (hx < 0) {
     320        9180 :     y[0] = -y[0];
     321        9180 :     y[1] = -y[1];
     322        9180 :     return -n;
     323             :   }
     324             :   return n;
     325             : }
     326             : 
     327             : /* __kernel_cos( x,  y )
     328             :  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
     329             :  * Input x is assumed to be bounded by ~pi/4 in magnitude.
     330             :  * Input y is the tail of x.
     331             :  *
     332             :  * Algorithm
     333             :  *      1. Since cos(-x) = cos(x), we need only to consider positive x.
     334             :  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
     335             :  *      3. cos(x) is approximated by a polynomial of degree 14 on
     336             :  *         [0,pi/4]
     337             :  *                                       4            14
     338             :  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
     339             :  *         where the remez error is
     340             :  *
     341             :  *      |              2     4     6     8     10    12     14 |     -58
     342             :  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
     343             :  *      |                                                      |
     344             :  *
     345             :  *                     4     6     8     10    12     14
     346             :  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
     347             :  *             cos(x) = 1 - x*x/2 + r
     348             :  *         since cos(x+y) ~ cos(x) - sin(x)*y
     349             :  *                        ~ cos(x) - x*y,
     350             :  *         a correction term is necessary in cos(x) and hence
     351             :  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
     352             :  *         For better accuracy when x > 0.3, let qx = |x|/4 with
     353             :  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
     354             :  *         Then
     355             :  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
     356             :  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
     357             :  *         magnitude of the latter is at least a quarter of x*x/2,
     358             :  *         thus, reducing the rounding error in the subtraction.
     359             :  */
     360             : V8_INLINE double __kernel_cos(double x, double y) {
     361             :   static const double
     362             :       one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
     363             :       C1 = 4.16666666666666019037e-02,  /* 0x3FA55555, 0x5555554C */
     364             :       C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
     365             :       C3 = 2.48015872894767294178e-05,  /* 0x3EFA01A0, 0x19CB1590 */
     366             :       C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
     367             :       C5 = 2.08757232129817482790e-09,  /* 0x3E21EE9E, 0xBDB4B1C4 */
     368             :       C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
     369             : 
     370             :   double a, iz, z, r, qx;
     371             :   int32_t ix;
     372    22780090 :   GET_HIGH_WORD(ix, x);
     373    22780090 :   ix &= 0x7fffffff;                           /* ix = |x|'s high word*/
     374    22780090 :   if (ix < 0x3e400000) {                      /* if x < 2**27 */
     375      525708 :     if (static_cast<int>(x) == 0) return one; /* generate inexact */
     376             :   }
     377    22254382 :   z = x * x;
     378    22254382 :   r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
     379    22254382 :   if (ix < 0x3FD33333) { /* if |x| < 0.3 */
     380    15707833 :     return one - (0.5 * z - (z * r - x * y));
     381             :   } else {
     382     6546549 :     if (ix > 0x3fe90000) { /* x > 0.78125 */
     383             :       qx = 0.28125;
     384             :     } else {
     385     6425336 :       INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
     386             :     }
     387     6546549 :     iz = 0.5 * z - qx;
     388     6546549 :     a = one - qx;
     389     6546549 :     return a - (iz - (z * r - x * y));
     390             :   }
     391             : }
     392             : 
     393             : /* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
     394             :  * double x[],y[]; int e0,nx,prec; int ipio2[];
     395             :  *
     396             :  * __kernel_rem_pio2 return the last three digits of N with
     397             :  *              y = x - N*pi/2
     398             :  * so that |y| < pi/2.
     399             :  *
     400             :  * The method is to compute the integer (mod 8) and fraction parts of
     401             :  * (2/pi)*x without doing the full multiplication. In general we
     402             :  * skip the part of the product that are known to be a huge integer (
     403             :  * more accurately, = 0 mod 8 ). Thus the number of operations are
     404             :  * independent of the exponent of the input.
     405             :  *
     406             :  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
     407             :  *
     408             :  * Input parameters:
     409             :  *      x[]     The input value (must be positive) is broken into nx
     410             :  *              pieces of 24-bit integers in double precision format.
     411             :  *              x[i] will be the i-th 24 bit of x. The scaled exponent
     412             :  *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
     413             :  *              match x's up to 24 bits.
     414             :  *
     415             :  *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
     416             :  *                      e0 = ilogb(z)-23
     417             :  *                      z  = scalbn(z,-e0)
     418             :  *              for i = 0,1,2
     419             :  *                      x[i] = floor(z)
     420             :  *                      z    = (z-x[i])*2**24
     421             :  *
     422             :  *
     423             :  *      y[]     output result in an array of double precision numbers.
     424             :  *              The dimension of y[] is:
     425             :  *                      24-bit  precision       1
     426             :  *                      53-bit  precision       2
     427             :  *                      64-bit  precision       2
     428             :  *                      113-bit precision       3
     429             :  *              The actual value is the sum of them. Thus for 113-bit
     430             :  *              precison, one may have to do something like:
     431             :  *
     432             :  *              long double t,w,r_head, r_tail;
     433             :  *              t = (long double)y[2] + (long double)y[1];
     434             :  *              w = (long double)y[0];
     435             :  *              r_head = t+w;
     436             :  *              r_tail = w - (r_head - t);
     437             :  *
     438             :  *      e0      The exponent of x[0]
     439             :  *
     440             :  *      nx      dimension of x[]
     441             :  *
     442             :  *      prec    an integer indicating the precision:
     443             :  *                      0       24  bits (single)
     444             :  *                      1       53  bits (double)
     445             :  *                      2       64  bits (extended)
     446             :  *                      3       113 bits (quad)
     447             :  *
     448             :  *      ipio2[]
     449             :  *              integer array, contains the (24*i)-th to (24*i+23)-th
     450             :  *              bit of 2/pi after binary point. The corresponding
     451             :  *              floating value is
     452             :  *
     453             :  *                      ipio2[i] * 2^(-24(i+1)).
     454             :  *
     455             :  * External function:
     456             :  *      double scalbn(), floor();
     457             :  *
     458             :  *
     459             :  * Here is the description of some local variables:
     460             :  *
     461             :  *      jk      jk+1 is the initial number of terms of ipio2[] needed
     462             :  *              in the computation. The recommended value is 2,3,4,
     463             :  *              6 for single, double, extended,and quad.
     464             :  *
     465             :  *      jz      local integer variable indicating the number of
     466             :  *              terms of ipio2[] used.
     467             :  *
     468             :  *      jx      nx - 1
     469             :  *
     470             :  *      jv      index for pointing to the suitable ipio2[] for the
     471             :  *              computation. In general, we want
     472             :  *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
     473             :  *              is an integer. Thus
     474             :  *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
     475             :  *              Hence jv = max(0,(e0-3)/24).
     476             :  *
     477             :  *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
     478             :  *
     479             :  *      q[]     double array with integral value, representing the
     480             :  *              24-bits chunk of the product of x and 2/pi.
     481             :  *
     482             :  *      q0      the corresponding exponent of q[0]. Note that the
     483             :  *              exponent for q[i] would be q0-24*i.
     484             :  *
     485             :  *      PIo2[]  double precision array, obtained by cutting pi/2
     486             :  *              into 24 bits chunks.
     487             :  *
     488             :  *      f[]     ipio2[] in floating point
     489             :  *
     490             :  *      iq[]    integer array by breaking up q[] in 24-bits chunk.
     491             :  *
     492             :  *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
     493             :  *
     494             :  *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
     495             :  *              it also indicates the *sign* of the result.
     496             :  *
     497             :  */
     498       38943 : int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
     499             :                       const int32_t *ipio2) {
     500             :   /* Constants:
     501             :    * The hexadecimal values are the intended ones for the following
     502             :    * constants. The decimal values may be used, provided that the
     503             :    * compiler will convert from decimal to binary accurately enough
     504             :    * to produce the hexadecimal values shown.
     505             :    */
     506             :   static const int init_jk[] = {2, 3, 4, 6}; /* initial value for jk */
     507             : 
     508             :   static const double PIo2[] = {
     509             :       1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
     510             :       7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
     511             :       5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
     512             :       3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
     513             :       1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
     514             :       1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
     515             :       2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
     516             :       2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
     517             :   };
     518             : 
     519             :   static const double
     520             :       zero = 0.0,
     521             :       one = 1.0,
     522             :       two24 = 1.67772160000000000000e+07,  /* 0x41700000, 0x00000000 */
     523             :       twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
     524             : 
     525             :   int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
     526             :   double z, fw, f[20], fq[20], q[20];
     527             : 
     528             :   /* initialize jk*/
     529       38943 :   jk = init_jk[prec];
     530             :   jp = jk;
     531             : 
     532             :   /* determine jx,jv,q0, note that 3>q0 */
     533       38943 :   jx = nx - 1;
     534       38943 :   jv = (e0 - 3) / 24;
     535       38943 :   if (jv < 0) jv = 0;
     536       38943 :   q0 = e0 - 24 * (jv + 1);
     537             : 
     538             :   /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
     539       38943 :   j = jv - jx;
     540       38943 :   m = jx + jk;
     541      262572 :   for (i = 0; i <= m; i++, j++) {
     542      223629 :     f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]);
     543             :   }
     544             : 
     545             :   /* compute q[0],q[1],...q[jk] */
     546      194715 :   for (i = 0; i <= jk; i++) {
     547      339285 :     for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
     548      194715 :     q[i] = fw;
     549             :   }
     550             : 
     551             :   jz = jk;
     552             : recompute:
     553             :   /* distill q[] into iq[] reversingly */
     554      213111 :   for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
     555      171102 :     fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
     556      171102 :     iq[i] = static_cast<int32_t>(z - two24 * fw);
     557      171102 :     z = q[j - 1] + fw;
     558             :   }
     559             : 
     560             :   /* compute n */
     561       42009 :   z = scalbn(z, q0);           /* actual value of z */
     562       42009 :   z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
     563       42009 :   n = static_cast<int32_t>(z);
     564       42009 :   z -= static_cast<double>(n);
     565             :   ih = 0;
     566       42009 :   if (q0 > 0) { /* need iq[jz-1] to determine n */
     567        2664 :     i = (iq[jz - 1] >> (24 - q0));
     568        2664 :     n += i;
     569        2664 :     iq[jz - 1] -= i << (24 - q0);
     570        2664 :     ih = iq[jz - 1] >> (23 - q0);
     571       39345 :   } else if (q0 == 0) {
     572        1242 :     ih = iq[jz - 1] >> 23;
     573       38103 :   } else if (z >= 0.5) {
     574             :     ih = 2;
     575             :   }
     576             : 
     577       42009 :   if (ih > 0) { /* q > 0.5 */
     578       21375 :     n += 1;
     579             :     carry = 0;
     580      108309 :     for (i = 0; i < jz; i++) { /* compute 1-q */
     581       86934 :       j = iq[i];
     582       86934 :       if (carry == 0) {
     583       25758 :         if (j != 0) {
     584             :           carry = 1;
     585       21375 :           iq[i] = 0x1000000 - j;
     586             :         }
     587             :       } else {
     588       61176 :         iq[i] = 0xffffff - j;
     589             :       }
     590             :     }
     591       21375 :     if (q0 > 0) { /* rare case: chance is 1 in 12 */
     592        1302 :       switch (q0) {
     593             :         case 1:
     594         792 :           iq[jz - 1] &= 0x7fffff;
     595         792 :           break;
     596             :         case 2:
     597         510 :           iq[jz - 1] &= 0x3fffff;
     598         510 :           break;
     599             :       }
     600             :     }
     601       21375 :     if (ih == 2) {
     602       19431 :       z = one - z;
     603       19431 :       if (carry != 0) z -= scalbn(one, q0);
     604             :     }
     605             :   }
     606             : 
     607             :   /* check if recomputation is needed */
     608       42009 :   if (z == zero) {
     609             :     j = 0;
     610        6132 :     for (i = jz - 1; i >= jk; i--) j |= iq[i];
     611        6132 :     if (j == 0) { /* need recomputation */
     612           0 :       for (k = 1; jk >= k && iq[jk - k] == 0; k++) {
     613             :         /* k = no. of terms needed */
     614             :       }
     615             : 
     616        6132 :       for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
     617        3066 :         f[jx + i] = ipio2[jv + i];
     618        3066 :         for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
     619        3066 :         q[i] = fw;
     620             :       }
     621             :       jz += k;
     622             :       goto recompute;
     623             :     }
     624             :   }
     625             : 
     626             :   /* chop off zero terms */
     627       38943 :   if (z == 0.0) {
     628        3066 :     jz -= 1;
     629        3066 :     q0 -= 24;
     630        6132 :     while (iq[jz] == 0) {
     631           0 :       jz--;
     632           0 :       q0 -= 24;
     633             :     }
     634             :   } else { /* break z into 24-bit if necessary */
     635       35877 :     z = scalbn(z, -q0);
     636       35877 :     if (z >= two24) {
     637         555 :       fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
     638         555 :       iq[jz] = z - two24 * fw;
     639         555 :       jz += 1;
     640         555 :       q0 += 24;
     641         555 :       iq[jz] = fw;
     642             :     } else {
     643       35322 :       iq[jz] = z;
     644             :     }
     645             :   }
     646             : 
     647             :   /* convert integer "bit" chunk to floating-point value */
     648       38943 :   fw = scalbn(one, q0);
     649      234213 :   for (i = jz; i >= 0; i--) {
     650      195270 :     q[i] = fw * iq[i];
     651      195270 :     fw *= twon24;
     652             :   }
     653             : 
     654             :   /* compute PIo2[0,...,jp]*q[jz,...,0] */
     655      195270 :   for (i = jz; i >= 0; i--) {
     656      586920 :     for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) fw += PIo2[k] * q[i + k];
     657      195270 :     fq[jz - i] = fw;
     658             :   }
     659             : 
     660             :   /* compress fq[] into y[] */
     661       38943 :   switch (prec) {
     662             :     case 0:
     663             :       fw = 0.0;
     664           0 :       for (i = jz; i >= 0; i--) fw += fq[i];
     665           0 :       y[0] = (ih == 0) ? fw : -fw;
     666           0 :       break;
     667             :     case 1:
     668             :     case 2:
     669             :       fw = 0.0;
     670      195270 :       for (i = jz; i >= 0; i--) fw += fq[i];
     671       38943 :       y[0] = (ih == 0) ? fw : -fw;
     672       38943 :       fw = fq[0] - fw;
     673       38943 :       for (i = 1; i <= jz; i++) fw += fq[i];
     674       38943 :       y[1] = (ih == 0) ? fw : -fw;
     675       38943 :       break;
     676             :     case 3: /* painful */
     677           0 :       for (i = jz; i > 0; i--) {
     678           0 :         fw = fq[i - 1] + fq[i];
     679           0 :         fq[i] += fq[i - 1] - fw;
     680           0 :         fq[i - 1] = fw;
     681             :       }
     682           0 :       for (i = jz; i > 1; i--) {
     683           0 :         fw = fq[i - 1] + fq[i];
     684           0 :         fq[i] += fq[i - 1] - fw;
     685           0 :         fq[i - 1] = fw;
     686             :       }
     687           0 :       for (fw = 0.0, i = jz; i >= 2; i--) fw += fq[i];
     688           0 :       if (ih == 0) {
     689           0 :         y[0] = fq[0];
     690           0 :         y[1] = fq[1];
     691           0 :         y[2] = fw;
     692             :       } else {
     693           0 :         y[0] = -fq[0];
     694           0 :         y[1] = -fq[1];
     695           0 :         y[2] = -fw;
     696             :       }
     697             :   }
     698       38943 :   return n & 7;
     699             : }
     700             : 
     701             : /* __kernel_sin( x, y, iy)
     702             :  * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
     703             :  * Input x is assumed to be bounded by ~pi/4 in magnitude.
