cosh.c

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MIT License
-----------
 
Copyright (c) 2002-2019 Advanced Micro Devices, Inc.
 
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#include "libm.h"
#include "libm_util.h"
 
#define USE_SPLITEXP
#define USE_SCALEDOUBLE_1
#define USE_SCALEDOUBLE_2
#define USE_INFINITY_WITH_FLAGS
#define USE_VAL_WITH_FLAGS
#define USE_HANDLE_ERROR
#include "libm_inlines.h"
#undef USE_SPLITEXP
#undef USE_SCALEDOUBLE_1
#undef USE_SCALEDOUBLE_2
#undef USE_INFINITY_WITH_FLAGS
#undef USE_VAL_WITH_FLAGS
#undef USE_HANDLE_ERROR
 
#ifdef _MSC_VER
#pragma function(cosh)
#endif
 
double cosh(double x)
{
  /*
    Derived from sinh subroutine
    
    After dealing with special cases the computation is split into
    regions as follows:
 
    abs(x) >= max_cosh_arg:
    cosh(x) = sign(x)*Inf
 
    abs(x) >= small_threshold:
    cosh(x) = sign(x)*exp(abs(x))/2 computed using the
    splitexp and scaleDouble functions as for exp_amd().
 
    abs(x) < small_threshold:
    compute p = exp(y) - 1 and then z = 0.5*(p+(p/(p+1.0)))
    cosh(x) is then sign(x)*z.                             */
 
  static const double
    max_cosh_arg = 7.10475860073943977113e+02, /* 0x408633ce8fb9f87e */
    thirtytwo_by_log2 = 4.61662413084468283841e+01, /* 0x40471547652b82fe */
    log2_by_32_lead = 2.16608493356034159660e-02, /* 0x3f962e42fe000000 */
    log2_by_32_tail = 5.68948749532545630390e-11, /* 0x3dcf473de6af278e */
//    small_threshold = 8*BASEDIGITS_DP64*0.30102999566398119521373889;
    small_threshold = 20.0;
  /* (8*BASEDIGITS_DP64*log10of2) ' exp(-x) insignificant c.f. exp(x) */
 
  /* Lead and tail tabulated values of sinh(i) and cosh(i) 
     for i = 0,...,36. The lead part has 26 leading bits. */
 
  static const double sinh_lead[   37] = {
    0.00000000000000000000e+00,  /* 0x0000000000000000 */
    1.17520117759704589844e+00,  /* 0x3ff2cd9fc0000000 */
    3.62686038017272949219e+00,  /* 0x400d03cf60000000 */
    1.00178747177124023438e+01,  /* 0x40240926e0000000 */
    2.72899169921875000000e+01,  /* 0x403b4a3800000000 */
    7.42032089233398437500e+01,  /* 0x40528d0160000000 */
    2.01713153839111328125e+02,  /* 0x406936d228000000 */
    5.48316116333007812500e+02,  /* 0x4081228768000000 */
    1.49047882080078125000e+03,  /* 0x409749ea50000000 */
    4.05154187011718750000e+03,  /* 0x40afa71570000000 */
    1.10132326660156250000e+04,  /* 0x40c5829dc8000000 */
    2.99370708007812500000e+04,  /* 0x40dd3c4488000000 */
    8.13773945312500000000e+04,  /* 0x40f3de1650000000 */
    2.21206695312500000000e+05,  /* 0x410b00b590000000 */
    6.01302140625000000000e+05,  /* 0x412259ac48000000 */
    1.63450865625000000000e+06,  /* 0x4138f0cca8000000 */
    4.44305525000000000000e+06,  /* 0x4150f2ebd0000000 */
    1.20774762500000000000e+07,  /* 0x4167093488000000 */
    3.28299845000000000000e+07,  /* 0x417f4f2208000000 */
    8.92411500000000000000e+07,  /* 0x419546d8f8000000 */
    2.42582596000000000000e+08,  /* 0x41aceb0888000000 */
    6.59407856000000000000e+08,  /* 0x41c3a6e1f8000000 */
    1.79245641600000000000e+09,  /* 0x41dab5adb8000000 */
    4.87240166400000000000e+09,  /* 0x41f226af30000000 */
    1.32445608960000000000e+10,  /* 0x4208ab7fb0000000 */
    3.60024494080000000000e+10,  /* 0x4220c3d390000000 */
    9.78648043520000000000e+10,  /* 0x4236c93268000000 */
    2.66024116224000000000e+11,  /* 0x424ef822f0000000 */
    7.23128516608000000000e+11,  /* 0x42650bba30000000 */
    1.96566712320000000000e+12,  /* 0x427c9aae40000000 */
    5.34323724288000000000e+12,  /* 0x4293704708000000 */
    1.45244246507520000000e+13,  /* 0x42aa6b7658000000 */
    3.94814795284480000000e+13,  /* 0x42c1f43fc8000000 */
    1.07321789251584000000e+14,  /* 0x42d866f348000000 */
    2.91730863685632000000e+14,  /* 0x42f0953e28000000 */
    7.93006722514944000000e+14,  /* 0x430689e220000000 */
    2.15561576592179200000e+15}; /* 0x431ea215a0000000 */
 
