src/math/tgammal.c

   1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_tgammal.c */
   2 /*
   3  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   4  *
   5  * Permission to use, copy, modify, and distribute this software for any
   6  * purpose with or without fee is hereby granted, provided that the above
   7  * copyright notice and this permission notice appear in all copies.
   8  *
   9  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  10  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  11  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  12  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  13  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  14  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  15  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  16  */
  17 /*
  18  *      Gamma function
  19  *
  20  *
  21  * SYNOPSIS:
  22  *
  23  * long double x, y, tgammal();
  24  *
  25  * y = tgammal( x );
  26  *
  27  *
  28  * DESCRIPTION:
  29  *
  30  * Returns gamma function of the argument.  The result is
  31  * correctly signed.
  32  *
  33  * Arguments |x| <= 13 are reduced by recurrence and the function
  34  * approximated by a rational function of degree 7/8 in the
  35  * interval (2,3).  Large arguments are handled by Stirling's
  36  * formula. Large negative arguments are made positive using
  37  * a reflection formula.
  38  *
  39  *
  40  * ACCURACY:
  41  *
  42  *                      Relative error:
  43  * arithmetic   domain     # trials      peak         rms
  44  *    IEEE     -40,+40      10000       3.6e-19     7.9e-20
  45  *    IEEE    -1755,+1755   10000       4.8e-18     6.5e-19
  46  *
  47  * Accuracy for large arguments is dominated by error in powl().
  48  *
  49  */
  50
  51 #include "libm.h"
  52
  53 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
  54 long double tgammal(long double x)
  55 {
  56         return tgamma(x);
  57 }
  58 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
  59 /*
  60 tgamma(x+2) = tgamma(x+2) P(x)/Q(x)
  61 0 <= x <= 1
  62 Relative error
  63 n=7, d=8
  64 Peak error =  1.83e-20
  65 Relative error spread =  8.4e-23
  66 */
  67 static const long double P[8] = {
  68  4.212760487471622013093E-5L,
  69  4.542931960608009155600E-4L,
  70  4.092666828394035500949E-3L,
  71  2.385363243461108252554E-2L,
  72  1.113062816019361559013E-1L,
  73  3.629515436640239168939E-1L,
  74  8.378004301573126728826E-1L,
  75  1.000000000000000000009E0L,
  76 };
  77 static const long double Q[9] = {
  78 -1.397148517476170440917E-5L,
  79  2.346584059160635244282E-4L,
  80 -1.237799246653152231188E-3L,
  81 -7.955933682494738320586E-4L,
  82  2.773706565840072979165E-2L,
  83 -4.633887671244534213831E-2L,
  84 -2.243510905670329164562E-1L,
  85  4.150160950588455434583E-1L,
  86  9.999999999999999999908E-1L,
  87 };
  88
  89 /*
  90 static const long double P[] = {
  91 -3.01525602666895735709e0L,
  92 -3.25157411956062339893e1L,
  93 -2.92929976820724030353e2L,
  94 -1.70730828800510297666e3L,
  95 -7.96667499622741999770e3L,
  96 -2.59780216007146401957e4L,
  97 -5.99650230220855581642e4L,
  98 -7.15743521530849602425e4L
  99 };
 100 static const long double Q[] = {
 101  1.00000000000000000000e0L,
 102 -1.67955233807178858919e1L,
 103  8.85946791747759881659e1L,
 104  5.69440799097468430177e1L,
 105 -1.98526250512761318471e3L,
 106  3.31667508019495079814e3L,
 107  1.60577839621734713377e4L,
 108 -2.97045081369399940529e4L,
 109 -7.15743521530849602412e4L
 110 };
 111 */
 112 #define MAXGAML 1755.455L
 113 /*static const long double LOGPI = 1.14472988584940017414L;*/
 114
 115 /* Stirling's formula for the gamma function
 116 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
 117 z(x) = x
 118 13 <= x <= 1024
 119 Relative error
 120 n=8, d=0
 121 Peak error =  9.44e-21
 122 Relative error spread =  8.8e-4
 123 */
 124 static const long double STIR[9] = {
 125  7.147391378143610789273E-4L,
 126 -2.363848809501759061727E-5L,
 127 -5.950237554056330156018E-4L,
 128  6.989332260623193171870E-5L,
 129  7.840334842744753003862E-4L,
 130 -2.294719747873185405699E-4L,
 131 -2.681327161876304418288E-3L,
 132  3.472222222230075327854E-3L,
 133  8.333333333333331800504E-2L,
 134 };
 135
 136 #define MAXSTIR 1024.0L
 137 static const long double SQTPI = 2.50662827463100050242E0L;
 138
 139 /* 1/tgamma(x) = z P(z)
 140  * z(x) = 1/x
 141  * 0 < x < 0.03125
 142  * Peak relative error 4.2e-23
 143  */