     704             :  * Input y is the tail of x.
     705             :  * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
     706             :  *
     707             :  * Algorithm
     708             :  *      1. Since sin(-x) = -sin(x), we need only to consider positive x.
     709             :  *      2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
     710             :  *      3. sin(x) is approximated by a polynomial of degree 13 on
     711             :  *         [0,pi/4]
     712             :  *                               3            13
     713             :  *              sin(x) ~ x + S1*x + ... + S6*x
     714             :  *         where
     715             :  *
     716             :  *      |sin(x)         2     4     6     8     10     12  |     -58
     717             :  *      |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
     718             :  *      |  x                                               |
     719             :  *
     720             :  *      4. sin(x+y) = sin(x) + sin'(x')*y
     721             :  *                  ~ sin(x) + (1-x*x/2)*y
     722             :  *         For better accuracy, let
     723             :  *                   3      2      2      2      2
     724             :  *              r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
     725             :  *         then                   3    2
     726             :  *              sin(x) = x + (S1*x + (x *(r-y/2)+y))
     727             :  */
     728             : V8_INLINE double __kernel_sin(double x, double y, int iy) {
     729             :   static const double
     730             :       half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
     731             :       S1 = -1.66666666666666324348e-01,  /* 0xBFC55555, 0x55555549 */
     732             :       S2 = 8.33333333332248946124e-03,   /* 0x3F811111, 0x1110F8A6 */
     733             :       S3 = -1.98412698298579493134e-04,  /* 0xBF2A01A0, 0x19C161D5 */
     734             :       S4 = 2.75573137070700676789e-06,   /* 0x3EC71DE3, 0x57B1FE7D */
     735             :       S5 = -2.50507602534068634195e-08,  /* 0xBE5AE5E6, 0x8A2B9CEB */
     736             :       S6 = 1.58969099521155010221e-10;   /* 0x3DE5D93A, 0x5ACFD57C */
     737             : 
     738             :   double z, r, v;
     739             :   int32_t ix;
     740    38756128 :   GET_HIGH_WORD(ix, x);
     741    38756128 :   ix &= 0x7fffffff;      /* high word of x */
     742    38756128 :   if (ix < 0x3e400000) { /* |x| < 2**-27 */
     743      600787 :     if (static_cast<int>(x) == 0) return x;
     744             :   } /* generate inexact */
     745    38155341 :   z = x * x;
     746    38155341 :   v = z * x;
     747    38155341 :   r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
     748             :   if (iy == 0) {
     749    37242225 :     return x + v * (S1 + z * r);
     750             :   } else {
     751      913116 :     return x - ((z * (half * y - v * r) - y) - v * S1);
     752             :   }
     753             : }
     754             : 
     755             : /* __kernel_tan( x, y, k )
     756             :  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
     757             :  * Input x is assumed to be bounded by ~pi/4 in magnitude.
     758             :  * Input y is the tail of x.
     759             :  * Input k indicates whether tan (if k=1) or
     760             :  * -1/tan (if k= -1) is returned.
     761             :  *
     762             :  * Algorithm
     763             :  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
     764             :  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
     765             :  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
     766             :  *         [0,0.67434]
     767             :  *                               3             27
     768             :  *              tan(x) ~ x + T1*x + ... + T13*x
     769             :  *         where
     770             :  *
     771             :  *              |tan(x)         2     4            26   |     -59.2
     772             :  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
     773             :  *              |  x                                    |
     774             :  *
     775             :  *         Note: tan(x+y) = tan(x) + tan'(x)*y
     776             :  *                        ~ tan(x) + (1+x*x)*y
     777             :  *         Therefore, for better accuracy in computing tan(x+y), let
     778             :  *                   3      2      2       2       2
     779             :  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
     780             :  *         then
     781             :  *                                  3    2
     782             :  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
     783             :  *
     784             :  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
     785             :  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
     786             :  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
     787             :  */
     788     1526857 : double __kernel_tan(double x, double y, int iy) {
     789             :   static const double xxx[] = {
     790             :       3.33333333333334091986e-01,             /* 3FD55555, 55555563 */
     791             :       1.33333333333201242699e-01,             /* 3FC11111, 1110FE7A */
     792             :       5.39682539762260521377e-02,             /* 3FABA1BA, 1BB341FE */
     793             :       2.18694882948595424599e-02,             /* 3F9664F4, 8406D637 */
     794             :       8.86323982359930005737e-03,             /* 3F8226E3, E96E8493 */
     795             :       3.59207910759131235356e-03,             /* 3F6D6D22, C9560328 */
     796             :       1.45620945432529025516e-03,             /* 3F57DBC8, FEE08315 */
     797             :       5.88041240820264096874e-04,             /* 3F4344D8, F2F26501 */
     798             :       2.46463134818469906812e-04,             /* 3F3026F7, 1A8D1068 */
     799             :       7.81794442939557092300e-05,             /* 3F147E88, A03792A6 */
     800             :       7.14072491382608190305e-05,             /* 3F12B80F, 32F0A7E9 */
     801             :       -1.85586374855275456654e-05,            /* BEF375CB, DB605373 */
     802             :       2.59073051863633712884e-05,             /* 3EFB2A70, 74BF7AD4 */
     803             :       /* one */ 1.00000000000000000000e+00,   /* 3FF00000, 00000000 */
     804             :       /* pio4 */ 7.85398163397448278999e-01,  /* 3FE921FB, 54442D18 */
     805             :       /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
     806             :   };
     807             : #define one xxx[13]
     808             : #define pio4 xxx[14]
     809             : #define pio4lo xxx[15]
     810             : #define T xxx
     811             : 
     812             :   double z, r, v, w, s;
     813             :   int32_t ix, hx;
     814             : 
     815     1526857 :   GET_HIGH_WORD(hx, x);             /* high word of x */
     816     1526857 :   ix = hx & 0x7fffffff;             /* high word of |x| */
     817     1526857 :   if (ix < 0x3e300000) {            /* x < 2**-28 */
     818        2875 :     if (static_cast<int>(x) == 0) { /* generate inexact */
     819             :       uint32_t low;
     820        2875 :       GET_LOW_WORD(low, x);
     821        2875 :       if (((ix | low) | (iy + 1)) == 0) {
     822           0 :         return one / fabs(x);
     823             :       } else {
     824        2875 :         if (iy == 1) {
     825             :           return x;
     826             :         } else { /* compute -1 / (x+y) carefully */
     827             :           double a, t;
     828             : 
     829         116 :           z = w = x + y;
     830         116 :           SET_LOW_WORD(z, 0);
     831         116 :           v = y - (z - x);
     832         116 :           t = a = -one / w;
     833         116 :           SET_LOW_WORD(t, 0);
     834         116 :           s = one + t * z;
     835         116 :           return t + a * (s + t * v);
     836             :         }
     837             :       }
     838             :     }
     839             :   }
     840     1523982 :   if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
     841      214692 :     if (hx < 0) {
     842      107806 :       x = -x;
     843      107806 :       y = -y;
     844             :     }
     845      214692 :     z = pio4 - x;
     846      214692 :     w = pio4lo - y;
     847      214692 :     x = z + w;
     848             :     y = 0.0;
     849             :   }
     850     1523982 :   z = x * x;
     851     1523982 :   w = z * z;
     852             :   /*
     853             :    * Break x^5*(T[1]+x^2*T[2]+...) into
     854             :    * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
     855             :    * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
     856             :    */
     857     1523982 :   r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
     858             :   v = z *
     859     1523982 :       (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
     860     1523982 :   s = z * x;
     861     1523982 :   r = y + z * (s * (r + v) + y);
     862     1523982 :   r += T[0] * s;
     863     1523982 :   w = x + r;
     864     1523982 :   if (ix >= 0x3FE59428) {
     865      214692 :     v = iy;
     866      214692 :     return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
     867             :   }
     868     1309290 :   if (iy == 1) {
     869             :     return w;
     870             :   } else {
     871             :     /*
     872             :      * if allow error up to 2 ulp, simply return
     873             :      * -1.0 / (x+r) here
     874             :      */
     875             :     /* compute -1.0 / (x+r) accurately */
     876             :     double a, t;
     877             :     z = w;
     878      654572 :     SET_LOW_WORD(z, 0);
     879      654572 :     v = r - (z - x);  /* z+v = r+x */
     880      654572 :     t = a = -1.0 / w; /* a = -1.0/w */
     881      654572 :     SET_LOW_WORD(t, 0);
     882      654572 :     s = 1.0 + t * z;
     883      654572 :     return t + a * (s + t * v);
     884             :   }
     885             : 
     886             : #undef one
     887             : #undef pio4
     888             : #undef pio4lo
     889             : #undef T
     890             : }
     891             : 
     892             : }  // namespace
     893             : 
     894             : /* acos(x)
     895             :  * Method :
     896             :  *      acos(x)  = pi/2 - asin(x)
     897             :  *      acos(-x) = pi/2 + asin(x)
     898             :  * For |x|<=0.5
     899             :  *      acos(x) = pi/2 - (x + x*x^2*R(x^2))     (see asin.c)
     900             :  * For x>0.5
     901             :  *      acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
     902             :  *              = 2asin(sqrt((1-x)/2))
     903             :  *              = 2s + 2s*z*R(z)        ...z=(1-x)/2, s=sqrt(z)
     904             :  *              = 2f + (2c + 2s*z*R(z))
     905             :  *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
     906             :  *     for f so that f+c ~ sqrt(z).
     907             :  * For x<-0.5
     908             :  *      acos(x) = pi - 2asin(sqrt((1-|x|)/2))
     909             :  *              = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
     910             :  *
     911             :  * Special cases:
     912             :  *      if x is NaN, return x itself;
     913             :  *      if |x|>1, return NaN with invalid signal.
     914             :  *
     915             :  * Function needed: sqrt
     916             :  */
     917       28966 : double acos(double x) {
     918             :   static const double
     919             :       one = 1.00000000000000000000e+00,     /* 0x3FF00000, 0x00000000 */
     920             :       pi = 3.14159265358979311600e+00,      /* 0x400921FB, 0x54442D18 */
     921             :       pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
     922             :       pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
     923             :       pS0 = 1.66666666666666657415e-01,     /* 0x3FC55555, 0x55555555 */
     924             :       pS1 = -3.25565818622400915405e-01,    /* 0xBFD4D612, 0x03EB6F7D */
     925             :       pS2 = 2.01212532134862925881e-01,     /* 0x3FC9C155, 0x0E884455 */
     926             :       pS3 = -4.00555345006794114027e-02,    /* 0xBFA48228, 0xB5688F3B */
     927             :       pS4 = 7.91534994289814532176e-04,     /* 0x3F49EFE0, 0x7501B288 */
     928             :       pS5 = 3.47933107596021167570e-05,     /* 0x3F023DE1, 0x0DFDF709 */
     929             :       qS1 = -2.40339491173441421878e+00,    /* 0xC0033A27, 0x1C8A2D4B */
     930             :       qS2 = 2.02094576023350569471e+00,     /* 0x40002AE5, 0x9C598AC8 */
     931             :       qS3 = -6.88283971605453293030e-01,    /* 0xBFE6066C, 0x1B8D0159 */
     932             :       qS4 = 7.70381505559019352791e-02;     /* 0x3FB3B8C5, 0xB12E9282 */
     933             : 
     934             :   double z, p, q, r, w, s, c, df;
     935             :   int32_t hx, ix;
     936       28966 :   GET_HIGH_WORD(hx, x);
     937       28966 :   ix = hx & 0x7fffffff;
     938       28966 :   if (ix >= 0x3ff00000) { /* |x| >= 1 */
     939             :     uint32_t lx;
     940       20239 :     GET_LOW_WORD(lx, x);
     941       20239 :     if (((ix - 0x3ff00000) | lx) == 0) { /* |x|==1 */
     942        1373 :       if (hx > 0)
     943             :         return 0.0; /* acos(1) = 0  */
     944             :       else
     945         672 :         return pi + 2.0 * pio2_lo; /* acos(-1)= pi */
     946             :     }
     947       18866 :     return (x - x) / (x - x); /* acos(|x|>1) is NaN */
     948             :   }
     949        8727 :   if (ix < 0x3fe00000) {                            /* |x| < 0.5 */
     950        6965 :     if (ix <= 0x3c600000) return pio2_hi + pio2_lo; /*if|x|<2**-57*/
     951        5233 :     z = x * x;
     952        5233 :     p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
     953        5233 :     q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
     954        5233 :     r = p / q;
     955        5233 :     return pio2_hi - (x - (pio2_lo - x * r));
     956        1762 :   } else if (hx < 0) { /* x < -0.5 */
     957         852 :     z = (one + x) * 0.5;
     958         852 :     p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
     959         852 :     q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
     960         852 :     s = sqrt(z);
     961         852 :     r = p / q;
     962         852 :     w = r * s - pio2_lo;
     963         852 :     return pi - 2.0 * (s + w);
     964             :   } else { /* x > 0.5 */
     965         910 :     z = (one - x) * 0.5;
     966         910 :     s = sqrt(z);
     967             :     df = s;
     968         910 :     SET_LOW_WORD(df, 0);
     969         910 :     c = (z - df * df) / (s + df);
     970         910 :     p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
     971         910 :     q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
     972         910 :     r = p / q;
     973         910 :     w = r * s + c;
     974         910 :     return 2.0 * (df + w);
     975             :   }
     976             : }
     977             : 
     978             : /* acosh(x)
     979             :  * Method :
     980             :  *      Based on
     981             :  *              acosh(x) = log [ x + sqrt(x*x-1) ]
     982             :  *      we have
     983             :  *              acosh(x) := log(x)+ln2, if x is large; else
     984             :  *              acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
     985             :  *              acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
     986             :  *
     987             :  * Special cases:
     988             :  *      acosh(x) is NaN with signal if x<1.
