compiler/stdc/math/bsdsrc/b_tgamma.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. Neither the name of the University nor the names of its contributors
  14  *    may be used to endorse or promote products derived from this software
  15  *    without specific prior written permission.
  16  *
  17  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  18  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  19  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  20  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  21  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  22  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  23  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  24  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  25  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  26  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  27  * SUCH DAMAGE.
  28  */
  29
  30 /* @(#)gamma.c  8.1 (Berkeley) 6/4/93 */
  31 //__FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_tgamma.c,v 1.10 2008/02/22 02:26:51 das Exp $");
  32
  33 /*
  34  * This code by P. McIlroy, Oct 1992;
  35  *
  36  * The financial support of UUNET Communications Services is greatfully
  37  * acknowledged.
  38  */
  39
  40 #include <float.h>
  41 #include "math.h"
  42
  43 #include "mathimpl.h"
  44
  45 /* METHOD:
  46  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
  47  *      At negative integers, return NaN and raise invalid.
  48  *
  49  * x < 6.5:
  50  *      Use argument reduction G(x+1) = xG(x) to reach the
  51  *      range [1.066124,2.066124].  Use a rational
  52  *      approximation centered at the minimum (x0+1) to
  53  *      ensure monotonicity.
  54  *
  55  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
  56  *      adjusted for equal-ripples:
  57  *
  58  *      log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
  59  *
  60  *      Keep extra precision in multiplying (x-.5)(log(x)-1), to
  61  *      avoid premature round-off.
  62  *
  63  * Special values:
  64  *      -Inf:                   return NaN and raise invalid;
  65  *      negative integer:       return NaN and raise invalid;
  66  *      other x ~< 177.79:      return +-0 and raise underflow;
  67  *      +-0:                    return +-Inf and raise divide-by-zero;
  68  *      finite x ~> 171.63:     return +Inf and raise overflow;
  69  *      +Inf:                   return +Inf;
  70  *      NaN:                    return NaN.
  71  *
  72  * Accuracy: tgamma(x) is accurate to within
  73  *      x > 0:  error provably < 0.9ulp.
  74  *      Maximum observed in 1,000,000 trials was .87ulp.
  75  *      x < 0:
  76  *      Maximum observed error < 4ulp in 1,000,000 trials.
  77  */
  78
  79 static double neg_gam(double);
  80 static double small_gam(double);
  81 static double smaller_gam(double);
  82 static struct Double large_gam(double);
  83 static struct Double ratfun_gam(double, double);
  84
  85 /*
  86  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
  87  * [1.066.., 2.066..] accurate to 4.25e-19.
  88  */
  89 #define LEFT -.3955078125       /* left boundary for rat. approx */
  90 #define x0 .461632144968362356785       /* xmin - 1 */
  91
  92 #define a0_hi 0.88560319441088874992
  93 #define a0_lo -.00000000000000004996427036469019695
  94 #define P0       6.21389571821820863029017800727e-01
  95 #define P1       2.65757198651533466104979197553e-01
  96 #define P2       5.53859446429917461063308081748e-03
  97 #define P3       1.38456698304096573887145282811e-03
  98 #define P4       2.40659950032711365819348969808e-03
  99 #define Q0       1.45019531250000000000000000000e+00
 100 #define Q1       1.06258521948016171343454061571e+00
 101 #define Q2      -2.07474561943859936441469926649e-01
 102 #define Q3      -1.46734131782005422506287573015e-01
 103 #define Q4       3.07878176156175520361557573779e-02
 104 #define Q5       5.12449347980666221336054633184e-03
 105 #define Q6      -1.76012741431666995019222898833e-03
 106 #define Q7       9.35021023573788935372153030556e-05
 107 #define Q8       6.13275507472443958924745652239e-06
 108 /*
 109  * Constants for large x approximation (x in [6, Inf])
 110  * (Accurate to 2.8*10^-19 absolute)
 111  */
 112 #define lns2pi_hi 0.418945312500000
 113 #define lns2pi_lo -.000006779295327258219670263595
 114 #define Pa0      8.33333333333333148296162562474e-02
 115 #define Pa1     -2.77777777774548123579378966497e-03
 116 #define Pa2      7.93650778754435631476282786423e-04
 117 #define Pa3     -5.95235082566672847950717262222e-04
