lib/libm/common_source/lgamma.c

   1 /*-
   2  * Copyright (c) 1992, 1993
   3  *      The Regents of the University of California.  All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  * 1. Redistributions of source code must retain the above copyright
   9  *    notice, this list of conditions and the following disclaimer.
  10  * 2. Redistributions in binary form must reproduce the above copyright
  11  *    notice, this list of conditions and the following disclaimer in the
  12  *    documentation and/or other materials provided with the distribution.
  13  * 3. All advertising materials mentioning features or use of this software
  14  *    must display the following acknowledgement:
  15  *      This product includes software developed by the University of
  16  *      California, Berkeley and its contributors.
  17  * 4. Neither the name of the University nor the names of its contributors
  18  *    may be used to endorse or promote products derived from this software
  19  *    without specific prior written permission.
  20  *
  21  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  22  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  23  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  24  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  25  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  26  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  27  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  28  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  29  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  30  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  31  * SUCH DAMAGE.
  32  */
  33
  34 #ifndef lint
  35 static char sccsid[] = "@(#)lgamma.c    8.2 (Berkeley) 11/30/93";
  36 #endif /* not lint */
  37
  38 /*
  39  * Coded by Peter McIlroy, Nov 1992;
  40  *
  41  * The financial support of UUNET Communications Services is greatfully
  42  * acknowledged.
  43  */
  44
  45 #include <math.h>
  46 #include <errno.h>
  47
  48 #include "mathimpl.h"
  49
  50 /* Log gamma function.
  51  * Error:  x > 0 error < 1.3ulp.
  52  *         x > 4, error < 1ulp.
  53  *         x > 9, error < .6ulp.
  54  *         x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
  55  * Method:
  56  *      x > 6:
  57  *              Use the asymptotic expansion (Stirling's Formula)
  58  *      0 < x < 6:
  59  *              Use gamma(x+1) = x*gamma(x) for argument reduction.
  60  *              Use rational approximation in
  61  *              the range 1.2, 2.5
  62  *              Two approximations are used, one centered at the
  63  *              minimum to ensure monotonicity; one centered at 2
  64  *              to maintain small relative error.
  65  *      x < 0:
  66  *              Use the reflection formula,
  67  *              G(1-x)G(x) = PI/sin(PI*x)
  68  * Special values:
  69  *      non-positive integer    returns +Inf.
  70  *      NaN                     returns NaN
  71 */
  72 static int endian;
  73 #if defined(vax) || defined(tahoe)
  74 #define _IEEE           0
  75 /* double and float have same size exponent field */
  76 #define TRUNC(x)        x = (double) (float) (x)
  77 #else
  78 #define _IEEE           1
  79 #define TRUNC(x)        *(((int *) &x) + endian) &= 0xf8000000
  80 #define infnan(x)       0.0
  81 #endif
  82
  83 static double small_lgam(double);
  84 static double large_lgam(double);
  85 static double neg_lgam(double);
  86 static double zero = 0.0, one = 1.0;
  87 int signgam;
  88
  89 #define UNDERFL (1e-1020 * 1e-1020)
  90
  91 #define LEFT    (1.0 - (x0 + .25))
  92 #define RIGHT   (x0 - .218)
  93 /*
  94  * Constants for approximation in [1.244,1.712]
  95 */
  96 #define x0      0.461632144968362356785
  97 #define x0_lo   -.000000000000000015522348162858676890521
  98 #define a0_hi   -0.12148629128932952880859
  99 #define a0_lo   .0000000007534799204229502
 100 #define r0      -2.771227512955130520e-002
 101 #define r1      -2.980729795228150847e-001
 102 #define r2      -3.257411333183093394e-001
 103 #define r3      -1.126814387531706041e-001
 104 #define r4      -1.129130057170225562e-002
 105 #define r5      -2.259650588213369095e-005
 106 #define s0       1.714457160001714442e+000
 107 #define s1       2.786469504618194648e+000
 108 #define s2       1.564546365519179805e+000
 109 #define s3       3.485846389981109850e-001
 110 #define s4       2.467759345363656348e-002
 111 /*
 112  * Constants for approximation in [1.71, 2.5]
 113 */
 114 #define a1_hi   4.227843350984671344505727574870e-01
 115 #define a1_lo   4.670126436531227189e-18
 116 #define p0      3.224670334241133695662995251041e-01
 117 #define p1      3.569659696950364669021382724168e-01
 118 #define p2      1.342918716072560025853732668111e-01
 119 #define p3      1.950702176409779831089963408886e-02
 120 #define p4      8.546740251667538090796227834289e-04
 121 #define q0      1.000000000000000444089209850062e+00
 122 #define q1      1.315850076960161985084596381057e+00
 123 #define q2      6.274644311862156431658377186977e-01
 124 #define q3      1.304706631926259297049597307705e-01
 125 #define q4      1.102815279606722369265536798366e-02
 126 #define q5      2.512690594856678929537585620579e-04
 127 #define q6      -1.003597548112371003358107325598e-06
 128 /*
 129  * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
 130 */
 131 #define lns2pi  .418938533204672741780329736405
 132 #define pb0      8.33333333333333148296162562474e-02
 133 #define pb1     -2.77777777774548123579378966497e-03
 134 #define pb2      7.93650778754435631476282786423e-04
 135 #define pb3     -5.95235082566672847950717262222e-04
 136 #define pb4      8.41428560346653702135821806252e-04
 137 #define pb5     -1.89773526463879200348872089421e-03