     989             :  *      acosh(NaN) is NaN without signal.
     990             :  */
     991        1246 : double acosh(double x) {
     992             :   static const double
     993             :       one = 1.0,
     994             :       ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
     995             :   double t;
     996             :   int32_t hx;
     997             :   uint32_t lx;
     998        1246 :   EXTRACT_WORDS(hx, lx, x);
     999        1246 :   if (hx < 0x3ff00000) { /* x < 1 */
    1000         642 :     return (x - x) / (x - x);
    1001         604 :   } else if (hx >= 0x41b00000) { /* x > 2**28 */
    1002         265 :     if (hx >= 0x7ff00000) {      /* x is inf of NaN */
    1003          99 :       return x + x;
    1004             :     } else {
    1005         166 :       return log(x) + ln2; /* acosh(huge)=log(2x) */
    1006             :     }
    1007         339 :   } else if (((hx - 0x3ff00000) | lx) == 0) {
    1008             :     return 0.0;                 /* acosh(1) = 0 */
    1009         312 :   } else if (hx > 0x40000000) { /* 2**28 > x > 2 */
    1010         163 :     t = x * x;
    1011         163 :     return log(2.0 * x - one / (x + sqrt(t - one)));
    1012             :   } else { /* 1<x<2 */
    1013         149 :     t = x - one;
    1014         149 :     return log1p(t + sqrt(2.0 * t + t * t));
    1015             :   }
    1016             : }
    1017             : 
    1018             : /* asin(x)
    1019             :  * Method :
    1020             :  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
    1021             :  *      we approximate asin(x) on [0,0.5] by
    1022             :  *              asin(x) = x + x*x^2*R(x^2)
    1023             :  *      where
    1024             :  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
    1025             :  *      and its remez error is bounded by
    1026             :  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
    1027             :  *
    1028             :  *      For x in [0.5,1]
    1029             :  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
    1030             :  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
    1031             :  *      then for x>0.98
    1032             :  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
    1033             :  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
    1034             :  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
    1035             :  *              f = hi part of s;
    1036             :  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
    1037             :  *      and
    1038             :  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
    1039             :  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
    1040             :  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
    1041             :  *
    1042             :  * Special cases:
    1043             :  *      if x is NaN, return x itself;
    1044             :  *      if |x|>1, return NaN with invalid signal.
    1045             :  */
    1046       28955 : double asin(double x) {
    1047             :   static const double
    1048             :       one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
    1049             :       huge = 1.000e+300,
    1050             :       pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
    1051             :       pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
    1052             :       pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
    1053             :                                             /* coefficient for R(x^2) */
    1054             :       pS0 = 1.66666666666666657415e-01,     /* 0x3FC55555, 0x55555555 */
    1055             :       pS1 = -3.25565818622400915405e-01,    /* 0xBFD4D612, 0x03EB6F7D */
    1056             :       pS2 = 2.01212532134862925881e-01,     /* 0x3FC9C155, 0x0E884455 */
    1057             :       pS3 = -4.00555345006794114027e-02,    /* 0xBFA48228, 0xB5688F3B */
    1058             :       pS4 = 7.91534994289814532176e-04,     /* 0x3F49EFE0, 0x7501B288 */
    1059             :       pS5 = 3.47933107596021167570e-05,     /* 0x3F023DE1, 0x0DFDF709 */
    1060             :       qS1 = -2.40339491173441421878e+00,    /* 0xC0033A27, 0x1C8A2D4B */
    1061             :       qS2 = 2.02094576023350569471e+00,     /* 0x40002AE5, 0x9C598AC8 */
    1062             :       qS3 = -6.88283971605453293030e-01,    /* 0xBFE6066C, 0x1B8D0159 */
    1063             :       qS4 = 7.70381505559019352791e-02;     /* 0x3FB3B8C5, 0xB12E9282 */
    1064             : 
    1065             :   double t, w, p, q, c, r, s;
    1066             :   int32_t hx, ix;
    1067             : 
    1068             :   t = 0;
    1069       28955 :   GET_HIGH_WORD(hx, x);
    1070       28955 :   ix = hx & 0x7fffffff;
    1071       28955 :   if (ix >= 0x3ff00000) { /* |x|>= 1 */
    1072             :     uint32_t lx;
    1073       20240 :     GET_LOW_WORD(lx, x);
    1074       20240 :     if (((ix - 0x3ff00000) | lx) == 0) /* asin(1)=+-pi/2 with inexact */
    1075        1372 :       return x * pio2_hi + x * pio2_lo;
    1076       18868 :     return (x - x) / (x - x);       /* asin(|x|>1) is NaN */
    1077        8715 :   } else if (ix < 0x3fe00000) {     /* |x|<0.5 */
    1078        6953 :     if (ix < 0x3e400000) {          /* if |x| < 2**-27 */
    1079        2636 :       if (huge + x > one) return x; /* return x with inexact if x!=0*/
    1080             :     } else {
    1081        4317 :       t = x * x;
    1082             :     }
    1083        4317 :     p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
    1084        4317 :     q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
    1085        4317 :     w = p / q;
    1086        4317 :     return x + x * w;
    1087             :   }
    1088             :   /* 1> |x|>= 0.5 */
    1089        1762 :   w = one - fabs(x);
    1090        1762 :   t = w * 0.5;
    1091        1762 :   p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
    1092        1762 :   q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
    1093        1762 :   s = sqrt(t);
    1094        1762 :   if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
    1095         120 :     w = p / q;
    1096         120 :     t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
    1097             :   } else {
    1098             :     w = s;
    1099        1642 :     SET_LOW_WORD(w, 0);
    1100        1642 :     c = (t - w * w) / (s + w);
    1101        1642 :     r = p / q;
    1102        1642 :     p = 2.0 * s * r - (pio2_lo - 2.0 * c);
    1103        1642 :     q = pio4_hi - 2.0 * w;
    1104        1642 :     t = pio4_hi - (p - q);
    1105             :   }
    1106        1762 :   if (hx > 0)
    1107             :     return t;
    1108             :   else
    1109         852 :     return -t;
    1110             : }
    1111             : /* asinh(x)
    1112             :  * Method :
    1113             :  *      Based on
    1114             :  *              asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
    1115             :  *      we have
    1116             :  *      asinh(x) := x  if  1+x*x=1,
    1117             :  *               := sign(x)*(log(x)+ln2)) for large |x|, else
    1118             :  *               := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
    1119             :  *               := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
    1120             :  */
    1121        1504 : double asinh(double x) {
    1122             :   static const double
    1123             :       one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
    1124             :       ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
    1125             :       huge = 1.00000000000000000000e+300;
    1126             : 
    1127             :   double t, w;
    1128             :   int32_t hx, ix;
    1129        1504 :   GET_HIGH_WORD(hx, x);
    1130        1504 :   ix = hx & 0x7fffffff;
    1131        1504 :   if (ix >= 0x7ff00000) return x + x; /* x is inf or NaN */
    1132        1338 :   if (ix < 0x3e300000) {              /* |x|<2**-28 */
    1133         290 :     if (huge + x > one) return x;     /* return x inexact except 0 */
    1134             :   }
    1135        1048 :   if (ix > 0x41b00000) { /* |x| > 2**28 */
    1136         430 :     w = log(fabs(x)) + ln2;
    1137         618 :   } else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
    1138         282 :     t = fabs(x);
    1139         282 :     w = log(2.0 * t + one / (sqrt(x * x + one) + t));
    1140             :   } else { /* 2.0 > |x| > 2**-28 */
    1141         336 :     t = x * x;
    1142         336 :     w = log1p(fabs(x) + t / (one + sqrt(one + t)));
    1143             :   }
    1144        1048 :   if (hx > 0) {
    1145             :     return w;
    1146             :   } else {
    1147         451 :     return -w;
    1148             :   }
    1149             : }
    1150             : 
    1151             : /* atan(x)
    1152             :  * Method
    1153             :  *   1. Reduce x to positive by atan(x) = -atan(-x).
    1154             :  *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
    1155             :  *      is further reduced to one of the following intervals and the
    1156             :  *      arctangent of t is evaluated by the corresponding formula:
    1157             :  *
    1158             :  *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
    1159             :  *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
    1160             :  *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
    1161             :  *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
    1162             :  *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
    1163             :  *
    1164             :  * Constants:
    1165             :  * The hexadecimal values are the intended ones for the following
    1166             :  * constants. The decimal values may be used, provided that the
    1167             :  * compiler will convert from decimal to binary accurately enough
    1168             :  * to produce the hexadecimal values shown.
    1169             :  */
    1170     1234342 : double atan(double x) {
    1171             :   static const double atanhi[] = {
    1172             :       4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
    1173             :       7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
    1174             :       9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
    1175             :       1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
    1176             :   };
    1177             : 
    1178             :   static const double atanlo[] = {
    1179             :       2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
    1180             :       3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
    1181             :       1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
    1182             :       6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
    1183             :   };
    1184             : 
    1185             :   static const double aT[] = {
    1186             :       3.33333333333329318027e-01,  /* 0x3FD55555, 0x5555550D */
    1187             :       -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
    1188             :       1.42857142725034663711e-01,  /* 0x3FC24924, 0x920083FF */
    1189             :       -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
    1190             :       9.09088713343650656196e-02,  /* 0x3FB745CD, 0xC54C206E */
    1191             :       -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
    1192             :       6.66107313738753120669e-02,  /* 0x3FB10D66, 0xA0D03D51 */
    1193             :       -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
    1194             :       4.97687799461593236017e-02,  /* 0x3FA97B4B, 0x24760DEB */
    1195             :       -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
    1196             :       1.62858201153657823623e-02,  /* 0x3F90AD3A, 0xE322DA11 */
    1197             :   };
    1198             : 
    1199             :   static const double one = 1.0, huge = 1.0e300;
    1200             : 
    1201             :   double w, s1, s2, z;
    1202             :   int32_t ix, hx, id;
    1203             : 
    1204     1234342 :   GET_HIGH_WORD(hx, x);
    1205     1234342 :   ix = hx & 0x7fffffff;
    1206     1234342 :   if (ix >= 0x44100000) { /* if |x| >= 2^66 */
    1207             :     uint32_t low;
    1208        2968 :     GET_LOW_WORD(low, x);
    1209        2968 :     if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0)))
    1210         832 :       return x + x; /* NaN */
    1211        2136 :     if (hx > 0)
    1212        1063 :       return atanhi[3] + *(volatile double *)&atanlo[3];
    1213             :     else
    1214        1073 :       return -atanhi[3] - *(volatile double *)&atanlo[3];
    1215             :   }
    1216     1231374 :   if (ix < 0x3fdc0000) {            /* |x| < 0.4375 */
    1217      598869 :     if (ix < 0x3e400000) {          /* |x| < 2^-27 */
    1218      171378 :       if (huge + x > one) return x; /* raise inexact */
    1219             :     }
    1220             :     id = -1;
    1221             :   } else {
    1222      632505 :     x = fabs(x);
    1223      632505 :     if (ix < 0x3ff30000) {   /* |x| < 1.1875 */
    1224       59727 :       if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
    1225             :         id = 0;
    1226       20186 :         x = (2.0 * x - one) / (2.0 + x);
    1227             :       } else { /* 11/16<=|x|< 19/16 */
    1228             :         id = 1;
    1229       39541 :         x = (x - one) / (x + one);
    1230             :       }
    1231             :     } else {
    1232      572778 :       if (ix < 0x40038000) { /* |x| < 2.4375 */
    1233             :         id = 2;
    1234       25328 :         x = (x - 1.5) / (one + 1.5 * x);
    1235             :       } else { /* 2.4375 <= |x| < 2^66 */
    1236             :         id = 3;
    1237      547450 :         x = -1.0 / x;
    1238             :       }
    1239             :     }
    1240             :   }
    1241             :   /* end of argument reduction */
    1242     1059996 :   z = x * x;
    1243     1059996 :   w = z * z;
    1244             :   /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
    1245     1059996 :   s1 = z * (aT[0] +
    1246     2119992 :             w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
    1247     1059996 :   s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
    1248     1059996 :   if (id < 0) {
    1249      427491 :     return x - x * (s1 + s2);
    1250             :   } else {
    1251      632505 :     z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
    1252      632505 :     return (hx < 0) ? -z : z;
    1253             :   }
    1254             : }
    1255             : 
    1256             : /* atan2(y,x)
    1257             :  * Method :
    1258             :  *  1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
    1259             :  *  2. Reduce x to positive by (if x and y are unexceptional):
    1260             :  *    ARG (x+iy) = arctan(y/x)       ... if x > 0,
    1261             :  *    ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
    1262             :  *
    1263             :  * Special cases:
    1264             :  *
    1265             :  *  ATAN2((anything), NaN ) is NaN;
    1266             :  *  ATAN2(NAN , (anything) ) is NaN;
    1267             :  *  ATAN2(+-0, +(anything but NaN)) is +-0  ;
    1268             :  *  ATAN2(+-0, -(anything but NaN)) is +-pi ;
    1269             :  *  ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
    1270             :  *  ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
    1271             :  *  ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
    1272             :  *  ATAN2(+-INF,+INF ) is +-pi/4 ;
    1273             :  *  ATAN2(+-INF,-INF ) is +-3pi/4;
    1274             :  *  ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
    1275             :  *
    1276             :  * Constants:
    1277             :  * The hexadecimal values are the intended ones for the following
    1278             :  * constants. The decimal values may be used, provided that the
    1279             :  * compiler will convert from decimal to binary accurately enough
    1280             :  * to produce the hexadecimal values shown.
