compiler/stdc/math/ld80/e_tgammal.c

   1 /*      $OpenBSD: e_tgammal.c,v 1.4 2013/11/12 20:35:19 martynas Exp $  */
   2
   3 /*
   4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   5  *
   6  * Permission to use, copy, modify, and distribute this software for any
   7  * purpose with or without fee is hereby granted, provided that the above
   8  * copyright notice and this permission notice appear in all copies.
   9  *
  10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  17  */
  18
  19 /*                                                      tgammal.c
  20  *
  21  *      Gamma function
  22  *
  23  *
  24  *
  25  * SYNOPSIS:
  26  *
  27  * long double x, y, tgammal();
  28  *
  29  * y = tgammal( x );
  30  *
  31  *
  32  *
  33  * DESCRIPTION:
  34  *
  35  * Returns gamma function of the argument.  The result is correctly
  36  * signed.  This variable is also filled in by the logarithmic gamma
  37  * function lgamma().
  38  *
  39  * Arguments |x| <= 13 are reduced by recurrence and the function
  40  * approximated by a rational function of degree 7/8 in the
  41  * interval (2,3).  Large arguments are handled by Stirling's
  42  * formula. Large negative arguments are made positive using
  43  * a reflection formula.
  44  *
  45  *
  46  * ACCURACY:
  47  *
  48  *                      Relative error:
  49  * arithmetic   domain     # trials      peak         rms
  50  *    IEEE     -40,+40      10000       3.6e-19     7.9e-20
  51  *    IEEE    -1755,+1755   10000       4.8e-18     6.5e-19
  52  *
  53  * Accuracy for large arguments is dominated by error in powl().
  54  *
  55  */
  56
  57 #include <float.h>
  58 #include "math.h"
  59
  60 #include "math_private.h"
  61
  62 /*
  63 tgamma(x+2)  = tgamma(x+2) P(x)/Q(x)
  64 0 <= x <= 1
  65 Relative error
  66 n=7, d=8
  67 Peak error =  1.83e-20
  68 Relative error spread =  8.4e-23
  69 */
  70
  71 static const long double P[8] = {
  72  4.212760487471622013093E-5L,
  73  4.542931960608009155600E-4L,
  74  4.092666828394035500949E-3L,
  75  2.385363243461108252554E-2L,
  76  1.113062816019361559013E-1L,
  77  3.629515436640239168939E-1L,
  78  8.378004301573126728826E-1L,
  79  1.000000000000000000009E0L,
  80 };
  81 static const long double Q[9] = {
  82 -1.397148517476170440917E-5L,
  83  2.346584059160635244282E-4L,
  84 -1.237799246653152231188E-3L,
  85 -7.955933682494738320586E-4L,
  86  2.773706565840072979165E-2L,
  87 -4.633887671244534213831E-2L,
  88 -2.243510905670329164562E-1L,
  89  4.150160950588455434583E-1L,
  90  9.999999999999999999908E-1L,
  91 };
  92
  93 /*
  94 static long double P[] = {
  95 -3.01525602666895735709e0L,
  96 -3.25157411956062339893e1L,
  97 -2.92929976820724030353e2L,
  98 -1.70730828800510297666e3L,
  99 -7.96667499622741999770e3L,
 100 -2.59780216007146401957e4L,
 101 -5.99650230220855581642e4L,
 102 -7.15743521530849602425e4L
 103 };
 104 static long double Q[] = {
 105  1.00000000000000000000e0L,
 106 -1.67955233807178858919e1L,
 107  8.85946791747759881659e1L,
 108  5.69440799097468430177e1L,
 109 -1.98526250512761318471e3L,
 110  3.31667508019495079814e3L,
 111  1.60577839621734713377e4L,
 112 -2.97045081369399940529e4L,
 113 -7.15743521530849602412e4L
 114 };
 115 */
 116 #define MAXGAML 1755.455L
 117 /*static const long double LOGPI = 1.14472988584940017414L;*/
 118
 119 /* Stirling's formula for the gamma function
 120 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
 121 z(x) = x
 122 13 <= x <= 1024
 123 Relative error
 124 n=8, d=0
 125 Peak error =  9.44e-21
 126 Relative error spread =  8.8e-4
 127 */
 128
 129 static const long double STIR[9] = {
 130  7.147391378143610789273E-4L,
 131 -2.363848809501759061727E-5L,
 132 -5.950237554056330156018E-4L,
 133  6.989332260623193171870E-5L,
 134  7.840334842744753003862E-4L,
 135 -2.294719747873185405699E-4L,
 136 -2.681327161876304418288E-3L,
 137  3.472222222230075327854E-3L,
 138  8.333333333333331800504E-2L,
 139 };
 140
 141 #define MAXSTIR 1024.0L
 142 static const long double SQTPI = 2.50662827463100050242E0L;
 143
 144 /* 1/tgamma(x) = z P(z)
 145  * z(x) = 1/x
 146  * 0 < x < 0.03125
 147  * Peak relative error 4.2e-23
 148  */
 149
 150 static const long double S[9] = {
 151 -1.193945051381510095614E-3L,
 152  7.220599478036909672331E-3L,
 153 -9.622023360406271645744E-3L,
 154 -4.219773360705915470089E-2L,
 155  1.665386113720805206758E-1L,
 156 -4.200263503403344054473E-2L,
 157 -6.558780715202540684668E-1L,
 158  5.772156649015328608253E-1L,
 159  1.000000000000000000000E0L,
 160 };
 161
 162 /* 1/tgamma(-x) = z P(z)
 163  * z(x) = 1/x
 164  * 0 < x < 0.03125
 165  * Peak relative error 5.16e-23
 166  * Relative error spread =  2.5e-24
 167  */
 168
 169 static const long double SN[9] = {
 170  1.133374167243894382010E-3L,
 171  7.220837261893170325704E-3L,
 172  9.621911155035976733706E-3L,
 173 -4.219773343731191721664E-2L,
 174 -1.665386113944413519335E-1L,
 175 -4.200263503402112910504E-2L,
 176  6.558780715202536547116E-1L,
 177  5.772156649015328608727E-1L,
 178 -1.000000000000000000000E0L,
 179 };
 180
 181 static const long double PIL = 3.1415926535897932384626L;
 182
 183 static long double stirf ( long double );
 184
 185 /* Gamma function computed by Stirling's formula.
