lib/libm/noieee_src/n_gamma.c

   1 /*      $NetBSD: n_gamma.c,v 1.6 2006/11/24 21:15:54 wiz 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[] = "@(#)gamma.c     8.1 (Berkeley) 6/4/93";
  34 #endif
  35 #endif /* not lint */
  36
  37 /*
  38  * This code by P. McIlroy, Oct 1992;
  39  *
  40  * The financial support of UUNET Communications Services is gratefully
  41  * acknowledged.
  42  */
  43
  44 #include <math.h>
  45 #include "mathimpl.h"
  46 #include <errno.h>
  47
  48 /* METHOD:
  49  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
  50  *      At negative integers, return +Inf, and set errno.
  51  *
  52  * x < 6.5:
  53  *      Use argument reduction G(x+1) = xG(x) to reach the
  54  *      range [1.066124,2.066124].  Use a rational
  55  *      approximation centered at the minimum (x0+1) to
  56  *      ensure monotonicity.
  57  *
  58  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
  59  *      adjusted for equal-ripples:
  60  *
  61  *      log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
  62  *
  63  *      Keep extra precision in multiplying (x-.5)(log(x)-1), to
  64  *      avoid premature round-off.
  65  *
  66  * Special values:
  67  *      non-positive integer:   Set overflow trap; return +Inf;
  68  *      x > 171.63:             Set overflow trap; return +Inf;
  69  *      NaN:                    Set invalid trap;  return NaN
  70  *
  71  * Accuracy: Gamma(x) is accurate to within
  72  *      x > 0:  error provably < 0.9ulp.
  73  *      Maximum observed in 1,000,000 trials was .87ulp.
  74  *      x < 0:
  75  *      Maximum observed error < 4ulp in 1,000,000 trials.
  76  */
  77
  78 static double neg_gam (double);
  79 static double small_gam (double);
  80 static double smaller_gam (double);
  81 static struct Double large_gam (double);
  82 static struct Double ratfun_gam (double, double);
  83
  84 /*
  85  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
  86  * [1.066.., 2.066..] accurate to 4.25e-19.
  87  */
  88 #define LEFT -.3955078125       /* left boundary for rat. approx */
  89 #define x0 .461632144968362356785       /* xmin - 1 */
  90
  91 #define a0_hi 0.88560319441088874992
  92 #define a0_lo -.00000000000000004996427036469019695
  93 #define P0       6.21389571821820863029017800727e-01
  94 #define P1       2.65757198651533466104979197553e-01
  95 #define P2       5.53859446429917461063308081748e-03
  96 #define P3       1.38456698304096573887145282811e-03
  97 #define P4       2.40659950032711365819348969808e-03
  98 #define Q0       1.45019531250000000000000000000e+00
  99 #define Q1       1.06258521948016171343454061571e+00
 100 #define Q2      -2.07474561943859936441469926649e-01
 101 #define Q3      -1.46734131782005422506287573015e-01
 102 #define Q4       3.07878176156175520361557573779e-02
 103 #define Q5       5.12449347980666221336054633184e-03
 104 #define Q6      -1.76012741431666995019222898833e-03
 105 #define Q7       9.35021023573788935372153030556e-05
 106 #define Q8       6.13275507472443958924745652239e-06
 107 /*
 108  * Constants for large x approximation (x in [6, Inf])
 109  * (Accurate to 2.8*10^-19 absolute)
 110  */
 111 #define lns2pi_hi 0.418945312500000
 112 #define lns2pi_lo -.000006779295327258219670263595
 113 #define Pa0      8.33333333333333148296162562474e-02
 114 #define Pa1     -2.77777777774548123579378966497e-03
 115 #define Pa2      7.93650778754435631476282786423e-04
 116 #define Pa3     -5.95235082566672847950717262222e-04
 117 #define Pa4      8.41428560346653702135821806252e-04
 118 #define Pa5     -1.89773526463879200348872089421e-03
 119 #define Pa6      5.69394463439411649408050664078e-03
 120 #define Pa7     -1.44705562421428915453880392761e-02
 121
 122 static const double zero = 0., one = 1.0, tiny = 1e-300;
 123 /*
 124  * TRUNC sets trailing bits in a floating-point number to zero.
 125  * is a temporary variable.
 126  */
 127 #if defined(__vax__) || defined(tahoe)
 128 #define _IEEE           0
 129 #define TRUNC(x)        x = (double) (float) (x)
 130 #else
 131 static int endian;
 132 #define _IEEE           1
 133 #define TRUNC(x)        *(((int *) &x) + endian) &= 0xf8000000
 134 #define infnan(x)       0.0
 135 #endif
 136
 137 double
 138 gamma(x)
 139         double x;
 140 {
 141         double b;
 142         struct Double u;
 143 #if _IEEE
 144         int endian = (*(int *) &one) ? 1 : 0;
 145 #endif
 146
 147         if (x >= 6) {
 148                 if(x > 171.63)
 149                         return(one/zero);
 150                 u = large_gam(x);
 151                 return(__exp__D(u.a, u.b));
 152         } else if (x >= 1.0 + LEFT + x0) {
 153                 return (small_gam(x));
 154         } else if (x > 1.e-17) {
 155                 return (smaller_gam(x));
 156         } else if (x > -1.e-17) {
 157                 if (x == 0.0) {
 158                         if (!_IEEE) return (infnan(ERANGE));
 159                         else return (one/x);
 160                 }
 161                 b =one+1e-20;           /* Raise inexact flag. ??? -ragge */
 162                 return (one/x);
 163         } else if (!finite(x)) {
 164                 if (_IEEE)              /* x = NaN, -Inf */
 165                         return (x*x);
 166                 else
 167                         return (infnan(EDOM));
 168          } else
 169                 return (neg_gam(x));
 170 }
 171 /*
 172  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
 173  */
 174 static struct Double
 175 large_gam(double x)
 176 {
 177         double z, p;
 178         struct Double t, u, v;
 179
 180         z = one/(x*x);
 181         p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
 182         p = p/x;
 183
 184         u = __log__D(x);
 185         u.a -= one;
 186         v.a = (x -= .5);
 187         TRUNC(v.a);
 188         v.b = x - v.a;
 189         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 190         t.b = v.b*u.a + x*u.b;
 191         /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
 192         t.b += lns2pi_lo; t.b += p;
 193         u.a = lns2pi_hi + t.b; u.a += t.a;
 194         u.b = t.a - u.a;
 195         u.b += lns2pi_hi; u.b += t.b;
 196         return (u);
 197 }
 198 /*
 199  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
 200  * It also has correct monotonicity.