    1281             :  */
    1282     1435367 : double atan2(double y, double x) {
    1283             :   static volatile double tiny = 1.0e-300;
    1284             :   static const double
    1285             :       zero = 0.0,
    1286             :       pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
    1287             :       pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
    1288             :       pi = 3.1415926535897931160E+00;     /* 0x400921FB, 0x54442D18 */
    1289             :   static volatile double pi_lo =
    1290             :       1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
    1291             : 
    1292             :   double z;
    1293             :   int32_t k, m, hx, hy, ix, iy;
    1294             :   uint32_t lx, ly;
    1295             : 
    1296     1435367 :   EXTRACT_WORDS(hx, lx, x);
    1297     1435367 :   ix = hx & 0x7fffffff;
    1298     1435367 :   EXTRACT_WORDS(hy, ly, y);
    1299     1435367 :   iy = hy & 0x7fffffff;
    1300     2851230 :   if (((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x7ff00000) ||
    1301     1415863 :       ((iy | ((ly | -static_cast<int32_t>(ly)) >> 31)) > 0x7ff00000)) {
    1302       38702 :     return x + y; /* x or y is NaN */
    1303             :   }
    1304     1396665 :   if (((hx - 0x3ff00000) | lx) == 0) return atan(y); /* x=1.0 */
    1305     1382601 :   m = ((hy >> 31) & 1) | ((hx >> 30) & 2);           /* 2*sign(x)+sign(y) */
    1306             : 
    1307             :   /* when y = 0 */
    1308     1382601 :   if ((iy | ly) == 0) {
    1309       37144 :     switch (m) {
    1310             :       case 0:
    1311             :       case 1:
    1312             :         return y; /* atan(+-0,+anything)=+-0 */
    1313             :       case 2:
    1314        5834 :         return pi + tiny; /* atan(+0,-anything) = pi */
    1315             :       case 3:
    1316        5834 :         return -pi - tiny; /* atan(-0,-anything) =-pi */
    1317             :     }
    1318             :   }
    1319             :   /* when x = 0 */
    1320     1345457 :   if ((ix | lx) == 0) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
    1321             : 
    1322             :   /* when x is INF */
    1323     1308945 :   if (ix == 0x7ff00000) {
    1324       36590 :     if (iy == 0x7ff00000) {
    1325        1080 :       switch (m) {
    1326             :         case 0:
    1327         270 :           return pi_o_4 + tiny; /* atan(+INF,+INF) */
    1328             :         case 1:
    1329         270 :           return -pi_o_4 - tiny; /* atan(-INF,+INF) */
    1330             :         case 2:
    1331         270 :           return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/
    1332             :         case 3:
    1333         270 :           return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/
    1334             :       }
    1335             :     } else {
    1336       35510 :       switch (m) {
    1337             :         case 0:
    1338             :           return zero; /* atan(+...,+INF) */
    1339             :         case 1:
    1340        7080 :           return -zero; /* atan(-...,+INF) */
    1341             :         case 2:
    1342       10682 :           return pi + tiny; /* atan(+...,-INF) */
    1343             :         case 3:
    1344        7066 :           return -pi - tiny; /* atan(-...,-INF) */
    1345             :       }
    1346             :     }
    1347             :   }
    1348             :   /* when y is INF */
    1349     1272355 :   if (iy == 0x7ff00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
    1350             : 
    1351             :   /* compute y/x */
    1352     1237243 :   k = (iy - ix) >> 20;
    1353     1237243 :   if (k > 60) { /* |y/x| >  2**60 */
    1354       35162 :     z = pi_o_2 + 0.5 * pi_lo;
    1355       35162 :     m &= 1;
    1356     1202081 :   } else if (hx < 0 && k < -60) {
    1357             :     z = 0.0; /* 0 > |y|/x > -2**-60 */
    1358             :   } else {
    1359     1191365 :     z = atan(fabs(y / x)); /* safe to do y/x */
    1360             :   }
    1361     1237243 :   switch (m) {
    1362             :     case 0:
    1363             :       return z; /* atan(+,+) */
    1364             :     case 1:
    1365      345412 :       return -z; /* atan(-,+) */
    1366             :     case 2:
    1367      222426 :       return pi - (z - pi_lo); /* atan(+,-) */
    1368             :     default:                   /* case 3 */
    1369      147536 :       return (z - pi_lo) - pi; /* atan(-,-) */
    1370             :   }
    1371             : }
    1372             : 
    1373             : /* cos(x)
    1374             :  * Return cosine function of x.
    1375             :  *
    1376             :  * kernel function:
    1377             :  *      __kernel_sin            ... sine function on [-pi/4,pi/4]
    1378             :  *      __kernel_cos            ... cosine function on [-pi/4,pi/4]
    1379             :  *      __ieee754_rem_pio2      ... argument reduction routine
    1380             :  *
    1381             :  * Method.
    1382             :  *      Let S,C and T denote the sin, cos and tan respectively on
    1383             :  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
    1384             :  *      in [-pi/4 , +pi/4], and let n = k mod 4.
    1385             :  *      We have
    1386             :  *
    1387             :  *          n        sin(x)      cos(x)        tan(x)
    1388             :  *     ----------------------------------------------------------
    1389             :  *          0          S           C             T
    1390             :  *          1          C          -S            -1/T
    1391             :  *          2         -S          -C             T
    1392             :  *          3         -C           S            -1/T
    1393             :  *     ----------------------------------------------------------
    1394             :  *
    1395             :  * Special cases:
    1396             :  *      Let trig be any of sin, cos, or tan.
    1397             :  *      trig(+-INF)  is NaN, with signals;
    1398             :  *      trig(NaN)    is that NaN;
    1399             :  *
    1400             :  * Accuracy:
    1401             :  *      TRIG(x) returns trig(x) nearly rounded
    1402             :  */
    1403    15908749 : double cos(double x) {
    1404             :   double y[2], z = 0.0;
    1405             :   int32_t n, ix;
    1406             : 
    1407             :   /* High word of x. */
    1408    15908749 :   GET_HIGH_WORD(ix, x);
    1409             : 
    1410             :   /* |x| ~< pi/4 */
    1411    15908749 :   ix &= 0x7fffffff;
    1412    15908749 :   if (ix <= 0x3fe921fb) {
    1413    15879289 :     return __kernel_cos(x, z);
    1414       29460 :   } else if (ix >= 0x7ff00000) {
    1415             :     /* cos(Inf or NaN) is NaN */
    1416        2480 :     return x - x;
    1417             :   } else {
    1418             :     /* argument reduction needed */
    1419       26980 :     n = __ieee754_rem_pio2(x, y);
    1420       26980 :     switch (n & 3) {
    1421             :       case 0:
    1422       19956 :         return __kernel_cos(y[0], y[1]);
    1423             :       case 1:
    1424       13558 :         return -__kernel_sin(y[0], y[1], 1);
    1425             :       case 2:
    1426        7604 :         return -__kernel_cos(y[0], y[1]);
    1427             :       default:
    1428       12842 :         return __kernel_sin(y[0], y[1], 1);
    1429             :     }
    1430             :   }
    1431             : }
    1432             : 
    1433             : /* exp(x)
    1434             :  * Returns the exponential of x.
    1435             :  *
    1436             :  * Method
    1437             :  *   1. Argument reduction:
    1438             :  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
    1439             :  *      Given x, find r and integer k such that
    1440             :  *
    1441             :  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
    1442             :  *
    1443             :  *      Here r will be represented as r = hi-lo for better
    1444             :  *      accuracy.
    1445             :  *
    1446             :  *   2. Approximation of exp(r) by a special rational function on
    1447             :  *      the interval [0,0.34658]:
    1448             :  *      Write
    1449             :  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
    1450             :  *      We use a special Remes algorithm on [0,0.34658] to generate
    1451             :  *      a polynomial of degree 5 to approximate R. The maximum error
    1452             :  *      of this polynomial approximation is bounded by 2**-59. In
    1453             :  *      other words,
    1454             :  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
    1455             :  *      (where z=r*r, and the values of P1 to P5 are listed below)
    1456             :  *      and
    1457             :  *          |                  5          |     -59
    1458             :  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
    1459             :  *          |                             |
    1460             :  *      The computation of exp(r) thus becomes
    1461             :  *                             2*r
    1462             :  *              exp(r) = 1 + -------
    1463             :  *                            R - r
    1464             :  *                                 r*R1(r)
    1465             :  *                     = 1 + r + ----------- (for better accuracy)
    1466             :  *                                2 - R1(r)
    1467             :  *      where
    1468             :  *                               2       4             10
    1469             :  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
    1470             :  *
    1471             :  *   3. Scale back to obtain exp(x):
    1472             :  *      From step 1, we have
    1473             :  *         exp(x) = 2^k * exp(r)
    1474             :  *
    1475             :  * Special cases:
    1476             :  *      exp(INF) is INF, exp(NaN) is NaN;
    1477             :  *      exp(-INF) is 0, and
    1478             :  *      for finite argument, only exp(0)=1 is exact.
    1479             :  *
    1480             :  * Accuracy:
    1481             :  *      according to an error analysis, the error is always less than
    1482             :  *      1 ulp (unit in the last place).
    1483             :  *
    1484             :  * Misc. info.
    1485             :  *      For IEEE double
    1486             :  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
    1487             :  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
    1488             :  *
    1489             :  * Constants:
    1490             :  * The hexadecimal values are the intended ones for the following
    1491             :  * constants. The decimal values may be used, provided that the
    1492             :  * compiler will convert from decimal to binary accurately enough
    1493             :  * to produce the hexadecimal values shown.
    1494             :  */
    1495      329321 : double exp(double x) {
    1496             :   static const double
    1497             :       one = 1.0,
    1498             :       halF[2] = {0.5, -0.5},
    1499             :       o_threshold = 7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
    1500             :       u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
    1501             :       ln2HI[2] = {6.93147180369123816490e-01,    /* 0x3fe62e42, 0xfee00000 */
    1502             :                   -6.93147180369123816490e-01},  /* 0xbfe62e42, 0xfee00000 */
    1503             :       ln2LO[2] = {1.90821492927058770002e-10,    /* 0x3dea39ef, 0x35793c76 */
    1504             :                   -1.90821492927058770002e-10},  /* 0xbdea39ef, 0x35793c76 */
    1505             :       invln2 = 1.44269504088896338700e+00,       /* 0x3ff71547, 0x652b82fe */
    1506             :       P1 = 1.66666666666666019037e-01,           /* 0x3FC55555, 0x5555553E */
    1507             :       P2 = -2.77777777770155933842e-03,          /* 0xBF66C16C, 0x16BEBD93 */
    1508             :       P3 = 6.61375632143793436117e-05,           /* 0x3F11566A, 0xAF25DE2C */
    1509             :       P4 = -1.65339022054652515390e-06,          /* 0xBEBBBD41, 0xC5D26BF1 */
    1510             :       P5 = 4.13813679705723846039e-08,           /* 0x3E663769, 0x72BEA4D0 */
    1511             :       E = 2.718281828459045;                     /* 0x4005bf0a, 0x8b145769 */
    1512             : 
    1513             :   static volatile double
    1514             :       huge = 1.0e+300,
    1515             :       twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
    1516             :       two1023 = 8.988465674311579539e307;     /* 0x1p1023 */
    1517             : 
    1518             :   double y, hi = 0.0, lo = 0.0, c, t, twopk;
    1519             :   int32_t k = 0, xsb;
    1520             :   uint32_t hx;
    1521             : 
    1522      329321 :   GET_HIGH_WORD(hx, x);
    1523      329321 :   xsb = (hx >> 31) & 1; /* sign bit of x */
    1524      329321 :   hx &= 0x7fffffff;     /* high word of |x| */
    1525             : 
    1526             :   /* filter out non-finite argument */
    1527      329321 :   if (hx >= 0x40862E42) { /* if |x|>=709.78... */
    1528       14490 :     if (hx >= 0x7ff00000) {
    1529             :       uint32_t lx;
    1530        2530 :       GET_LOW_WORD(lx, x);
    1531        2530 :       if (((hx & 0xfffff) | lx) != 0)
    1532         866 :         return x + x; /* NaN */
    1533             :       else
    1534        1664 :         return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
    1535             :     }
    1536       11960 :     if (x > o_threshold) return huge * huge;         /* overflow */
    1537        4679 :     if (x < u_threshold) return twom1000 * twom1000; /* underflow */
    1538             :   }
    1539             : 
    1540             :   /* argument reduction */
    1541      314834 :   if (hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
    1542      307877 :     if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
    1543             :       /* TODO(rtoy): We special case exp(1) here to return the correct
    1544             :        * value of E, as the computation below would get the last bit
    1545             :        * wrong. We should probably fix the algorithm instead.
