lib/libm/noieee_src/n_lgamma.c

   1 /*      $NetBSD: n_lgamma.c,v 1.5 2003/08/07 16:44:51 agc Exp $ */
   2 /*-
   3  * Copyright (c) 1992, 1993
   4  *      The Regents of the University of California.  All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  * 3. Neither the name of the University nor the names of its contributors
  15  *    may be used to endorse or promote products derived from this software
  16  *    without specific prior written permission.
  17  *
  18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  28  * SUCH DAMAGE.
  29  */
  30
  31 #ifndef lint
  32 #if 0
  33 static char sccsid[] = "@(#)lgamma.c    8.2 (Berkeley) 11/30/93";
  34 #endif
  35 #endif /* not lint */
  36
  37 /*
  38  * Coded by Peter McIlroy, Nov 1992;
  39  *
  40  * The financial support of UUNET Communications Services is gratefully
  41  * acknowledged.
  42  */
  43
  44 #include <math.h>
  45 #include <errno.h>
  46
  47 #include "mathimpl.h"
  48
  49 /* Log gamma function.
  50  * Error:  x > 0 error < 1.3ulp.
  51  *         x > 4, error < 1ulp.
  52  *         x > 9, error < .6ulp.
  53  *         x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
  54  * Method:
  55  *      x > 6:
  56  *              Use the asymptotic expansion (Stirling's Formula)
  57  *      0 < x < 6:
  58  *              Use gamma(x+1) = x*gamma(x) for argument reduction.
  59  *              Use rational approximation in
  60  *              the range 1.2, 2.5
  61  *              Two approximations are used, one centered at the
  62  *              minimum to ensure monotonicity; one centered at 2
  63  *              to maintain small relative error.
  64  *      x < 0:
  65  *              Use the reflection formula,
  66  *              G(1-x)G(x) = PI/sin(PI*x)
  67  * Special values:
  68  *      non-positive integer    returns +Inf.
  69  *      NaN                     returns NaN
  70 */
  71 #if defined(__vax__) || defined(tahoe)
  72 #define _IEEE           0
  73 /* double and float have same size exponent field */
  74 #define TRUNC(x)        x = (double) (float) (x)
  75 #else
  76 static int endian;
  77 #define _IEEE           1
  78 #define TRUNC(x)        *(((int *) &x) + endian) &= 0xf8000000
  79 #define infnan(x)       0.0
  80 #endif
  81
  82 static double small_lgam(double);
  83 static double large_lgam(double);
  84 static double neg_lgam(double);
  85 static const double one = 1.0;
  86 int signgam;
  87
  88 #define UNDERFL (1e-1020 * 1e-1020)
  89
  90 #define LEFT    (1.0 - (x0 + .25))
  91 #define RIGHT   (x0 - .218)
  92 /*
  93  * Constants for approximation in [1.244,1.712]
  94 */
  95 #define x0      0.461632144968362356785
  96 #define x0_lo   -.000000000000000015522348162858676890521
  97 #define a0_hi   -0.12148629128932952880859
  98 #define a0_lo   .0000000007534799204229502
  99 #define r0      -2.771227512955130520e-002
 100 #define r1      -2.980729795228150847e-001
 101 #define r2      -3.257411333183093394e-001
 102 #define r3      -1.126814387531706041e-001
 103 #define r4      -1.129130057170225562e-002
 104 #define r5      -2.259650588213369095e-005
 105 #define s0       1.714457160001714442e+000
 106 #define s1       2.786469504618194648e+000
 107 #define s2       1.564546365519179805e+000
 108 #define s3       3.485846389981109850e-001
 109 #define s4       2.467759345363656348e-002
 110 /*
 111  * Constants for approximation in [1.71, 2.5]
 112 */
 113 #define a1_hi   4.227843350984671344505727574870e-01
 114 #define a1_lo   4.670126436531227189e-18
 115 #define p0      3.224670334241133695662995251041e-01
 116 #define p1      3.569659696950364669021382724168e-01
 117 #define p2      1.342918716072560025853732668111e-01
 118 #define p3      1.950702176409779831089963408886e-02
 119 #define p4      8.546740251667538090796227834289e-04
 120 #define q0      1.000000000000000444089209850062e+00
 121 #define q1      1.315850076960161985084596381057e+00
 122 #define q2      6.274644311862156431658377186977e-01
 123 #define q3      1.304706631926259297049597307705e-01
 124 #define q4      1.102815279606722369265536798366e-02
 125 #define q5      2.512690594856678929537585620579e-04
 126 #define q6      -1.003597548112371003358107325598e-06
 127 /*
 128  * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
 129 */
 130 #define lns2pi  .418938533204672741780329736405
 131 #define pb0      8.33333333333333148296162562474e-02
 132 #define pb1     -2.77777777774548123579378966497e-03
 133 #define pb2      7.93650778754435631476282786423e-04
 134 #define pb3     -5.95235082566672847950717262222e-04
 135 #define pb4      8.41428560346653702135821806252e-04
 136 #define pb5     -1.89773526463879200348872089421e-03
 137 #define pb6      5.69394463439411649408050664078e-03
 138 #define pb7     -1.44705562421428915453880392761e-02
 139
 140 __pure double
 141 lgamma(double x)
 142 {
 143         double r;
 144
 145         signgam = 1;
 146 #if _IEEE
 147         endian = ((*(int *) &one)) ? 1 : 0;
 148 #endif
 149
 150         if (!finite(x)) {
 151                 if (_IEEE)
 152                         return (x+x);
 153                 else return (infnan(EDOM));
 154         }
 155
 156         if (x > 6 + RIGHT) {
 157                 r = large_lgam(x);
 158                 return (r);
 159         } else if (x > 1e-16)
 160                 return (small_lgam(x));
 161         else if (x > -1e-16) {
 162                 if (x < 0)
 163                         signgam = -1, x = -x;