  static const double sinh_tail[   37] = {
    0.00000000000000000000e+00,  /* 0x0000000000000000 */
    1.60467555584448807892e-08,  /* 0x3e513ae6096a0092 */
    2.76742892754807136947e-08,  /* 0x3e5db70cfb79a640 */
    2.09697499555224576530e-07,  /* 0x3e8c2526b66dc067 */
    2.04940252448908240062e-07,  /* 0x3e8b81b18647f380 */
    1.65444891522700935932e-06,  /* 0x3ebbc1cdd1e1eb08 */
    3.53116789999998198721e-06,  /* 0x3ecd9f201534fb09 */
    6.94023870987375490695e-06,  /* 0x3edd1c064a4e9954 */
    4.98876893611587449271e-06,  /* 0x3ed4eca65d06ea74 */
    3.19656024605152215752e-05,  /* 0x3f00c259bcc0ecc5 */
    2.08687768377236501204e-04,  /* 0x3f2b5a6647cf9016 */
    4.84668088325403796299e-05,  /* 0x3f09691adefb0870 */
    1.17517985422733832468e-03,  /* 0x3f53410fc29cde38 */
    6.90830086959560562415e-04,  /* 0x3f46a31a50b6fb3c */
    1.45697262451506548420e-03,  /* 0x3f57defc71805c40 */
    2.99859023684906737806e-02,  /* 0x3f9eb49fd80e0bab */
    1.02538800507941396667e-02,  /* 0x3f84fffc7bcd5920 */
    1.26787628407699110022e-01,  /* 0x3fc03a93b6c63435 */
    6.86652479544033744752e-02,  /* 0x3fb1940bb255fd1c */
    4.81593627621056619148e-01,  /* 0x3fded26e14260b50 */
    1.70489513795397629181e+00,  /* 0x3ffb47401fc9f2a2 */
    1.12416073482258713767e+01,  /* 0x40267bb3f55634f1 */
    7.06579578070110514432e+00,  /* 0x401c435ff8194ddc */
    5.91244512999659974639e+01,  /* 0x404d8fee052ba63a */
    1.68921736147050694399e+02,  /* 0x40651d7edccde3f6 */
    2.60692936262073658327e+02,  /* 0x40704b1644557d1a */
    3.62419382134885609048e+02,  /* 0x4076a6b5ca0a9dc4 */
    4.07689930834187271103e+03,  /* 0x40afd9cc72249aba */
    1.55377375868385224749e+04,  /* 0x40ce58de693edab5 */
    2.53720210371943067003e+04,  /* 0x40d8c70158ac6363 */
    4.78822310734952334315e+04,  /* 0x40e7614764f43e20 */
    1.81871712615542812273e+05,  /* 0x4106337db36fc718 */
    5.62892347580489004031e+05,  /* 0x41212d98b1f611e2 */
    6.41374032312148716301e+05,  /* 0x412392bc108b37cc */
    7.57809544070145115256e+06,  /* 0x415ce87bdc3473dc */
    3.64177136406482197344e+06,  /* 0x414bc8d5ae99ad14 */
    7.63580561355670914054e+06}; /* 0x415d20d76744835c */
 