    1546             :        */
    1547        3765 :       if (x == 1.0) return E;
    1548        2909 :       hi = x - ln2HI[xsb];
    1549        2909 :       lo = ln2LO[xsb];
    1550        2909 :       k = 1 - xsb - xsb;
    1551             :     } else {
    1552      304112 :       k = static_cast<int>(invln2 * x + halF[xsb]);
    1553      304112 :       t = k;
    1554      304112 :       hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
    1555      304112 :       lo = t * ln2LO[0];
    1556             :     }
    1557      307021 :     STRICT_ASSIGN(double, x, hi - lo);
    1558        6957 :   } else if (hx < 0x3e300000) {         /* when |x|<2**-28 */
    1559        2668 :     if (huge + x > one) return one + x; /* trigger inexact */
    1560             :   } else {
    1561             :     k = 0;
    1562             :   }
    1563             : 
    1564             :   /* x is now in primary range */
    1565      311310 :   t = x * x;
    1566      311310 :   if (k >= -1021) {
    1567      311308 :     INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0);
    1568             :   } else {
    1569           2 :     INSERT_WORDS(twopk, 0x3ff00000 + ((k + 1000) << 20), 0);
    1570             :   }
    1571      311310 :   c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
    1572      311310 :   if (k == 0) {
    1573        4289 :     return one - ((x * c) / (c - 2.0) - x);
    1574             :   } else {
    1575      307021 :     y = one - ((lo - (x * c) / (2.0 - c)) - hi);
    1576             :   }
    1577      307021 :   if (k >= -1021) {
    1578      307019 :     if (k == 1024) return y * 2.0 * two1023;
    1579      306961 :     return y * twopk;
    1580             :   } else {
    1581           2 :     return y * twopk * twom1000;
    1582             :   }
    1583             : }
    1584             : 
    1585             : /*
    1586             :  * Method :
    1587             :  *    1.Reduced x to positive by atanh(-x) = -atanh(x)
    1588             :  *    2.For x>=0.5
    1589             :  *              1              2x                          x
    1590             :  *  atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
    1591             :  *              2             1 - x                      1 - x
    1592             :  *
    1593             :  *   For x<0.5
    1594             :  *  atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
    1595             :  *
    1596             :  * Special cases:
    1597             :  *  atanh(x) is NaN if |x| > 1 with signal;
    1598             :  *  atanh(NaN) is that NaN with no signal;
    1599             :  *  atanh(+-1) is +-INF with signal.
    1600             :  *
    1601             :  */
    1602        1586 : double atanh(double x) {
    1603             :   static const double one = 1.0, huge = 1e300;
    1604             :   static const double zero = 0.0;
    1605             : 
    1606             :   double t;
    1607             :   int32_t hx, ix;
    1608             :   uint32_t lx;
    1609        1586 :   EXTRACT_WORDS(hx, lx, x);
    1610        1586 :   ix = hx & 0x7fffffff;
    1611        1586 :   if ((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x3ff00000) /* |x|>1 */
    1612         903 :     return (x - x) / (x - x);
    1613         683 :   if (ix == 0x3ff00000) return x / zero;
    1614         617 :   if (ix < 0x3e300000 && (huge + x) > zero) return x; /* x<2**-28 */
    1615         319 :   SET_HIGH_WORD(x, ix);
    1616         319 :   if (ix < 0x3fe00000) { /* x < 0.5 */
    1617         126 :     t = x + x;
    1618         126 :     t = 0.5 * log1p(t + t * x / (one - x));
    1619             :   } else {
    1620         193 :     t = 0.5 * log1p((x + x) / (one - x));
    1621             :   }
    1622         319 :   if (hx >= 0)
    1623             :     return t;
    1624             :   else
    1625         128 :     return -t;
    1626             : }
    1627             : 
    1628             : /* log(x)
    1629             :  * Return the logrithm of x
    1630             :  *
    1631             :  * Method :
    1632             :  *   1. Argument Reduction: find k and f such that
    1633             :  *     x = 2^k * (1+f),
    1634             :  *     where  sqrt(2)/2 < 1+f < sqrt(2) .
    1635             :  *
    1636             :  *   2. Approximation of log(1+f).
    1637             :  *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    1638             :  *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    1639             :  *         = 2s + s*R
    1640             :  *      We use a special Reme algorithm on [0,0.1716] to generate
    1641             :  *  a polynomial of degree 14 to approximate R The maximum error
    1642             :  *  of this polynomial approximation is bounded by 2**-58.45. In
    1643             :  *  other words,
    1644             :  *            2      4      6      8      10      12      14
    1645             :  *      R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
    1646             :  *    (the values of Lg1 to Lg7 are listed in the program)
    1647             :  *  and
    1648             :  *      |      2          14          |     -58.45
    1649             :  *      | Lg1*s +...+Lg7*s    -  R(z) | <= 2
    1650             :  *      |                             |
    1651             :  *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    1652             :  *  In order to guarantee error in log below 1ulp, we compute log
    1653             :  *  by
    1654             :  *    log(1+f) = f - s*(f - R)  (if f is not too large)
    1655             :  *    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
    1656             :  *
    1657             :  *  3. Finally,  log(x) = k*ln2 + log(1+f).
    1658             :  *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
    1659             :  *     Here ln2 is split into two floating point number:
    1660             :  *      ln2_hi + ln2_lo,
    1661             :  *     where n*ln2_hi is always exact for |n| < 2000.
    1662             :  *
    1663             :  * Special cases:
    1664             :  *  log(x) is NaN with signal if x < 0 (including -INF) ;
    1665             :  *  log(+INF) is +INF; log(0) is -INF with signal;
    1666             :  *  log(NaN) is that NaN with no signal.
    1667             :  *
    1668             :  * Accuracy:
    1669             :  *  according to an error analysis, the error is always less than
    1670             :  *  1 ulp (unit in the last place).
    1671             :  *
    1672             :  * Constants:
    1673             :  * The hexadecimal values are the intended ones for the following
    1674             :  * constants. The decimal values may be used, provided that the
    1675             :  * compiler will convert from decimal to binary accurately enough
    1676             :  * to produce the hexadecimal values shown.
    1677             :  */
    1678     1265373 : double log(double x) {
    1679             :   static const double                      /* -- */
    1680             :       ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
    1681             :       ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
    1682             :       two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
    1683             :       Lg1 = 6.666666666666735130e-01,      /* 3FE55555 55555593 */
    1684             :       Lg2 = 3.999999999940941908e-01,      /* 3FD99999 9997FA04 */
    1685             :       Lg3 = 2.857142874366239149e-01,      /* 3FD24924 94229359 */
    1686             :       Lg4 = 2.222219843214978396e-01,      /* 3FCC71C5 1D8E78AF */
    1687             :       Lg5 = 1.818357216161805012e-01,      /* 3FC74664 96CB03DE */
    1688             :       Lg6 = 1.531383769920937332e-01,      /* 3FC39A09 D078C69F */
    1689             :       Lg7 = 1.479819860511658591e-01;      /* 3FC2F112 DF3E5244 */
    1690             : 
    1691             :   static const double zero = 0.0;
    1692             :   static volatile double vzero = 0.0;
    1693             : 
    1694             :   double hfsq, f, s, z, R, w, t1, t2, dk;
    1695             :   int32_t k, hx, i, j;
    1696             :   uint32_t lx;
    1697             : 
    1698     1265373 :   EXTRACT_WORDS(hx, lx, x);
    1699             : 
    1700             :   k = 0;
    1701     1265373 :   if (hx < 0x00100000) { /* x < 2**-1022  */
    1702       43843 :     if (((hx & 0x7fffffff) | lx) == 0)
    1703       18197 :       return -two54 / vzero;           /* log(+-0)=-inf */
    1704       25646 :     if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
    1705             :     k -= 54;
    1706           2 :     x *= two54; /* subnormal number, scale up x */
    1707           2 :     GET_HIGH_WORD(hx, x);
    1708             :   }
    1709     1221532 :   if (hx >= 0x7ff00000) return x + x;
    1710     1204861 :   k += (hx >> 20) - 1023;
    1711     1204861 :   hx &= 0x000fffff;
    1712     1204861 :   i = (hx + 0x95f64) & 0x100000;
    1713     1204861 :   SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
    1714     1204861 :   k += (i >> 20);
    1715     1204861 :   f = x - 1.0;
    1716     1204861 :   if ((0x000fffff & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
    1717        8921 :     if (f == zero) {
    1718        7426 :       if (k == 0) {
    1719             :         return zero;
    1720             :       } else {
    1721        6442 :         dk = static_cast<double>(k);
    1722        6442 :         return dk * ln2_hi + dk * ln2_lo;
    1723             :       }
    1724             :     }
    1725        1495 :     R = f * f * (0.5 - 0.33333333333333333 * f);
    1726        1495 :     if (k == 0) {
    1727          74 :       return f - R;
    1728             :     } else {
    1729        1421 :       dk = static_cast<double>(k);
    1730        1421 :       return dk * ln2_hi - ((R - dk * ln2_lo) - f);
    1731             :     }
    1732             :   }
    1733     1195940 :   s = f / (2.0 + f);
    1734     1195940 :   dk = static_cast<double>(k);
    1735     1195940 :   z = s * s;
    1736     1195940 :   i = hx - 0x6147a;
    1737     1195940 :   w = z * z;
    1738     1195940 :   j = 0x6b851 - hx;
    1739     1195940 :   t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
    1740     1195940 :   t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
    1741     1195940 :   i |= j;
    1742     1195940 :   R = t2 + t1;
    1743     1195940 :   if (i > 0) {
    1744       48268 :     hfsq = 0.5 * f * f;
    1745       48268 :     if (k == 0)
    1746         904 :       return f - (hfsq - s * (hfsq + R));
    1747             :     else
    1748       47364 :       return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
    1749             :   } else {
    1750     1147672 :     if (k == 0)
    1751        9360 :       return f - s * (f - R);
    1752             :     else
    1753     1138312 :       return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
    1754             :   }
    1755             : }
    1756             : 
    1757             : /* double log1p(double x)
    1758             :  *
    1759             :  * Method :
    1760             :  *   1. Argument Reduction: find k and f such that
    1761             :  *      1+x = 2^k * (1+f),
    1762             :  *     where  sqrt(2)/2 < 1+f < sqrt(2) .
    1763             :  *
    1764             :  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
    1765             :  *  may not be representable exactly. In that case, a correction
    1766             :  *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
    1767             :  *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
    1768             :  *  and add back the correction term c/u.
    1769             :  *  (Note: when x > 2**53, one can simply return log(x))
    1770             :  *
    1771             :  *   2. Approximation of log1p(f).
    1772             :  *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    1773             :  *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    1774             :  *         = 2s + s*R
    1775             :  *      We use a special Reme algorithm on [0,0.1716] to generate
    1776             :  *  a polynomial of degree 14 to approximate R The maximum error
    1777             :  *  of this polynomial approximation is bounded by 2**-58.45. In
    1778             :  *  other words,
    1779             :  *            2      4      6      8      10      12      14
    1780             :  *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
    1781             :  *    (the values of Lp1 to Lp7 are listed in the program)
    1782             :  *  and
    1783             :  *      |      2          14          |     -58.45
    1784             :  *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
    1785             :  *      |                             |
    1786             :  *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    1787             :  *  In order to guarantee error in log below 1ulp, we compute log
    1788             :  *  by
    1789             :  *    log1p(f) = f - (hfsq - s*(hfsq+R)).
    1790             :  *
    1791             :  *  3. Finally, log1p(x) = k*ln2 + log1p(f).
    1792             :  *           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
    1793             :  *     Here ln2 is split into two floating point number:
    1794             :  *      ln2_hi + ln2_lo,
    1795             :  *     where n*ln2_hi is always exact for |n| < 2000.
    1796             :  *
    1797             :  * Special cases:
    1798             :  *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
    1799             :  *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
    1800             :  *  log1p(NaN) is that NaN with no signal.
    1801             :  *
    1802             :  * Accuracy:
    1803             :  *  according to an error analysis, the error is always less than
    1804             :  *  1 ulp (unit in the last place).
    1805             :  *
    1806             :  * Constants:
    1807             :  * The hexadecimal values are the intended ones for the following
    1808             :  * constants. The decimal values may be used, provided that the
    1809             :  * compiler will convert from decimal to binary accurately enough
    1810             :  * to produce the hexadecimal values shown.
    1811             :  *
    1812             :  * Note: Assuming log() return accurate answer, the following
    1813             :  *   algorithm can be used to compute log1p(x) to within a few ULP:
    1814             :  *
    1815             :  *    u = 1+x;
    1816             :  *    if(u==1.0) return x ; else
    1817             :  *         return log(u)*(x/(u-1.0));
    1818             :  *
    1819             :  *   See HP-15C Advanced Functions Handbook, p.193.
    1820             :  */
    1821       86257 : double log1p(double x) {
    1822             :   static const double                      /* -- */
    1823             :       ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
    1824             :       ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
    1825             :       two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
    1826             :       Lp1 = 6.666666666666735130e-01,      /* 3FE55555 55555593 */
    1827             :       Lp2 = 3.999999999940941908e-01,      /* 3FD99999 9997FA04 */
    1828             :       Lp3 = 2.857142874366239149e-01,      /* 3FD24924 94229359 */
    1829             :       Lp4 = 2.222219843214978396e-01,      /* 3FCC71C5 1D8E78AF */
    1830             :       Lp5 = 1.818357216161805012e-01,      /* 3FC74664 96CB03DE */
    1831             :       Lp6 = 1.531383769920937332e-01,      /* 3FC39A09 D078C69F */
    1832             :       Lp7 = 1.479819860511658591e-01;      /* 3FC2F112 DF3E5244 */
    1833             : 
    1834             :   static const double zero = 0.0;
    1835             :   static volatile double vzero = 0.0;
    1836             : 
    1837             :   double hfsq, f, c, s, z, R, u;
    1838             :   int32_t k, hx, hu, ax;
    1839             : 
    1840       86257 :   GET_HIGH_WORD(hx, x);
    1841       86257 :   ax = hx & 0x7fffffff;
    1842             : 
    1843             :   k = 1;
    1844       86257 :   if (hx < 0x3FDA827A) {    /* 1+x < sqrt(2)+ */
    1845       44025 :     if (ax >= 0x3ff00000) { /* x <= -1.0 */
    1846         302 :       if (x == -1.0)
    1847          33 :         return -two54 / vzero; /* log1p(-1)=+inf */
    1848             :       else
    1849         269 :         return (x - x) / (x - x); /* log1p(x<-1)=NaN */
    1850             :     }
    1851       43723 :     if (ax < 0x3e200000) {    /* |x| < 2**-29 */
    1852       42302 :       if (two54 + x > zero    /* raise inexact */
    1853       42302 :           && ax < 0x3c900000) /* |x| < 2**-54 */
    1854             :         return x;
    1855             :       else
    1856        1123 :         return x - x * x * 0.5;
    1857             :     }
    1858        1421 :     if (hx > 0 || hx <= static_cast<int32_t>(0xbfd2bec4)) {
    1859             :       k = 0;
    1860             :       f = x;
    1861             :       hu = 1;
    1862             :     } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
    1863             :   }
    1864       43653 :   if (hx >= 0x7ff00000) return x + x;
    1865       43536 :   if (k != 0) {
    1866       42127 :     if (hx < 0x43400000) {
    1867         919 :       STRICT_ASSIGN(double, u, 1.0 + x);
    1868         919 :       GET_HIGH_WORD(hu, u);
    1869         919 :       k = (hu >> 20) - 1023;
    1870         919 :       c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
    1871         919 :       c /= u;
    1872             :     } else {
    1873             :       u = x;
    1874       41208 :       GET_HIGH_WORD(hu, u);
    1875       41208 :       k = (hu >> 20) - 1023;
    1876             :       c = 0;
    1877             :     }
    1878       42127 :     hu &= 0x000fffff;
    1879             :     /*
    1880             :      * The approximation to sqrt(2) used in thresholds is not
    1881             :      * critical.  However, the ones used above must give less
    1882             :      * strict bounds than the one here so that the k==0 case is
    1883             :      * never reached from here, since here we have committed to
    1884             :      * using the correction term but don't use it if k==0.