 144 static const long double S[9] = {
 145 -1.193945051381510095614E-3L,
 146  7.220599478036909672331E-3L,
 147 -9.622023360406271645744E-3L,
 148 -4.219773360705915470089E-2L,
 149  1.665386113720805206758E-1L,
 150 -4.200263503403344054473E-2L,
 151 -6.558780715202540684668E-1L,
 152  5.772156649015328608253E-1L,
 153  1.000000000000000000000E0L,
 154 };
 155
 156 /* 1/tgamma(-x) = z P(z)
 157  * z(x) = 1/x
 158  * 0 < x < 0.03125
 159  * Peak relative error 5.16e-23
 160  * Relative error spread =  2.5e-24
 161  */
 162 static const long double SN[9] = {
 163  1.133374167243894382010E-3L,
 164  7.220837261893170325704E-3L,
 165  9.621911155035976733706E-3L,
 166 -4.219773343731191721664E-2L,
 167 -1.665386113944413519335E-1L,
 168 -4.200263503402112910504E-2L,
 169  6.558780715202536547116E-1L,
 170  5.772156649015328608727E-1L,
 171 -1.000000000000000000000E0L,
 172 };
 173
 174 static const long double PIL = 3.1415926535897932384626L;
 175
 176 /* Gamma function computed by Stirling's formula.
 177  */
 178 static long double stirf(long double x)
 179 {
 180         long double y, w, v;
 181
 182         w = 1.0/x;
 183         /* For large x, use rational coefficients from the analytical expansion.  */
 184         if (x > 1024.0)
 185                 w = (((((6.97281375836585777429E-5L * w
 186                  + 7.84039221720066627474E-4L) * w
 187                  - 2.29472093621399176955E-4L) * w
 188                  - 2.68132716049382716049E-3L) * w
 189                  + 3.47222222222222222222E-3L) * w
 190                  + 8.33333333333333333333E-2L) * w
 191                  + 1.0;
 192         else
 193                 w = 1.0 + w * __polevll(w, STIR, 8);
 194         y = expl(x);
 195         if (x > MAXSTIR) { /* Avoid overflow in pow() */
 196                 v = powl(x, 0.5L * x - 0.25L);
 197                 y = v * (v / y);
 198         } else {
 199                 y = powl(x, x - 0.5L) / y;
 200         }
 201         y = SQTPI * y * w;
 202         return y;
 203 }
 204
 205 long double tgammal(long double x)
 206 {
 207         long double p, q, z;
 208
 209         if (!isfinite(x))
 210                 return x + INFINITY;
 211
 212         q = fabsl(x);
 213         if (q > 13.0) {
 214                 if (x < 0.0) {
 215                         p = floorl(q);
 216                         z = q - p;
 217                         if (z == 0)
 218                                 return 0 / z;
 219                         if (q > MAXGAML) {
 220                                 z = 0;
 221                         } else {
 222                                 if (z > 0.5) {
 223                                         p += 1.0;
 224                                         z = q - p;
 225                                 }
 226                                 z = q * sinl(PIL * z);
 227                                 z = fabsl(z) * stirf(q);
 228                                 z = PIL/z;
 229                         }
 230                         if (0.5 * p == floorl(q * 0.5))
 231                                 z = -z;
 232                 } else if (x > MAXGAML) {
 233                         z = x * 0x1p16383L;
 234                 } else {
 235                         z = stirf(x);
 236                 }
 237                 return z;
 238         }
 239
 240         z = 1.0;
 241         while (x >= 3.0) {
 242                 x -= 1.0;
 243                 z *= x;
 244         }
 245         while (x < -0.03125L) {
 246                 z /= x;
 247                 x += 1.0;
 248         }
 249         if (x <= 0.03125L)
 250                 goto small;
 251         while (x < 2.0) {
 252                 z /= x;
 253                 x += 1.0;
 254         }
 255         if (x == 2.0)
 256                 return z;
 257
 258         x -= 2.0;
 259         p = __polevll(x, P, 7);
 260         q = __polevll(x, Q, 8);
 261         z = z * p / q;
 262         return z;
 263
 264 small:
 265         /* z==1 if x was originally +-0 */
 266         if (x == 0 && z != 1)
 267                 return x / x;
 268         if (x < 0.0) {
 269                 x = -x;
 270                 q = z / (x * __polevll(x, SN, 8));
 271         } else
 272                 q = z / (x * __polevll(x, S, 8));
 273         return q;
 274 }
 275 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
 276 // TODO: broken implementation to make things compile
 277 long double tgammal(long double x)
 278 {
 279         return tgamma(x);
 280 }
 281 #endif