 138 #define pb6      5.69394463439411649408050664078e-03
 139 #define pb7     -1.44705562421428915453880392761e-02
 140
 141 /* __pure double */
 142 double
 143 lgamma(double x)
 144 {
 145         double r;
 146
 147         signgam = 1;
 148         endian = ((*(int *) &one)) ? 1 : 0;
 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
 170 static double
 171 large_lgam(double x)
 172 {
 173         double z, p, x1;
 174         int i;
 175         struct Double t, u, v;
 176         u = __log__D(x);
 177         u.a -= 1.0;
 178         if (x > 1e15) {
 179                 v.a = x - 0.5;
 180                 TRUNC(v.a);
 181                 v.b = (x - v.a) - 0.5;
 182                 t.a = u.a*v.a;
 183                 t.b = x*u.b + v.b*u.a;
 184                 if (_IEEE == 0 && !finite(t.a))
 185                         return(infnan(ERANGE));
 186                 return(t.a + t.b);
 187         }
 188         x1 = 1./x;
 189         z = x1*x1;
 190         p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
 191                                         /* error in approximation = 2.8e-19 */
 192
 193         p = p*x1;                       /* error < 2.3e-18 absolute */
 194                                         /* 0 < p < 1/64 (at x = 5.5) */
 195         v.a = x = x - 0.5;
 196         TRUNC(v.a);                     /* truncate v.a to 26 bits. */
 197         v.b = x - v.a;
 198         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 199         t.b = v.b*u.a + x*u.b;
 200         t.b += p; t.b += lns2pi;        /* return t + lns2pi + p */
 201         return (t.a + t.b);
 202 }
 203
 204 static double
 205 small_lgam(double x)
 206 {
 207         int x_int;
 208         double y, z, t, r = 0, p, q, hi, lo;
 209         struct Double rr;
 210         x_int = (x + .5);
 211         y = x - x_int;
 212         if (x_int <= 2 && y > RIGHT) {
 213                 t = y - x0;
 214                 y--; x_int++;
 215                 goto CONTINUE;
 216         } else if (y < -LEFT) {
 217                 t = y +(1.0-x0);
 218 CONTINUE:
 219                 z = t - x0_lo;
 220                 p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
 221                 q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
 222                 r = t*(z*(p/q) - x0_lo);
 223                 t = .5*t*t;
 224                 z = 1.0;
 225                 switch (x_int) {
 226                 case 6: z  = (y + 5);
 227                 case 5: z *= (y + 4);
 228                 case 4: z *= (y + 3);
 229                 case 3: z *= (y + 2);
 230                         rr = __log__D(z);
 231                         rr.b += a0_lo; rr.a += a0_hi;
 232                         return(((r+rr.b)+t+rr.a));
 233                 case 2: return(((r+a0_lo)+t)+a0_hi);
 234                 case 0: r -= log1p(x);
 235                 default: rr = __log__D(x);
 236                         rr.a -= a0_hi; rr.b -= a0_lo;
 237                         return(((r - rr.b) + t) - rr.a);
 238                 }
 239         } else {
 240                 p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
 241                 q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
 242                 p = p*(y/q);
 243                 t = (double)(float) y;
 244                 z = y-t;
 245                 hi = (double)(float) (p+a1_hi);
 246                 lo = a1_hi - hi; lo += p; lo += a1_lo;
 247                 r = lo*y + z*hi;        /* q + r = y*(a0+p/q) */
 248                 q = hi*t;
 249                 z = 1.0;
 250                 switch (x_int) {
 251                 case 6: z  = (y + 5);
 252                 case 5: z *= (y + 4);
 253                 case 4: z *= (y + 3);
 254                 case 3: z *= (y + 2);
 255                         rr = __log__D(z);
 256                         r += rr.b; r += q;
 257                         return(rr.a + r);
 258                 case 2: return (q+ r);
 259                 case 0: rr = __log__D(x);
 260                         r -= rr.b; r -= log1p(x);
 261                         r += q; r-= rr.a;
 262                         return(r);
 263                 default: rr = __log__D(x);
 264                         r -= rr.b;
 265                         q -= rr.a;
 266                         return (r+q);
 267                 }
 268         }
 269 }
 270
 271 static double
 272 neg_lgam(double x)
 273 {
 274         int xi;
 275         double y, z, one = 1.0, zero = 0.0;
 276         extern double gamma();
 277
 278         /* avoid destructive cancellation as much as possible */
 279         if (x > -170) {
 280                 xi = x;
 281                 if (xi == x) {
 282                         if (_IEEE)
 283                                 return(one/zero);
 284                         else
 285                                 return(infnan(ERANGE));
 286                 }
 287                 y = gamma(x);
 288                 if (y < 0)
 289                         y = -y, signgam = -1;
 290                 return (log(y));
 291         }
 292         z = floor(x + .5);
 293         if (z == x) {           /* convention: G(-(integer)) -> +Inf */
 294                 if (_IEEE)
 295                         return (one/zero);
 296                 else
 297                         return (infnan(ERANGE));
 298         }
 299         y = .5*ceil(x);
 300         if (y == ceil(y))
 301                 signgam = -1;
 302         x = -x;
 303         z = fabs(x + z);        /* 0 < z <= .5 */
 304         if (z < .25)
 305                 z = sin(M_PI*z);
 306         else
 307                 z = cos(M_PI*(0.5-z));
 308         z = log(M_PI/(z*x));
 309         y = large_lgam(x);
 310         return (z - y);
 311 }