 201  */
 202 static double
 203 small_gam(double x)
 204 {
 205         double y, ym1, t;
 206         struct Double yy, r;
 207         y = x - one;
 208         ym1 = y - one;
 209         if (y <= 1.0 + (LEFT + x0)) {
 210                 yy = ratfun_gam(y - x0, 0);
 211                 return (yy.a + yy.b);
 212         }
 213         r.a = y;
 214         TRUNC(r.a);
 215         yy.a = r.a - one;
 216         y = ym1;
 217         yy.b = r.b = y - yy.a;
 218         /* Argument reduction: G(x+1) = x*G(x) */
 219         for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
 220                 t = r.a*yy.a;
 221                 r.b = r.a*yy.b + y*r.b;
 222                 r.a = t;
 223                 TRUNC(r.a);
 224                 r.b += (t - r.a);
 225         }
 226         /* Return r*gamma(y). */
 227         yy = ratfun_gam(y - x0, 0);
 228         y = r.b*(yy.a + yy.b) + r.a*yy.b;
 229         y += yy.a*r.a;
 230         return (y);
 231 }
 232 /*
 233  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
 234  */
 235 static double
 236 smaller_gam(double x)
 237 {
 238         double t, d;
 239         struct Double r, xx;
 240         if (x < x0 + LEFT) {
 241                 t = x, TRUNC(t);
 242                 d = (t+x)*(x-t);
 243                 t *= t;
 244                 xx.a = (t + x), TRUNC(xx.a);
 245                 xx.b = x - xx.a; xx.b += t; xx.b += d;
 246                 t = (one-x0); t += x;
 247                 d = (one-x0); d -= t; d += x;
 248                 x = xx.a + xx.b;
 249         } else {
 250                 xx.a =  x, TRUNC(xx.a);
 251                 xx.b = x - xx.a;
 252                 t = x - x0;
 253                 d = (-x0 -t); d += x;
 254         }
 255         r = ratfun_gam(t, d);
 256         d = r.a/x, TRUNC(d);
 257         r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
 258         return (d + r.a/x);
 259 }
 260 /*
 261  * returns (z+c)^2 * P(z)/Q(z) + a0
 262  */
 263 static struct Double
 264 ratfun_gam(double z, double c)
 265 {
 266         double p, q;
 267         struct Double r, t;
 268
 269         q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
 270         p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
 271
 272         /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
 273         p = p/q;
 274         t.a = z, TRUNC(t.a);            /* t ~= z + c */
 275         t.b = (z - t.a) + c;
 276         t.b *= (t.a + z);
 277         q = (t.a *= t.a);               /* t = (z+c)^2 */
 278         TRUNC(t.a);
 279         t.b += (q - t.a);
 280         r.a = p, TRUNC(r.a);            /* r = P/Q */
 281         r.b = p - r.a;
 282         t.b = t.b*p + t.a*r.b + a0_lo;
 283         t.a *= r.a;                     /* t = (z+c)^2*(P/Q) */
 284         r.a = t.a + a0_hi, TRUNC(r.a);
 285         r.b = ((a0_hi-r.a) + t.a) + t.b;
 286         return (r);                     /* r = a0 + t */
 287 }
 288
 289 static double
 290 neg_gam(double x)
 291 {
 292         int sgn = 1;
 293         struct Double lg, lsine;
 294         double y, z;
 295
 296         y = floor(x + .5);
 297         if (y == x) {           /* Negative integer. */
 298                 if(!_IEEE)
 299                         return (infnan(ERANGE));
 300                 else
 301                         return (one/zero);
 302         }
 303         z = fabs(x - y);
 304         y = .5*ceil(x);
 305         if (y == ceil(y))
 306                 sgn = -1;
 307         if (z < .25)
 308                 z = sin(M_PI*z);
 309         else
 310                 z = cos(M_PI*(0.5-z));
 311         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
 312         if (x < -170) {
 313                 if (x < -190)
 314                         return ((double)sgn*tiny*tiny);
 315                 y = one - x;            /* exact: 128 < |x| < 255 */
 316                 lg = large_gam(y);
 317                 lsine = __log__D(M_PI/z);       /* = TRUNC(log(u)) + small */
 318                 lg.a -= lsine.a;                /* exact (opposite signs) */
 319                 lg.b -= lsine.b;
 320                 y = -(lg.a + lg.b);
 321                 z = (y + lg.a) + lg.b;
 322                 y = __exp__D(y, z);
 323                 if (sgn < 0) y = -y;
 324                 return (y);
 325         }
 326         y = one-x;
 327         if (one-y == x)
 328                 y = gamma(y);
 329         else            /* 1-x is inexact */
 330                 y = -x*gamma(-x);
 331         if (sgn < 0) y = -y;
 332         return (M_PI / (y*z));
 333 }