 164                 return (-log(x));
 165         } else
 166                 return (neg_lgam(x));
 167 }
 168
 169 static double
 170 large_lgam(double x)
 171 {
 172         double z, p, x1;
 173         struct Double t, u, v;
 174         u = __log__D(x);
 175         u.a -= 1.0;
 176         if (x > 1e15) {
 177                 v.a = x - 0.5;
 178                 TRUNC(v.a);
 179                 v.b = (x - v.a) - 0.5;
 180                 t.a = u.a*v.a;
 181                 t.b = x*u.b + v.b*u.a;
 182                 if (_IEEE == 0 && !finite(t.a))
 183                         return(infnan(ERANGE));
 184                 return(t.a + t.b);
 185         }
 186         x1 = 1./x;
 187         z = x1*x1;
 188         p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
 189                                         /* error in approximation = 2.8e-19 */
 190
 191         p = p*x1;                       /* error < 2.3e-18 absolute */
 192                                         /* 0 < p < 1/64 (at x = 5.5) */
 193         v.a = x = x - 0.5;
 194         TRUNC(v.a);                     /* truncate v.a to 26 bits. */
 195         v.b = x - v.a;
 196         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 197         t.b = v.b*u.a + x*u.b;
 198         t.b += p; t.b += lns2pi;        /* return t + lns2pi + p */
 199         return (t.a + t.b);
 200 }
 201
 202 static double
 203 small_lgam(double x)
 204 {
 205         int x_int;
 206         double y, z, t, r = 0, p, q, hi, lo;
 207         struct Double rr;
 208         x_int = (x + .5);
 209         y = x - x_int;
 210         if (x_int <= 2 && y > RIGHT) {
 211                 t = y - x0;
 212                 y--; x_int++;
 213                 goto CONTINUE;
 214         } else if (y < -LEFT) {
 215                 t = y +(1.0-x0);
 216 CONTINUE:
 217                 z = t - x0_lo;
 218                 p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
 219                 q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
 220                 r = t*(z*(p/q) - x0_lo);
 221                 t = .5*t*t;
 222                 z = 1.0;
 223                 switch (x_int) {
 224                 case 6: z  = (y + 5);
 225                 case 5: z *= (y + 4);
 226                 case 4: z *= (y + 3);
 227                 case 3: z *= (y + 2);
 228                         rr = __log__D(z);
 229                         rr.b += a0_lo; rr.a += a0_hi;
 230                         return(((r+rr.b)+t+rr.a));
 231                 case 2: return(((r+a0_lo)+t)+a0_hi);
 232                 case 0: r -= log1p(x);
 233                 default: rr = __log__D(x);
 234                         rr.a -= a0_hi; rr.b -= a0_lo;
 235                         return(((r - rr.b) + t) - rr.a);
 236                 }
 237         } else {
 238                 p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
 239                 q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
 240                 p = p*(y/q);
 241                 t = (double)(float) y;
 242                 z = y-t;
 243                 hi = (double)(float) (p+a1_hi);
 244                 lo = a1_hi - hi; lo += p; lo += a1_lo;
 245                 r = lo*y + z*hi;        /* q + r = y*(a0+p/q) */
 246                 q = hi*t;
 247                 z = 1.0;
 248                 switch (x_int) {
 249                 case 6: z  = (y + 5);
 250                 case 5: z *= (y + 4);
 251                 case 4: z *= (y + 3);
 252                 case 3: z *= (y + 2);
 253                         rr = __log__D(z);
 254                         r += rr.b; r += q;
 255                         return(rr.a + r);
 256                 case 2: return (q+ r);
 257                 case 0: rr = __log__D(x);
 258                         r -= rr.b; r -= log1p(x);
 259                         r += q; r-= rr.a;
 260                         return(r);
 261                 default: rr = __log__D(x);
 262                         r -= rr.b;
 263                         q -= rr.a;
 264                         return (r+q);
 265                 }
 266         }
 267 }
 268
 269 static double
 270 neg_lgam(double x)
 271 {
 272         int xi;
 273         double y, z, zero = 0.0;
 274
 275         /* avoid destructive cancellation as much as possible */
 276         if (x > -170) {
 277                 xi = x;
 278                 if (xi == x) {
 279                         if (_IEEE)
 280                                 return(one/zero);
 281                         else
 282                                 return(infnan(ERANGE));
 283                 }
 284                 y = gamma(x);
 285                 if (y < 0)
 286                         y = -y, signgam = -1;
 287                 return (log(y));
 288         }
 289         z = floor(x + .5);
 290         if (z == x) {           /* convention: G(-(integer)) -> +Inf */
 291                 if (_IEEE)
 292                         return (one/zero);
 293                 else
 294                         return (infnan(ERANGE));
 295         }
 296         y = .5*ceil(x);
 297         if (y == ceil(y))
 298                 signgam = -1;
 299         x = -x;
 300         z = fabs(x + z);        /* 0 < z <= .5 */
 301         if (z < .25)
 302                 z = sin(M_PI*z);
 303         else
 304                 z = cos(M_PI*(0.5-z));
 305         z = log(M_PI/(z*x));
 306         y = large_lgam(x);
 307         return (z - y);
 308 }