  static const double cosh_lead[   37] = {
    1.00000000000000000000e+00,  /* 0x3ff0000000000000 */
    1.54308062791824340820e+00,  /* 0x3ff8b07550000000 */
    3.76219564676284790039e+00,  /* 0x400e18fa08000000 */
    1.00676617622375488281e+01,  /* 0x402422a490000000 */
    2.73082327842712402344e+01,  /* 0x403b4ee858000000 */
    7.42099475860595703125e+01,  /* 0x40528d6fc8000000 */
    2.01715633392333984375e+02,  /* 0x406936e678000000 */
    5.48317031860351562500e+02,  /* 0x4081228948000000 */
    1.49047915649414062500e+03,  /* 0x409749eaa8000000 */
    4.05154199218750000000e+03,  /* 0x40afa71580000000 */
    1.10132329101562500000e+04,  /* 0x40c5829dd0000000 */
    2.99370708007812500000e+04,  /* 0x40dd3c4488000000 */
    8.13773945312500000000e+04,  /* 0x40f3de1650000000 */
    2.21206695312500000000e+05,  /* 0x410b00b590000000 */
    6.01302140625000000000e+05,  /* 0x412259ac48000000 */
    1.63450865625000000000e+06,  /* 0x4138f0cca8000000 */
    4.44305525000000000000e+06,  /* 0x4150f2ebd0000000 */
    1.20774762500000000000e+07,  /* 0x4167093488000000 */
    3.28299845000000000000e+07,  /* 0x417f4f2208000000 */
    8.92411500000000000000e+07,  /* 0x419546d8f8000000 */
    2.42582596000000000000e+08,  /* 0x41aceb0888000000 */
    6.59407856000000000000e+08,  /* 0x41c3a6e1f8000000 */
    1.79245641600000000000e+09,  /* 0x41dab5adb8000000 */
    4.87240166400000000000e+09,  /* 0x41f226af30000000 */
    1.32445608960000000000e+10,  /* 0x4208ab7fb0000000 */
    3.60024494080000000000e+10,  /* 0x4220c3d390000000 */
    9.78648043520000000000e+10,  /* 0x4236c93268000000 */
    2.66024116224000000000e+11,  /* 0x424ef822f0000000 */
    7.23128516608000000000e+11,  /* 0x42650bba30000000 */
    1.96566712320000000000e+12,  /* 0x427c9aae40000000 */
    5.34323724288000000000e+12,  /* 0x4293704708000000 */
    1.45244246507520000000e+13,  /* 0x42aa6b7658000000 */
    3.94814795284480000000e+13,  /* 0x42c1f43fc8000000 */
    1.07321789251584000000e+14,  /* 0x42d866f348000000 */
    2.91730863685632000000e+14,  /* 0x42f0953e28000000 */
    7.93006722514944000000e+14,  /* 0x430689e220000000 */
    2.15561576592179200000e+15}; /* 0x431ea215a0000000 */
 
  static const double cosh_tail[   37] = {
    0.00000000000000000000e+00,  /* 0x0000000000000000 */
    6.89700037027478056904e-09,  /* 0x3e3d9f5504c2bd28 */
    4.43207835591715833630e-08,  /* 0x3e67cb66f0a4c9fd */
    2.33540217013828929694e-07,  /* 0x3e8f58617928e588 */
    5.17452463948269748331e-08,  /* 0x3e6bc7d000c38d48 */
    9.38728274131605919153e-07,  /* 0x3eaf7f9d4e329998 */
    2.73012191010840495544e-06,  /* 0x3ec6e6e464885269 */
    3.29486051438996307950e-06,  /* 0x3ecba3a8b946c154 */
    4.75803746362771416375e-06,  /* 0x3ed3f4e76110d5a4 */
    3.33050940471947692369e-05,  /* 0x3f017622515a3e2b */
    9.94707313972136215365e-06,  /* 0x3ee4dc4b528af3d0 */
    6.51685096227860253398e-05,  /* 0x3f11156278615e10 */
    1.18132406658066663359e-03,  /* 0x3f535ad50ed821f5 */
    6.93090416366541877541e-04,  /* 0x3f46b61055f2935c */
    1.45780415323416845386e-03,  /* 0x3f57e2794a601240 */
    2.99862082708111758744e-02,  /* 0x3f9eb4b45f6aadd3 */
    1.02539925859688602072e-02,  /* 0x3f85000b967b3698 */
    1.26787669807076286421e-01,  /* 0x3fc03a940fadc092 */
    6.86652631843830962843e-02,  /* 0x3fb1940bf3bf874c */
    4.81593633223853068159e-01,  /* 0x3fded26e1a2a2110 */
    1.70489514001513020602e+00,  /* 0x3ffb4740205796d6 */
    1.12416073489841270572e+01,  /* 0x40267bb3f55cb85d */
    7.06579578098005001152e+00,  /* 0x401c435ff81e18ac */
    5.91244513000686140458e+01,  /* 0x404d8fee052bdea4 */
    1.68921736147088438429e+02,  /* 0x40651d7edccde926 */
    2.60692936262087528121e+02,  /* 0x40704b1644557e0e */
    3.62419382134890611269e+02,  /* 0x4076a6b5ca0a9e1c */
    4.07689930834187453002e+03,  /* 0x40afd9cc72249abe */
    1.55377375868385224749e+04,  /* 0x40ce58de693edab5 */
    2.53720210371943103382e+04,  /* 0x40d8c70158ac6364 */
    4.78822310734952334315e+04,  /* 0x40e7614764f43e20 */
    1.81871712615542812273e+05,  /* 0x4106337db36fc718 */
    5.62892347580489004031e+05,  /* 0x41212d98b1f611e2 */
    6.41374032312148716301e+05,  /* 0x412392bc108b37cc */
    7.57809544070145115256e+06,  /* 0x415ce87bdc3473dc */
    3.64177136406482197344e+06,  /* 0x414bc8d5ae99ad14 */
    7.63580561355670914054e+06}; /* 0x415d20d76744835c */
 
  unsigned long long ux, aux, xneg;
  double y, z, z1, z2;
  int m;
  