 186  */
 187 static long double stirf(long double x)
 188 {
 189 long double y, w, v;
 190
 191 w = 1.0L/x;
 192 /* For large x, use rational coefficients from the analytical expansion.  */
 193 if( x > 1024.0L )
 194         w = (((((6.97281375836585777429E-5L * w
 195                 + 7.84039221720066627474E-4L) * w
 196                 - 2.29472093621399176955E-4L) * w
 197                 - 2.68132716049382716049E-3L) * w
 198                 + 3.47222222222222222222E-3L) * w
 199                 + 8.33333333333333333333E-2L) * w
 200                 + 1.0L;
 201 else
 202         w = 1.0L + w * __polevll( w, (void *)STIR, 8 );
 203 y = expl(x);
 204 if( x > MAXSTIR )
 205         { /* Avoid overflow in pow() */
 206         v = powl( x, 0.5L * x - 0.25L );
 207         y = v * (v / y);
 208         }
 209 else
 210         {
 211         y = powl( x, x - 0.5L ) / y;
 212         }
 213 y = SQTPI * y * w;
 214 return( y );
 215 }
 216
 217 long double
 218 tgammal(long double x)
 219 {
 220 long double p, q, z;
 221 int i;
 222
 223 if( isnan(x) )
 224         return(NAN);
 225 if(x == INFINITY)
 226         return(INFINITY);
 227 if(x == -INFINITY)
 228         return(x - x);
 229 if( x == 0.0L )
 230         return( 1.0L / x );
 231 q = fabsl(x);
 232
 233 if( q > 13.0L )
 234         {
 235         int sign = 1;
 236         if( q > MAXGAML )
 237                 goto goverf;
 238         if( x < 0.0L )
 239                 {
 240                 p = floorl(q);
 241                 if( p == q )
 242                         return (x - x) / (x - x);
 243                 i = p;
 244                 if( (i & 1) == 0 )
 245                         sign = -1;
 246                 z = q - p;
 247                 if( z > 0.5L )
 248                         {
 249                         p += 1.0L;
 250                         z = q - p;
 251                         }
 252                 z = q * sinl( PIL * z );
 253                 z = fabsl(z) * stirf(q);
 254                 if( z <= PIL/LDBL_MAX )
 255                         {
 256 goverf:
 257                         return( sign * INFINITY);
 258                         }
 259                 z = PIL/z;
 260                 }
 261         else
 262                 {
 263                 z = stirf(x);
 264                 }
 265         return( sign * z );
 266         }
 267
 268 z = 1.0L;
 269 while( x >= 3.0L )
 270         {
 271         x -= 1.0L;
 272         z *= x;
 273         }
 274
 275 while( x < -0.03125L )
 276         {
 277         z /= x;
 278         x += 1.0L;
 279         }
 280
 281 if( x <= 0.03125L )
 282         goto small;
 283
 284 while( x < 2.0L )
 285         {
 286         z /= x;
 287         x += 1.0L;
 288         }
 289
 290 if( x == 2.0L )
 291         return(z);
 292
 293 x -= 2.0L;
 294 p = __polevll( x, (void *)P, 7 );
 295 q = __polevll( x, (void *)Q, 8 );
 296 z = z * p / q;
 297 return z;
 298
 299 small:
 300 if( x == 0.0L )
 301         return (x - x) / (x - x);
 302 else
 303         {
 304         if( x < 0.0L )
 305                 {
 306                 x = -x;
 307                 q = z / (x * __polevll( x, (void *)SN, 8 ));
 308                 }
 309         else
 310                 q = z / (x * __polevll( x, (void *)S, 8 ));
 311         }
 312 return q;
 313 }