usr/src/lib/libast/common/uwin/gamma.c

   1 #include "FEATURE/uwin"
   2
   3 #if !_UWIN || _lib_gamma
   4
   5 void _STUB_gamma(){}
   6
   7 #else
   8
   9 /*-
  10  * Copyright (c) 1992, 1993
  11  *      The Regents of the University of California.  All rights reserved.
  12  *
  13  * Redistribution and use in source and binary forms, with or without
  14  * modification, are permitted provided that the following conditions
  15  * are met:
  16  * 1. Redistributions of source code must retain the above copyright
  17  *    notice, this list of conditions and the following disclaimer.
  18  * 2. Redistributions in binary form must reproduce the above copyright
  19  *    notice, this list of conditions and the following disclaimer in the
  20  *    documentation and/or other materials provided with the distribution.
  21  * 3. Neither the name of the University nor the names of its contributors
  22  *    may be used to endorse or promote products derived from this software
  23  *    without specific prior written permission.
  24  *
  25  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  26  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  27  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  28  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  29  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  30  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  31  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  32  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  33  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  34  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  35  * SUCH DAMAGE.
  36  */
  37
  38 static char sccsid[] = "@(#)gamma.c     8.1 (Berkeley) 6/4/93";
  39
  40 /*
  41  * This code by P. McIlroy, Oct 1992;
  42  *
  43  * The financial support of UUNET Communications Services is greatfully
  44  * acknowledged.
  45  */
  46
  47 #define gamma   ______gamma
  48
  49 #include <math.h>
  50 #include <errno.h>
  51 #include "mathimpl.h"
  52
  53 #undef  gamma
  54
  55 /* METHOD:
  56  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
  57  *      At negative integers, return +Inf, and set errno.
  58  *
  59  * x < 6.5:
  60  *      Use argument reduction G(x+1) = xG(x) to reach the
  61  *      range [1.066124,2.066124].  Use a rational
  62  *      approximation centered at the minimum (x0+1) to
  63  *      ensure monotonicity.
  64  *
  65  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
  66  *      adjusted for equal-ripples:
  67  *
  68  *      log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
  69  *
  70  *      Keep extra precision in multiplying (x-.5)(log(x)-1), to
  71  *      avoid premature round-off.
  72  *
  73  * Special values:
  74  *      non-positive integer:   Set overflow trap; return +Inf;
  75  *      x > 171.63:             Set overflow trap; return +Inf;
  76  *      NaN:                    Set invalid trap;  return NaN
  77  *
  78  * Accuracy: Gamma(x) is accurate to within
  79  *      x > 0:  error provably < 0.9ulp.
  80  *      Maximum observed in 1,000,000 trials was .87ulp.
  81  *      x < 0:
  82  *      Maximum observed error < 4ulp in 1,000,000 trials.
  83  */
  84
  85 static double neg_gam __P((double));
  86 static double small_gam __P((double));
  87 static double smaller_gam __P((double));
  88 static struct Double large_gam __P((double));
  89 static struct Double ratfun_gam __P((double, double));
  90
  91 /*
  92  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
  93  * [1.066.., 2.066..] accurate to 4.25e-19.