    1885             :      */
    1886       42127 :     if (hu < 0x6a09e) {                  /* u ~< sqrt(2) */
    1887       21047 :       SET_HIGH_WORD(u, hu | 0x3ff00000); /* normalize u */
    1888             :     } else {
    1889       21080 :       k += 1;
    1890       21080 :       SET_HIGH_WORD(u, hu | 0x3fe00000); /* normalize u/2 */
    1891       21080 :       hu = (0x00100000 - hu) >> 2;
    1892             :     }
    1893       42127 :     f = u - 1.0;
    1894             :   }
    1895       43536 :   hfsq = 0.5 * f * f;
    1896       43536 :   if (hu == 0) { /* |f| < 2**-20 */
    1897         243 :     if (f == zero) {
    1898          87 :       if (k == 0) {
    1899             :         return zero;
    1900             :       } else {
    1901          87 :         c += k * ln2_lo;
    1902          87 :         return k * ln2_hi + c;
    1903             :       }
    1904             :     }
    1905         156 :     R = hfsq * (1.0 - 0.66666666666666666 * f);
    1906         156 :     if (k == 0)
    1907           0 :       return f - R;
    1908             :     else
    1909         156 :       return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
    1910             :   }
    1911       43293 :   s = f / (2.0 + f);
    1912       43293 :   z = s * s;
    1913       43293 :   R = z * (Lp1 +
    1914       86586 :            z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
    1915       43293 :   if (k == 0)
    1916        1409 :     return f - (hfsq - s * (hfsq + R));
    1917             :   else
    1918       41884 :     return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
    1919             : }
    1920             : 
    1921             : /*
    1922             :  * k_log1p(f):
    1923             :  * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
    1924             :  *
    1925             :  * The following describes the overall strategy for computing
    1926             :  * logarithms in base e.  The argument reduction and adding the final
    1927             :  * term of the polynomial are done by the caller for increased accuracy
    1928             :  * when different bases are used.
    1929             :  *
    1930             :  * Method :
    1931             :  *   1. Argument Reduction: find k and f such that
    1932             :  *         x = 2^k * (1+f),
    1933             :  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
    1934             :  *
    1935             :  *   2. Approximation of log(1+f).
    1936             :  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    1937             :  *            = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    1938             :  *            = 2s + s*R
    1939             :  *      We use a special Reme algorithm on [0,0.1716] to generate
    1940             :  *      a polynomial of degree 14 to approximate R The maximum error
    1941             :  *      of this polynomial approximation is bounded by 2**-58.45. In
    1942             :  *      other words,
    1943             :  *          2      4      6      8      10      12      14
    1944             :  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
    1945             :  *      (the values of Lg1 to Lg7 are listed in the program)
    1946             :  *      and
    1947             :  *          |      2          14          |     -58.45
    1948             :  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
    1949             :  *          |                             |
    1950             :  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    1951             :  *      In order to guarantee error in log below 1ulp, we compute log
    1952             :  *      by
    1953             :  *          log(1+f) = f - s*(f - R)            (if f is not too large)
    1954             :  *          log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
    1955             :  *
    1956             :  *   3. Finally,  log(x) = k*ln2 + log(1+f).
    1957             :  *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
    1958             :  *      Here ln2 is split into two floating point number:
    1959             :  *          ln2_hi + ln2_lo,
    1960             :  *      where n*ln2_hi is always exact for |n| < 2000.
    1961             :  *
    1962             :  * Special cases:
    1963             :  *      log(x) is NaN with signal if x < 0 (including -INF) ;
    1964             :  *      log(+INF) is +INF; log(0) is -INF with signal;
    1965             :  *      log(NaN) is that NaN with no signal.
    1966             :  *
    1967             :  * Accuracy:
    1968             :  *      according to an error analysis, the error is always less than
    1969             :  *      1 ulp (unit in the last place).
    1970             :  *
    1971             :  * Constants:
    1972             :  * The hexadecimal values are the intended ones for the following
    1973             :  * constants. The decimal values may be used, provided that the
    1974             :  * compiler will convert from decimal to binary accurately enough
    1975             :  * to produce the hexadecimal values shown.
    1976             :  */
    1977             : 
    1978             : static const double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
    1979             :     Lg2 = 3.999999999940941908e-01,                 /* 3FD99999 9997FA04 */
    1980             :     Lg3 = 2.857142874366239149e-01,                 /* 3FD24924 94229359 */
    1981             :     Lg4 = 2.222219843214978396e-01,                 /* 3FCC71C5 1D8E78AF */
    1982             :     Lg5 = 1.818357216161805012e-01,                 /* 3FC74664 96CB03DE */
    1983             :     Lg6 = 1.531383769920937332e-01,                 /* 3FC39A09 D078C69F */
    1984             :     Lg7 = 1.479819860511658591e-01;                 /* 3FC2F112 DF3E5244 */
    1985             : 
    1986             : /*
    1987             :  * We always inline k_log1p(), since doing so produces a
    1988             :  * substantial performance improvement (~40% on amd64).
    1989             :  */
    1990       89096 : static inline double k_log1p(double f) {
    1991             :   double hfsq, s, z, R, w, t1, t2;
    1992             : 
    1993       89096 :   s = f / (2.0 + f);
    1994       89096 :   z = s * s;
    1995       89096 :   w = z * z;
    1996       89096 :   t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
    1997       89096 :   t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
    1998       89096 :   R = t2 + t1;
    1999       89096 :   hfsq = 0.5 * f * f;
    2000       89096 :   return s * (hfsq + R);
    2001             : }
    2002             : 
    2003             : /*
    2004             :  * Return the base 2 logarithm of x.  See e_log.c and k_log.h for most
    2005             :  * comments.
    2006             :  *
    2007             :  * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
    2008             :  * then does the combining and scaling steps
    2009             :  *    log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
    2010             :  * in not-quite-routine extra precision.
    2011             :  */
    2012       89637 : double log2(double x) {
    2013             :   static const double
    2014             :       two54 = 1.80143985094819840000e+16,   /* 0x43500000, 0x00000000 */
    2015             :       ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
    2016             :       ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
    2017             : 
    2018             :   static const double zero = 0.0;
    2019             :   static volatile double vzero = 0.0;
    2020             : 
    2021             :   double f, hfsq, hi, lo, r, val_hi, val_lo, w, y;
    2022             :   int32_t i, k, hx;
    2023             :   uint32_t lx;
    2024             : 
    2025       89637 :   EXTRACT_WORDS(hx, lx, x);
    2026             : 
    2027             :   k = 0;
    2028       89637 :   if (hx < 0x00100000) { /* x < 2**-1022  */
    2029        1066 :     if (((hx & 0x7fffffff) | lx) == 0)
    2030          66 :       return -two54 / vzero;           /* log(+-0)=-inf */
    2031        1000 :     if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
    2032             :     k -= 54;
    2033         728 :     x *= two54; /* subnormal number, scale up x */
    2034         728 :     GET_HIGH_WORD(hx, x);
    2035             :   }
    2036       89299 :   if (hx >= 0x7ff00000) return x + x;
    2037       89184 :   if (hx == 0x3ff00000 && lx == 0) return zero; /* log(1) = +0 */
    2038       89096 :   k += (hx >> 20) - 1023;
    2039       89096 :   hx &= 0x000fffff;
    2040       89096 :   i = (hx + 0x95f64) & 0x100000;
    2041       89096 :   SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
    2042       89096 :   k += (i >> 20);
    2043       89096 :   y = static_cast<double>(k);
    2044       89096 :   f = x - 1.0;
    2045       89096 :   hfsq = 0.5 * f * f;
    2046       89096 :   r = k_log1p(f);
    2047             : 
    2048             :   /*
    2049             :    * f-hfsq must (for args near 1) be evaluated in extra precision
    2050             :    * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
    2051             :    * This is fairly efficient since f-hfsq only depends on f, so can
    2052             :    * be evaluated in parallel with R.  Not combining hfsq with R also
    2053             :    * keeps R small (though not as small as a true `lo' term would be),
    2054             :    * so that extra precision is not needed for terms involving R.
    2055             :    *
    2056             :    * Compiler bugs involving extra precision used to break Dekker's
    2057             :    * theorem for spitting f-hfsq as hi+lo, unless double_t was used
    2058             :    * or the multi-precision calculations were avoided when double_t
    2059             :    * has extra precision.  These problems are now automatically
    2060             :    * avoided as a side effect of the optimization of combining the
    2061             :    * Dekker splitting step with the clear-low-bits step.
    2062             :    *
    2063             :    * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
    2064             :    * precision to avoid a very large cancellation when x is very near
    2065             :    * these values.  Unlike the above cancellations, this problem is
    2066             :    * specific to base 2.  It is strange that adding +-1 is so much
    2067             :    * harder than adding +-ln2 or +-log10_2.
    2068             :    *
    2069             :    * This uses Dekker's theorem to normalize y+val_hi, so the
    2070             :    * compiler bugs are back in some configurations, sigh.  And I
    2071             :    * don't want to used double_t to avoid them, since that gives a
    2072             :    * pessimization and the support for avoiding the pessimization
    2073             :    * is not yet available.
    2074             :    *
    2075             :    * The multi-precision calculations for the multiplications are
    2076             :    * routine.
    2077             :    */
    2078       89096 :   hi = f - hfsq;
    2079       89096 :   SET_LOW_WORD(hi, 0);
    2080       89096 :   lo = (f - hi) - hfsq + r;
    2081       89096 :   val_hi = hi * ivln2hi;
    2082       89096 :   val_lo = (lo + hi) * ivln2lo + lo * ivln2hi;
    2083             : 
    2084             :   /* spadd(val_hi, val_lo, y), except for not using double_t: */
    2085       89096 :   w = y + val_hi;
    2086       89096 :   val_lo += (y - w) + val_hi;
    2087             :   val_hi = w;
    2088             : 
    2089       89096 :   return val_lo + val_hi;
    2090             : }
    2091             : 
    2092             : /*
    2093             :  * Return the base 10 logarithm of x
    2094             :  *
    2095             :  * Method :
    2096             :  *      Let log10_2hi = leading 40 bits of log10(2) and
    2097             :  *          log10_2lo = log10(2) - log10_2hi,
    2098             :  *          ivln10   = 1/log(10) rounded.
    2099             :  *      Then
    2100             :  *              n = ilogb(x),
    2101             :  *              if(n<0)  n = n+1;
    2102             :  *              x = scalbn(x,-n);
    2103             :  *              log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
    2104             :  *
    2105             :  *  Note 1:
    2106             :  *     To guarantee log10(10**n)=n, where 10**n is normal, the rounding
    2107             :  *     mode must set to Round-to-Nearest.
    2108             :  *  Note 2:
    2109             :  *      [1/log(10)] rounded to 53 bits has error .198 ulps;
    2110             :  *      log10 is monotonic at all binary break points.
    2111             :  *
    2112             :  *  Special cases:
    2113             :  *      log10(x) is NaN if x < 0;
    2114             :  *      log10(+INF) is +INF; log10(0) is -INF;
    2115             :  *      log10(NaN) is that NaN;
    2116             :  *      log10(10**N) = N  for N=0,1,...,22.
    2117             :  */
    2118       18265 : double log10(double x) {
    2119             :   static const double
    2120             :       two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
    2121             :       ivln10 = 4.34294481903251816668e-01,
    2122             :       log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
    2123             :       log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
    2124             : 
    2125             :   static const double zero = 0.0;
    2126             :   static volatile double vzero = 0.0;
    2127             : 
    2128             :   double y;
    2129             :   int32_t i, k, hx;
    2130             :   uint32_t lx;
    2131             : 
    2132       18265 :   EXTRACT_WORDS(hx, lx, x);
    2133             : 
    2134             :   k = 0;
    2135       18265 :   if (hx < 0x00100000) { /* x < 2**-1022  */
    2136         422 :     if (((hx & 0x7fffffff) | lx) == 0)
    2137          66 :       return -two54 / vzero;           /* log(+-0)=-inf */
    2138         356 :     if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
    2139             :     k -= 54;
    2140          84 :     x *= two54; /* subnormal number, scale up x */
    2141          84 :     GET_HIGH_WORD(hx, x);
    2142          84 :     GET_LOW_WORD(lx, x);
    2143             :   }
    2144       17927 :   if (hx >= 0x7ff00000) return x + x;
    2145       17812 :   if (hx == 0x3ff00000 && lx == 0) return zero; /* log(1) = +0 */
    2146       17772 :   k += (hx >> 20) - 1023;
    2147             : 
    2148       17772 :   i = (k & 0x80000000) >> 31;
    2149       17772 :   hx = (hx & 0x000fffff) | ((0x3ff - i) << 20);
    2150       17772 :   y = k + i;
    2151       17772 :   SET_HIGH_WORD(x, hx);
    2152       17772 :   SET_LOW_WORD(x, lx);
    2153             : 
    2154       17772 :   double z = y * log10_2lo + ivln10 * log(x);
    2155       17772 :   return z + y * log10_2hi;
    2156             : }
    2157             : 
    2158             : /* expm1(x)
    2159             :  * Returns exp(x)-1, the exponential of x minus 1.