  /* Special cases */
 
  GET_BITS_DP64(x, ux);
  aux = ux & ~SIGNBIT_DP64;
  if (aux < 0x3e30000000000000) /* |x| small enough that cosh(x) = 1 */
  {
      if (aux == 0)
        /* with no inexact */
        return 1.0;
      else
        return val_with_flags(1.0, AMD_F_INEXACT);
  }
  else if (aux >= PINFBITPATT_DP64) /* |x| is NaN or Inf */
  {
      if (aux > PINFBITPATT_DP64) /* x is NaN */
        return _handle_error("cosh", OP_COSH, ux|0x0008000000000000,_DOMAIN, 
                            0,EDOM, x, 0.0, 1);
      else     /* x is infinity */
        return infinity_with_flags(0);
  }
 
  xneg = (aux != ux);
 
  y = x;
  if (xneg) y = -x;
 
  if (y >= max_cosh_arg)
      {
             return _handle_error("cosh", OP_COSH, PINFBITPATT_DP64,_OVERFLOW, 
                        AMD_F_INEXACT|AMD_F_OVERFLOW,ERANGE, x, 0.0, 1);
        
//    z = infinity_with_flags(AMD_F_OVERFLOW);
      }
  else if (y >= small_threshold)
    {
      /* In this range y is large enough so that
         the negative exponential is negligible,
         so cosh(y) is approximated by sign(x)*exp(y)/2. The
         code below is an inlined version of that from
         exp() with two changes (it operates on
         y instead of x, and the division by 2 is
         done by reducing m by 1). */
 
      splitexp(y, 1.0, thirtytwo_by_log2, log2_by_32_lead,
               log2_by_32_tail, &m, &z1, &z2);
      m -= 1;
 
      if (m >= EMIN_DP64 && m <= EMAX_DP64)
        z = scaleDouble_1((z1+z2),m);
      else
        z = scaleDouble_2((z1+z2),m);
    }
  else
    {
      /* In this range we find the integer part y0 of y 
         and the increment dy = y - y0. We then compute
 
         z = sinh(y) = sinh(y0)cosh(dy) + cosh(y0)sinh(dy)
         z = cosh(y) = cosh(y0)cosh(dy) + sinh(y0)sinh(dy)
 
         where sinh(y0) and cosh(y0) are tabulated above. */
 
      int ind;
      double dy, dy2, sdy, cdy;
 
      ind = (int)y;
      dy = y - ind;
 
      dy2 = dy*dy;
      sdy = dy*dy2*(0.166666666666666667013899e0 +
                    (0.833333333333329931873097e-2 +
                     (0.198412698413242405162014e-3 +
                      (0.275573191913636406057211e-5 +
                       (0.250521176994133472333666e-7 +
                        (0.160576793121939886190847e-9 +
                         0.7746188980094184251527126e-12*dy2)*dy2)*dy2)*dy2)*dy2)*dy2);
 
      cdy = dy2*(0.500000000000000005911074e0 +
                 (0.416666666666660876512776e-1 +
                  (0.138888888889814854814536e-2 +
                   (0.248015872460622433115785e-4 +
                    (0.275573350756016588011357e-6 +
                     (0.208744349831471353536305e-8 +
                      0.1163921388172173692062032e-10*dy2)*dy2)*dy2)*dy2)*dy2)*dy2);
 
      /* At this point sinh(dy) is approximated by dy + sdy, and cosh(dy) is approximated by 1 + cdy.
     Shift some significant bits from dy to cdy. */
#if 0 
    double  sdy1,sdy2;
      GET_BITS_DP64(dy, ux);
      ux &= 0xfffffffff8000000;
      PUT_BITS_DP64(ux, sdy1);    // sdy1 is  upper 53-27=26 significant bits of dy.
      sdy2 = sdy + (dy - sdy1);   // sdy2 is  sdy + lower bits of dy
 
      z = ((((((cosh_tail[ind]*cdy + sinh_tail[ind]*sdy2) 
           + sinh_tail[ind]*sdy1) + cosh_tail[ind])  
         + cosh_lead[ind]*cdy) + sinh_lead[ind]*sdy2) 
       + sinh_lead[ind]*sdy1) + cosh_lead[ind];
#else
      z = ((((((cosh_tail[ind]*cdy + sinh_tail[ind]*sdy) 
           + sinh_tail[ind]*dy) + cosh_tail[ind])  
         + cosh_lead[ind]*cdy) + sinh_lead[ind]*sdy) 
       + sinh_lead[ind]*dy) + cosh_lead[ind];
#endif
    }
 
  return z;
}