  94  */
  95 #define LEFT -.3955078125       /* left boundary for rat. approx */
  96 #define x0 .461632144968362356785       /* xmin - 1 */
  97
  98 #define a0_hi 0.88560319441088874992
  99 #define a0_lo -.00000000000000004996427036469019695
 100 #define P0       6.21389571821820863029017800727e-01
 101 #define P1       2.65757198651533466104979197553e-01
 102 #define P2       5.53859446429917461063308081748e-03
 103 #define P3       1.38456698304096573887145282811e-03
 104 #define P4       2.40659950032711365819348969808e-03
 105 #define Q0       1.45019531250000000000000000000e+00
 106 #define Q1       1.06258521948016171343454061571e+00
 107 #define Q2      -2.07474561943859936441469926649e-01
 108 #define Q3      -1.46734131782005422506287573015e-01
 109 #define Q4       3.07878176156175520361557573779e-02
 110 #define Q5       5.12449347980666221336054633184e-03
 111 #define Q6      -1.76012741431666995019222898833e-03
 112 #define Q7       9.35021023573788935372153030556e-05
 113 #define Q8       6.13275507472443958924745652239e-06
 114 /*
 115  * Constants for large x approximation (x in [6, Inf])
 116  * (Accurate to 2.8*10^-19 absolute)
 117  */
 118 #define lns2pi_hi 0.418945312500000
 119 #define lns2pi_lo -.000006779295327258219670263595
 120 #define Pa0      8.33333333333333148296162562474e-02
 121 #define Pa1     -2.77777777774548123579378966497e-03
 122 #define Pa2      7.93650778754435631476282786423e-04
 123 #define Pa3     -5.95235082566672847950717262222e-04
 124 #define Pa4      8.41428560346653702135821806252e-04
 125 #define Pa5     -1.89773526463879200348872089421e-03
 126 #define Pa6      5.69394463439411649408050664078e-03
 127 #define Pa7     -1.44705562421428915453880392761e-02
 128
 129 static const double zero = 0., one = 1.0, tiny = 1e-300;
 130 static int endian;
 131 /*
 132  * TRUNC sets trailing bits in a floating-point number to zero.
 133  * is a temporary variable.
 134  */
 135 #if defined(vax) || defined(tahoe)
 136 #define _IEEE           0
 137 #define TRUNC(x)        x = (double) (float) (x)
 138 #else
 139 #define _IEEE           1
 140 #define TRUNC(x)        *(((int *) &x) + endian) &= 0xf8000000
 141 #define infnan(x)       0.0
 142 #endif
 143
 144 extern double gamma(x)
 145         double x;
 146 {
 147         struct Double u;
 148         endian = (*(int *) &one) ? 1 : 0;
 149
 150         if (x >= 6) {
 151                 if(x > 171.63)
 152                         return(one/zero);
 153                 u = large_gam(x);
 154                 return(__exp__D(u.a, u.b));
 155         } else if (x >= 1.0 + LEFT + x0)
 156                 return (small_gam(x));
 157         else if (x > 1.e-17)
 158                 return (smaller_gam(x));
 159         else if (x > -1.e-17) {
 160                 if (x == 0.0)
 161                         if (!_IEEE) return (infnan(ERANGE));
 162                         else return (one/x);
 163                 one+1e-20;              /* Raise inexact flag. */
 164                 return (one/x);
 165         } else if (!finite(x)) {
 166                 if (_IEEE)              /* x = NaN, -Inf */
 167                         return (x*x);
 168                 else
 169                         return (infnan(EDOM));
 170          } else
 171                 return (neg_gam(x));
 172 }
 173 /*
 174  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
 175  */
 176 static struct Double
 177 large_gam(x)
 178         double x;
 179 {
 180         double z, p;
 181         struct Double t, u, v;
 182
 183         z = one/(x*x);
 184         p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
 185         p = p/x;
 186
 187         u = __log__D(x);
 188         u.a -= one;
 189         v.a = (x -= .5);
 190         TRUNC(v.a);
 191         v.b = x - v.a;
 192         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 193         t.b = v.b*u.a + x*u.b;
 194         /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
 195         t.b += lns2pi_lo; t.b += p;
 196         u.a = lns2pi_hi + t.b; u.a += t.a;
 197         u.b = t.a - u.a;
 198         u.b += lns2pi_hi; u.b += t.b;
 199         return (u);
 200 }
 201 /*
 202  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
 203  * It also has correct monotonicity.