    2160             :  *
    2161             :  * Method
    2162             :  *   1. Argument reduction:
    2163             :  *  Given x, find r and integer k such that
    2164             :  *
    2165             :  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
    2166             :  *
    2167             :  *      Here a correction term c will be computed to compensate
    2168             :  *  the error in r when rounded to a floating-point number.
    2169             :  *
    2170             :  *   2. Approximating expm1(r) by a special rational function on
    2171             :  *  the interval [0,0.34658]:
    2172             :  *  Since
    2173             :  *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
    2174             :  *  we define R1(r*r) by
    2175             :  *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
    2176             :  *  That is,
    2177             :  *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
    2178             :  *         = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
    2179             :  *         = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
    2180             :  *      We use a special Reme algorithm on [0,0.347] to generate
    2181             :  *   a polynomial of degree 5 in r*r to approximate R1. The
    2182             :  *  maximum error of this polynomial approximation is bounded
    2183             :  *  by 2**-61. In other words,
    2184             :  *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
    2185             :  *  where   Q1  =  -1.6666666666666567384E-2,
    2186             :  *     Q2  =   3.9682539681370365873E-4,
    2187             :  *     Q3  =  -9.9206344733435987357E-6,
    2188             :  *     Q4  =   2.5051361420808517002E-7,
    2189             :  *     Q5  =  -6.2843505682382617102E-9;
    2190             :  *    z   =  r*r,
    2191             :  *  with error bounded by
    2192             :  *      |                  5           |     -61
    2193             :  *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
    2194             :  *      |                              |
    2195             :  *
    2196             :  *  expm1(r) = exp(r)-1 is then computed by the following
    2197             :  *   specific way which minimize the accumulation rounding error:
    2198             :  *             2     3
    2199             :  *            r     r    [ 3 - (R1 + R1*r/2)  ]
    2200             :  *        expm1(r) = r + --- + --- * [--------------------]
    2201             :  *                  2     2    [ 6 - r*(3 - R1*r/2) ]
    2202             :  *
    2203             :  *  To compensate the error in the argument reduction, we use
    2204             :  *    expm1(r+c) = expm1(r) + c + expm1(r)*c
    2205             :  *         ~ expm1(r) + c + r*c
    2206             :  *  Thus c+r*c will be added in as the correction terms for
    2207             :  *  expm1(r+c). Now rearrange the term to avoid optimization
    2208             :  *   screw up:
    2209             :  *            (      2                                    2 )
    2210             :  *            ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
    2211             :  *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
    2212             :  *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
    2213             :  *                      (                                             )
    2214             :  *
    2215             :  *       = r - E
    2216             :  *   3. Scale back to obtain expm1(x):
    2217             :  *  From step 1, we have
    2218             :  *     expm1(x) = either 2^k*[expm1(r)+1] - 1
    2219             :  *        = or     2^k*[expm1(r) + (1-2^-k)]
    2220             :  *   4. Implementation notes:
    2221             :  *  (A). To save one multiplication, we scale the coefficient Qi
    2222             :  *       to Qi*2^i, and replace z by (x^2)/2.
    2223             :  *  (B). To achieve maximum accuracy, we compute expm1(x) by
    2224             :  *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
    2225             :  *    (ii)  if k=0, return r-E
    2226             :  *    (iii) if k=-1, return 0.5*(r-E)-0.5
    2227             :  *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
    2228             :  *                  else       return  1.0+2.0*(r-E);
    2229             :  *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
    2230             :  *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
    2231             :  *    (vii) return 2^k(1-((E+2^-k)-r))
    2232             :  *
    2233             :  * Special cases:
    2234             :  *  expm1(INF) is INF, expm1(NaN) is NaN;
    2235             :  *  expm1(-INF) is -1, and
    2236             :  *  for finite argument, only expm1(0)=0 is exact.
    2237             :  *
    2238             :  * Accuracy:
    2239             :  *  according to an error analysis, the error is always less than
    2240             :  *  1 ulp (unit in the last place).
    2241             :  *
    2242             :  * Misc. info.
    2243             :  *  For IEEE double
    2244             :  *      if x >  7.09782712893383973096e+02 then expm1(x) overflow
    2245             :  *
    2246             :  * Constants:
    2247             :  * The hexadecimal values are the intended ones for the following
    2248             :  * constants. The decimal values may be used, provided that the
    2249             :  * compiler will convert from decimal to binary accurately enough
    2250             :  * to produce the hexadecimal values shown.
    2251             :  */
    2252      124001 : double expm1(double x) {
    2253             :   static const double
    2254             :       one = 1.0,
    2255             :       tiny = 1.0e-300,
    2256             :       o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
    2257             :       ln2_hi = 6.93147180369123816490e-01,      /* 0x3fe62e42, 0xfee00000 */
    2258             :       ln2_lo = 1.90821492927058770002e-10,      /* 0x3dea39ef, 0x35793c76 */
    2259             :       invln2 = 1.44269504088896338700e+00,      /* 0x3ff71547, 0x652b82fe */
    2260             :       /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs =
    2261             :          x*x/2: */
    2262             :       Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
    2263             :       Q2 = 1.58730158725481460165e-03,  /* 3F5A01A0 19FE5585 */
    2264             :       Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
    2265             :       Q4 = 4.00821782732936239552e-06,  /* 3ED0CFCA 86E65239 */
    2266             :       Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
    2267             : 
    2268             :   static volatile double huge = 1.0e+300;
    2269             : 
    2270             :   double y, hi, lo, c, t, e, hxs, hfx, r1, twopk;
    2271             :   int32_t k, xsb;
    2272             :   uint32_t hx;
    2273             : 
    2274      124001 :   GET_HIGH_WORD(hx, x);
    2275      124001 :   xsb = hx & 0x80000000; /* sign bit of x */
    2276      124001 :   hx &= 0x7fffffff;      /* high word of |x| */
    2277             : 
    2278             :   /* filter out huge and non-finite argument */
    2279      124001 :   if (hx >= 0x4043687A) {   /* if |x|>=56*ln2 */
    2280       74553 :     if (hx >= 0x40862E42) { /* if |x|>=709.78... */
    2281         477 :       if (hx >= 0x7ff00000) {
    2282             :         uint32_t low;
    2283         160 :         GET_LOW_WORD(low, x);
    2284         160 :         if (((hx & 0xfffff) | low) != 0)
    2285          94 :           return x + x; /* NaN */
    2286             :         else
    2287          66 :           return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
    2288             :       }
    2289         317 :       if (x > o_threshold) return huge * huge; /* overflow */
    2290             :     }
    2291       74222 :     if (xsb != 0) {        /* x < -56*ln2, return -1.0 with inexact */
    2292       37190 :       if (x + tiny < 0.0)  /* raise inexact */
    2293             :         return tiny - one; /* return -1 */
    2294             :     }
    2295             :   }
    2296             : 
    2297             :   /* argument reduction */
    2298       86480 :   if (hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
    2299       42643 :     if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
    2300         460 :       if (xsb == 0) {
    2301         262 :         hi = x - ln2_hi;
    2302             :         lo = ln2_lo;
    2303             :         k = 1;
    2304             :       } else {
    2305         198 :         hi = x + ln2_hi;
    2306             :         lo = -ln2_lo;
    2307             :         k = -1;
    2308             :       }
    2309             :     } else {
    2310       42183 :       k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
    2311       42183 :       t = k;
    2312       42183 :       hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
    2313       42183 :       lo = t * ln2_lo;
    2314             :     }
    2315       42643 :     STRICT_ASSIGN(double, x, hi - lo);
    2316       42643 :     c = (hi - x) - lo;
    2317       43837 :   } else if (hx < 0x3c900000) { /* when |x|<2**-54, return x */
    2318       41341 :     t = huge + x;               /* return x with inexact flags when x!=0 */
    2319       41341 :     return x - (t - (huge + x));
    2320             :   } else {
    2321             :     k = 0;
    2322             :   }
    2323             : 
    2324             :   /* x is now in primary range */
    2325       45139 :   hfx = 0.5 * x;
    2326       45139 :   hxs = x * hfx;
    2327       45139 :   r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
    2328       45139 :   t = 3.0 - r1 * hfx;
    2329       45139 :   e = hxs * ((r1 - t) / (6.0 - x * t));
    2330       45139 :   if (k == 0) {
    2331        2496 :     return x - (x * e - hxs); /* c is 0 */
    2332             :   } else {
    2333       42643 :     INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0); /* 2^k */
    2334       42643 :     e = (x * (e - c) - c);
    2335       42643 :     e -= hxs;
    2336       42643 :     if (k == -1) return 0.5 * (x - e) - 0.5;
    2337       42445 :     if (k == 1) {
    2338         262 :       if (x < -0.25)
    2339          24 :         return -2.0 * (e - (x + 0.5));
    2340             :       else
    2341         238 :         return one + 2.0 * (x - e);
    2342             :     }
    2343       42183 :     if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
    2344       39198 :       y = one - (e - x);
    2345             :       // TODO(mvstanton): is this replacement for the hex float
    2346             :       // sufficient?
    2347             :       // if (k == 1024) y = y*2.0*0x1p1023;
    2348       39198 :       if (k == 1024)
    2349           0 :         y = y * 2.0 * 8.98846567431158e+307;
    2350             :       else
    2351       39198 :         y = y * twopk;
    2352       39198 :       return y - one;
    2353             :     }
    2354             :     t = one;
    2355        2985 :     if (k < 20) {
    2356        1543 :       SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
    2357        1543 :       y = t - (e - x);
    2358        1543 :       y = y * twopk;
    2359             :     } else {
    2360        1442 :       SET_HIGH_WORD(t, ((0x3ff - k) << 20)); /* 2^-k */
    2361        1442 :       y = x - (e + t);
    2362        1442 :       y += one;
    2363        1442 :       y = y * twopk;
    2364             :     }
    2365             :   }
    2366        2985 :   return y;
    2367             : }
    2368             : 
    2369      152827 : double cbrt(double x) {
    2370             :   static const uint32_t
    2371             :       B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
    2372             :       B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
    2373             : 
    2374             :   /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
    2375             :   static const double P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
    2376             :       P1 = -1.88497979543377169875,                /* 0xbffe28e0, 0x92f02420 */
    2377             :       P2 = 1.621429720105354466140,                /* 0x3ff9f160, 0x4a49d6c2 */
    2378             :       P3 = -0.758397934778766047437,               /* 0xbfe844cb, 0xbee751d9 */
    2379             :       P4 = 0.145996192886612446982;                /* 0x3fc2b000, 0xd4e4edd7 */
    2380             : 
    2381             :   int32_t hx;
    2382             :   union {
    2383             :     double value;
    2384             :     uint64_t bits;
    2385             :   } u;
    2386             :   double r, s, t = 0.0, w;
    2387             :   uint32_t sign;
    2388             :   uint32_t high, low;
    2389             : 
    2390      152827 :   EXTRACT_WORDS(hx, low, x);
    2391      152827 :   sign = hx & 0x80000000; /* sign= sign(x) */
    2392      152827 :   hx ^= sign;
    2393      152827 :   if (hx >= 0x7ff00000) return (x + x); /* cbrt(NaN,INF) is itself */
    2394             : 
    2395             :   /*
    2396             :    * Rough cbrt to 5 bits:
    2397             :    *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
    2398             :    * where e is integral and >= 0, m is real and in [0, 1), and "/" and
    2399             :    * "%" are integer division and modulus with rounding towards minus
    2400             :    * infinity.  The RHS is always >= the LHS and has a maximum relative
    2401             :    * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
    2402             :    * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
    2403             :    * floating point representation, for finite positive normal values,
    2404             :    * ordinary integer divison of the value in bits magically gives
    2405             :    * almost exactly the RHS of the above provided we first subtract the
    2406             :    * exponent bias (1023 for doubles) and later add it back.  We do the
    2407             :    * subtraction virtually to keep e >= 0 so that ordinary integer
    2408             :    * division rounds towards minus infinity; this is also efficient.
    2409             :    */
    2410      152667 :   if (hx < 0x00100000) {             /* zero or subnormal? */
    2411          52 :     if ((hx | low) == 0) return (x); /* cbrt(0) is itself */
    2412           0 :     SET_HIGH_WORD(t, 0x43500000);    /* set t= 2**54 */
    2413           0 :     t *= x;
    2414           0 :     GET_HIGH_WORD(high, t);
    2415           0 :     INSERT_WORDS(t, sign | ((high & 0x7fffffff) / 3 + B2), 0);
    2416             :   } else {
    2417      152615 :     INSERT_WORDS(t, sign | (hx / 3 + B1), 0);
    2418             :   }
    2419             : 
    2420             :   /*
    2421             :    * New cbrt to 23 bits:
    2422             :    *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
    2423             :    * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
    2424             :    * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
    2425             :    * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
    2426             :    * gives us bounds for r = t**3/x.
    2427             :    *
    2428             :    * Try to optimize for parallel evaluation as in k_tanf.c.
    2429             :    */
    2430      152615 :   r = (t * t) * (t / x);
    2431      152615 :   t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));
    2432             : 
    2433             :   /*
    2434             :    * Round t away from zero to 23 bits (sloppily except for ensuring that
    2435             :    * the result is larger in magnitude than cbrt(x) but not much more than
    2436             :    * 2 23-bit ulps larger).  With rounding towards zero, the error bound
    2437             :    * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
    2438             :    * in the rounded t, the infinite-precision error in the Newton
    2439             :    * approximation barely affects third digit in the final error
    2440             :    * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
    2441             :    * before the final error is larger than 0.667 ulps.