 204  */
 205 static double
 206 small_gam(x)
 207         double x;
 208 {
 209         double y, ym1, t;
 210         struct Double yy, r;
 211         y = x - one;
 212         ym1 = y - one;
 213         if (y <= 1.0 + (LEFT + x0)) {
 214                 yy = ratfun_gam(y - x0, 0);
 215                 return (yy.a + yy.b);
 216         }
 217         r.a = y;
 218         TRUNC(r.a);
 219         yy.a = r.a - one;
 220         y = ym1;
 221         yy.b = r.b = y - yy.a;
 222         /* Argument reduction: G(x+1) = x*G(x) */
 223         for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
 224                 t = r.a*yy.a;
 225                 r.b = r.a*yy.b + y*r.b;
 226                 r.a = t;
 227                 TRUNC(r.a);
 228                 r.b += (t - r.a);
 229         }
 230         /* Return r*gamma(y). */
 231         yy = ratfun_gam(y - x0, 0);
 232         y = r.b*(yy.a + yy.b) + r.a*yy.b;
 233         y += yy.a*r.a;
 234         return (y);
 235 }
 236 /*
 237  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
 238  */
 239 static double
 240 smaller_gam(x)
 241         double x;
 242 {
 243         double t, d;
 244         struct Double r, xx;
 245         if (x < x0 + LEFT) {
 246                 t = x, TRUNC(t);
 247                 d = (t+x)*(x-t);
 248                 t *= t;
 249                 xx.a = (t + x), TRUNC(xx.a);
 250                 xx.b = x - xx.a; xx.b += t; xx.b += d;
 251                 t = (one-x0); t += x;
 252                 d = (one-x0); d -= t; d += x;
 253                 x = xx.a + xx.b;
 254         } else {
 255                 xx.a =  x, TRUNC(xx.a);
 256                 xx.b = x - xx.a;
 257                 t = x - x0;
 258                 d = (-x0 -t); d += x;
 259         }
 260         r = ratfun_gam(t, d);
 261         d = r.a/x, TRUNC(d);
 262         r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
 263         return (d + r.a/x);
 264 }
 265 /*
 266  * returns (z+c)^2 * P(z)/Q(z) + a0
 267  */
 268 static struct Double
 269 ratfun_gam(z, c)
 270         double z, c;
 271 {
 272         double p, q;
 273         struct Double r, t;
 274
 275         q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
 276         p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
 277
 278         /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
 279         p = p/q;
 280         t.a = z, TRUNC(t.a);            /* t ~= z + c */
 281         t.b = (z - t.a) + c;
 282         t.b *= (t.a + z);
 283         q = (t.a *= t.a);               /* t = (z+c)^2 */
 284         TRUNC(t.a);
 285         t.b += (q - t.a);
 286         r.a = p, TRUNC(r.a);            /* r = P/Q */
 287         r.b = p - r.a;
 288         t.b = t.b*p + t.a*r.b + a0_lo;
 289         t.a *= r.a;                     /* t = (z+c)^2*(P/Q) */
 290         r.a = t.a + a0_hi, TRUNC(r.a);
 291         r.b = ((a0_hi-r.a) + t.a) + t.b;
 292         return (r);                     /* r = a0 + t */
 293 }
 294
 295 static double
 296 neg_gam(x)
 297         double x;
 298 {
 299         int sgn = 1;
 300         struct Double lg, lsine;
 301         double y, z;
 302
 303         y = floor(x + .5);
 304         if (y == x)             /* Negative integer. */
 305                 if(!_IEEE)
 306                         return (infnan(ERANGE));
 307                 else
 308                         return (one/zero);
 309         z = fabs(x - y);
 310         y = .5*ceil(x);
 311         if (y == ceil(y))
 312                 sgn = -1;
 313         if (z < .25)
 314                 z = sin(M_PI*z);
 315         else
 316                 z = cos(M_PI*(0.5-z));
 317         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
 318         if (x < -170) {
 319                 if (x < -190)
 320                         return ((double)sgn*tiny*tiny);
 321                 y = one - x;            /* exact: 128 < |x| < 255 */
 322                 lg = large_gam(y);
 323                 lsine = __log__D(M_PI/z);       /* = TRUNC(log(u)) + small */
 324                 lg.a -= lsine.a;                /* exact (opposite signs) */
 325                 lg.b -= lsine.b;
 326                 y = -(lg.a + lg.b);
 327                 z = (y + lg.a) + lg.b;
 328                 y = __exp__D(y, z);
 329                 if (sgn < 0) y = -y;
 330                 return (y);
 331         }
 332         y = one-x;
 333         if (one-y == x)
 334                 y = gamma(y);
 335         else            /* 1-x is inexact */
 336                 y = -x*gamma(-x);
 337         if (sgn < 0) y = -y;
 338         return (M_PI / (y*z));
 339 }
 340
 341 #endif