    2442             :    */
    2443             :   u.value = t;
    2444      152615 :   u.bits = (u.bits + 0x80000000) & 0xffffffffc0000000ULL;
    2445             :   t = u.value;
    2446             : 
    2447             :   /* one step Newton iteration to 53 bits with error < 0.667 ulps */
    2448      152615 :   s = t * t;             /* t*t is exact */
    2449      152615 :   r = x / s;             /* error <= 0.5 ulps; |r| < |t| */
    2450      152615 :   w = t + t;             /* t+t is exact */
    2451      152615 :   r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
    2452      152615 :   t = t + t * r;         /* error <= 0.5 + 0.5/3 + epsilon */
    2453             : 
    2454      152615 :   return (t);
    2455             : }
    2456             : 
    2457             : /* sin(x)
    2458             :  * Return sine function of x.
    2459             :  *
    2460             :  * kernel function:
    2461             :  *      __kernel_sin            ... sine function on [-pi/4,pi/4]
    2462             :  *      __kernel_cos            ... cose function on [-pi/4,pi/4]
    2463             :  *      __ieee754_rem_pio2      ... argument reduction routine
    2464             :  *
    2465             :  * Method.
    2466             :  *      Let S,C and T denote the sin, cos and tan respectively on
    2467             :  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
    2468             :  *      in [-pi/4 , +pi/4], and let n = k mod 4.
    2469             :  *      We have
    2470             :  *
    2471             :  *          n        sin(x)      cos(x)        tan(x)
    2472             :  *     ----------------------------------------------------------
    2473             :  *          0          S           C             T
    2474             :  *          1          C          -S            -1/T
    2475             :  *          2         -S          -C             T
    2476             :  *          3         -C           S            -1/T
    2477             :  *     ----------------------------------------------------------
    2478             :  *
    2479             :  * Special cases:
    2480             :  *      Let trig be any of sin, cos, or tan.
    2481             :  *      trig(+-INF)  is NaN, with signals;
    2482             :  *      trig(NaN)    is that NaN;
    2483             :  *
    2484             :  * Accuracy:
    2485             :  *      TRIG(x) returns trig(x) nearly rounded
    2486             :  */
    2487    45632414 : double sin(double x) {
    2488             :   double y[2], z = 0.0;
    2489             :   int32_t n, ix;
    2490             : 
    2491             :   /* High word of x. */
    2492    45632414 :   GET_HIGH_WORD(ix, x);
    2493             : 
    2494             :   /* |x| ~< pi/4 */
    2495    45632414 :   ix &= 0x7fffffff;
    2496    45632414 :   if (ix <= 0x3fe921fb) {
    2497    37842753 :     return __kernel_sin(x, z, 0);
    2498     7789661 :   } else if (ix >= 0x7ff00000) {
    2499             :     /* sin(Inf or NaN) is NaN */
    2500        2465 :     return x - x;
    2501             :   } else {
    2502             :     /* argument reduction needed */
    2503     7787196 :     n = __ieee754_rem_pio2(x, y);
    2504     7787196 :     switch (n & 3) {
    2505             :       case 0:
    2506      900480 :         return __kernel_sin(y[0], y[1], 1);
    2507             :       case 1:
    2508    12872038 :         return __kernel_cos(y[0], y[1]);
    2509             :       case 2:
    2510      899870 :         return -__kernel_sin(y[0], y[1], 1);
    2511             :       default:
    2512      902004 :         return -__kernel_cos(y[0], y[1]);
    2513             :     }
    2514             :   }
    2515             : }
    2516             : 
    2517             : /* tan(x)
    2518             :  * Return tangent function of x.
    2519             :  *
    2520             :  * kernel function:
    2521             :  *      __kernel_tan            ... tangent function on [-pi/4,pi/4]
    2522             :  *      __ieee754_rem_pio2      ... argument reduction routine
    2523             :  *
    2524             :  * Method.
    2525             :  *      Let S,C and T denote the sin, cos and tan respectively on
    2526             :  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
    2527             :  *      in [-pi/4 , +pi/4], and let n = k mod 4.
    2528             :  *      We have
    2529             :  *
    2530             :  *          n        sin(x)      cos(x)        tan(x)
    2531             :  *     ----------------------------------------------------------
    2532             :  *          0          S           C             T
    2533             :  *          1          C          -S            -1/T
    2534             :  *          2         -S          -C             T
    2535             :  *          3         -C           S            -1/T
    2536             :  *     ----------------------------------------------------------
    2537             :  *
    2538             :  * Special cases:
    2539             :  *      Let trig be any of sin, cos, or tan.
    2540             :  *      trig(+-INF)  is NaN, with signals;
    2541             :  *      trig(NaN)    is that NaN;
    2542             :  *
    2543             :  * Accuracy:
    2544             :  *      TRIG(x) returns trig(x) nearly rounded
    2545             :  */
    2546     1529307 : double tan(double x) {
    2547             :   double y[2], z = 0.0;
    2548             :   int32_t n, ix;
    2549             : 
    2550             :   /* High word of x. */
    2551     1529307 :   GET_HIGH_WORD(ix, x);
    2552             : 
    2553             :   /* |x| ~< pi/4 */
    2554     1529307 :   ix &= 0x7fffffff;
    2555     1529307 :   if (ix <= 0x3fe921fb) {
    2556        7350 :     return __kernel_tan(x, z, 1);
    2557     1521957 :   } else if (ix >= 0x7ff00000) {
    2558             :     /* tan(Inf or NaN) is NaN */
    2559        2450 :     return x - x; /* NaN */
    2560             :   } else {
    2561             :     /* argument reduction needed */
    2562     1519507 :     n = __ieee754_rem_pio2(x, y);
    2563             :     /* 1 -> n even, -1 -> n odd */
    2564     1519507 :     return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1));
    2565             :   }
    2566             : }
    2567             : 
    2568             : /*
    2569             :  * ES6 draft 09-27-13, section 20.2.2.12.
    2570             :  * Math.cosh
    2571             :  * Method :
    2572             :  * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
    2573             :  *      1. Replace x by |x| (cosh(x) = cosh(-x)).
    2574             :  *      2.
    2575             :  *                                                      [ exp(x) - 1 ]^2
    2576             :  *          0        <= x <= ln2/2  :  cosh(x) := 1 + -------------------
    2577             :  *                                                         2*exp(x)
    2578             :  *
    2579             :  *                                                 exp(x) + 1/exp(x)
    2580             :  *          ln2/2    <= x <= 22     :  cosh(x) := -------------------
    2581             :  *                                                        2
    2582             :  *          22       <= x <= lnovft :  cosh(x) := exp(x)/2
    2583             :  *          lnovft   <= x <= ln2ovft:  cosh(x) := exp(x/2)/2 * exp(x/2)
    2584             :  *          ln2ovft  <  x           :  cosh(x) := huge*huge (overflow)
    2585             :  *
    2586             :  * Special cases:
    2587             :  *      cosh(x) is |x| if x is +INF, -INF, or NaN.
    2588             :  *      only cosh(0)=1 is exact for finite x.
    2589             :  */
    2590        1453 : double cosh(double x) {
    2591             :   static const double KCOSH_OVERFLOW = 710.4758600739439;
    2592             :   static const double one = 1.0, half = 0.5;
    2593             :   static volatile double huge = 1.0e+300;
    2594             : 
    2595             :   int32_t ix;
    2596             : 
    2597             :   /* High word of |x|. */
    2598        1453 :   GET_HIGH_WORD(ix, x);
    2599        1453 :   ix &= 0x7fffffff;
    2600             : 
    2601             :   // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
    2602        1453 :   if (ix < 0x3fd62e43) {
    2603         333 :     double t = expm1(fabs(x));
    2604         333 :     double w = one + t;
    2605             :     // For |x| < 2^-55, cosh(x) = 1
    2606         333 :     if (ix < 0x3c800000) return w;
    2607         102 :     return one + (t * t) / (w + w);
    2608             :   }
    2609             : 
    2610             :   // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
    2611        1120 :   if (ix < 0x40360000) {
    2612         439 :     double t = exp(fabs(x));
    2613         439 :     return half * t + half / t;
    2614             :   }
    2615             : 
    2616             :   // |x| in [22, log(maxdouble)], return half*exp(|x|)
    2617         681 :   if (ix < 0x40862e42) return half * exp(fabs(x));
    2618             : 
    2619             :   // |x| in [log(maxdouble), overflowthreshold]
    2620         641 :   if (fabs(x) <= KCOSH_OVERFLOW) {
    2621          28 :     double w = exp(half * fabs(x));
    2622          28 :     double t = half * w;
    2623          28 :     return t * w;
    2624             :   }
    2625             : 
    2626             :   /* x is INF or NaN */
    2627         613 :   if (ix >= 0x7ff00000) return x * x;
    2628             : 
    2629             :   // |x| > overflowthreshold.
    2630         418 :   return huge * huge;
    2631             : }
    2632             : 
    2633             : /*
    2634             :  * ES6 draft 09-27-13, section 20.2.2.30.
    2635             :  * Math.sinh
    2636             :  * Method :
    2637             :  * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
    2638             :  *      1. Replace x by |x| (sinh(-x) = -sinh(x)).
    2639             :  *      2.
    2640             :  *                                                  E + E/(E+1)
    2641             :  *          0        <= x <= 22     :  sinh(x) := --------------, E=expm1(x)
    2642             :  *                                                      2
    2643             :  *
    2644             :  *          22       <= x <= lnovft :  sinh(x) := exp(x)/2
    2645             :  *          lnovft   <= x <= ln2ovft:  sinh(x) := exp(x/2)/2 * exp(x/2)
    2646             :  *          ln2ovft  <  x           :  sinh(x) := x*shuge (overflow)
    2647             :  *
    2648             :  * Special cases:
    2649             :  *      sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
    2650             :  *      only sinh(0)=0 is exact for finite x.
    2651             :  */
    2652        1607 : double sinh(double x) {
    2653             :   static const double KSINH_OVERFLOW = 710.4758600739439,
    2654             :                       TWO_M28 =
    2655             :                           3.725290298461914e-9,  // 2^-28, empty lower half
    2656             :       LOG_MAXD = 709.7822265625;  // 0x40862e42 00000000, empty lower half
    2657             :   static const double shuge = 1.0e307;
    2658             : 
    2659        1607 :   double h = (x < 0) ? -0.5 : 0.5;
    2660             :   // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
    2661        1607 :   double ax = fabs(x);
    2662        1607 :   if (ax < 22) {
    2663             :     // For |x| < 2^-28, sinh(x) = x
    2664         870 :     if (ax < TWO_M28) return x;
    2665         595 :     double t = expm1(ax);
    2666         595 :     if (ax < 1) {
    2667         178 :       return h * (2 * t - t * t / (t + 1));
    2668             :     }
    2669         417 :     return h * (t + t / (t + 1));
    2670             :   }
    2671             :   // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
    2672         737 :   if (ax < LOG_MAXD) return h * exp(ax);
    2673             :   // |x| in [log(maxdouble), overflowthreshold]
    2674             :   // overflowthreshold = 710.4758600739426
    2675         669 :   if (ax <= KSINH_OVERFLOW) {
    2676          28 :     double w = exp(0.5 * ax);
    2677          28 :     double t = h * w;
    2678          28 :     return t * w;
    2679             :   }
    2680             :   // |x| > overflowthreshold or is NaN.
    2681             :   // Return Infinity of the appropriate sign or NaN.
    2682         641 :   return x * shuge;
    2683             : }
    2684             : 
    2685             : /* Tanh(x)
    2686             :  * Return the Hyperbolic Tangent of x
    2687             :  *
    2688             :  * Method :
    2689             :  *                                 x    -x
    2690             :  *                                e  - e
    2691             :  *  0. tanh(x) is defined to be -----------
    2692             :  *                                 x    -x
    2693             :  *                                e  + e
    2694             :  *  1. reduce x to non-negative by tanh(-x) = -tanh(x).
    2695             :  *  2.  0      <= x <  2**-28 : tanh(x) := x with inexact if x != 0
    2696             :  *                                          -t
    2697             :  *      2**-28 <= x <  1      : tanh(x) := -----; t = expm1(-2x)
    2698             :  *                                         t + 2
    2699             :  *                                               2
    2700             :  *      1      <= x <  22     : tanh(x) := 1 - -----; t = expm1(2x)
    2701             :  *                                             t + 2
    2702             :  *      22     <= x <= INF    : tanh(x) := 1.
    2703             :  *
    2704             :  * Special cases:
    2705             :  *      tanh(NaN) is NaN;
    2706             :  *      only tanh(0)=0 is exact for finite argument.
    2707             :  */
    2708        1467 : double tanh(double x) {
    2709             :   static const volatile double tiny = 1.0e-300;
    2710             :   static const double one = 1.0, two = 2.0, huge = 1.0e300;
    2711             :   double t, z;
    2712             :   int32_t jx, ix;
    2713             : 
    2714        1467 :   GET_HIGH_WORD(jx, x);
    2715        1467 :   ix = jx & 0x7fffffff;
    2716             : 
    2717             :   /* x is INF or NaN */
    2718        1467 :   if (ix >= 0x7ff00000) {
    2719         167 :     if (jx >= 0)
    2720         126 :       return one / x + one; /* tanh(+-inf)=+-1 */
    2721             :     else
    2722          41 :       return one / x - one; /* tanh(NaN) = NaN */
    2723             :   }
    2724             : 
    2725             :   /* |x| < 22 */
    2726        1300 :   if (ix < 0x40360000) {            /* |x|<22 */
    2727         842 :     if (ix < 0x3e300000) {          /* |x|<2**-28 */
    2728         247 :       if (huge + x > one) return x; /* tanh(tiny) = tiny with inexact */
    2729             :     }
    2730         595 :     if (ix >= 0x3ff00000) { /* |x|>=1  */
    2731         389 :       t = expm1(two * fabs(x));
    2732         389 :       z = one - two / (t + two);
    2733             :     } else {
    2734         206 :       t = expm1(-two * fabs(x));
    2735         206 :       z = -t / (t + two);
    2736             :     }
    2737             :     /* |x| >= 22, return +-1 */
    2738             :   } else {
    2739         458 :     z = one - tiny; /* raise inexact flag */
    2740             :   }
    2741        1053 :   return (jx >= 0) ? z : -z;
    2742             : }
    2743             : 
    2744             : }  // namespace ieee754
    2745             : }  // namespace base
    2746             : }  // namespace v8

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