 118 #define Pa4      8.41428560346653702135821806252e-04
 119 #define Pa5     -1.89773526463879200348872089421e-03
 120 #define Pa6      5.69394463439411649408050664078e-03
 121 #define Pa7     -1.44705562421428915453880392761e-02
 122
 123 static const double zero = 0., one = 1.0, tiny = 1e-300;
 124
 125 double
 126 tgamma(x)
 127         double x;
 128 {
 129         struct Double u;
 130
 131         if (x >= 6) {
 132                 if(x > 171.63)
 133                         return (x / zero);
 134                 u = large_gam(x);
 135                 return(__exp__D(u.a, u.b));
 136         } else if (x >= 1.0 + LEFT + x0)
 137                 return (small_gam(x));
 138         else if (x > 1.e-17)
 139                 return (smaller_gam(x));
 140         else if (x > -1.e-17) {
 141                 if (x != 0.0)
 142                         u.a = one - tiny;       /* raise inexact */
 143                 return (one/x);
 144         } else if (!isfinite(x))
 145                 return (x - x);         /* x is NaN or -Inf */
 146         else
 147                 return (neg_gam(x));
 148 }
 149 /*
 150  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
 151  */
 152 static struct Double
 153 large_gam(x)
 154         double x;
 155 {
 156         double z, p;
 157         struct Double t, u, v;
 158
 159         z = one/(x*x);
 160         p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
 161         p = p/x;
 162
 163         u = __log__D(x);
 164         u.a -= one;
 165         v.a = (x -= .5);
 166         TRUNC(v.a);
 167         v.b = x - v.a;
 168         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 169         t.b = v.b*u.a + x*u.b;
 170         /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
 171         t.b += lns2pi_lo; t.b += p;
 172         u.a = lns2pi_hi + t.b; u.a += t.a;
 173         u.b = t.a - u.a;
 174         u.b += lns2pi_hi; u.b += t.b;
 175         return (u);
 176 }
 177 /*
 178  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
 179  * It also has correct monotonicity.
 180  */
 181 static double
 182 small_gam(x)
 183         double x;
 184 {
 185         double y, ym1, t;
 186         struct Double yy, r;
 187         y = x - one;
 188         ym1 = y - one;
 189         if (y <= 1.0 + (LEFT + x0)) {
 190                 yy = ratfun_gam(y - x0, 0);
 191                 return (yy.a + yy.b);
 192         }
 193         r.a = y;
 194         TRUNC(r.a);
 195         yy.a = r.a - one;
 196         y = ym1;
 197         yy.b = r.b = y - yy.a;
 198         /* Argument reduction: G(x+1) = x*G(x) */
 199         for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
 200                 t = r.a*yy.a;
 201                 r.b = r.a*yy.b + y*r.b;
 202                 r.a = t;
 203                 TRUNC(r.a);
 204                 r.b += (t - r.a);
 205         }
 206         /* Return r*tgamma(y). */
 207         yy = ratfun_gam(y - x0, 0);
 208         y = r.b*(yy.a + yy.b) + r.a*yy.b;
 209         y += yy.a*r.a;
 210         return (y);
 211 }
 212 /*
 213  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
 214  */
 215 static double
 216 smaller_gam(x)
 217         double x;
 218 {
 219         double t, d;
 220         struct Double r, xx;
 221         if (x < x0 + LEFT) {
 222                 t = x, TRUNC(t);
 223                 d = (t+x)*(x-t);
 224                 t *= t;
 225                 xx.a = (t + x), TRUNC(xx.a);
 226                 xx.b = x - xx.a; xx.b += t; xx.b += d;
 227                 t = (one-x0); t += x;
 228                 d = (one-x0); d -= t; d += x;
 229                 x = xx.a + xx.b;
 230         } else {
 231                 xx.a =  x, TRUNC(xx.a);
 232                 xx.b = x - xx.a;
 233                 t = x - x0;
 234                 d = (-x0 -t); d += x;
 235         }
 236         r = ratfun_gam(t, d);
 237         d = r.a/x, TRUNC(d);
 238         r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
 239         return (d + r.a/x);
 240 }
 241 /*
 242  * returns (z+c)^2 * P(z)/Q(z) + a0
 243  */
 244 static struct Double
 245 ratfun_gam(z, c)
 246         double z, c;
 247 {
 248         double p, q;
 249         struct Double r, t;
 250
 251         q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
 252         p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
 253
 254         /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
 255         p = p/q;
 256         t.a = z, TRUNC(t.a);            /* t ~= z + c */
 257         t.b = (z - t.a) + c;
 258         t.b *= (t.a + z);
 259         q = (t.a *= t.a);               /* t = (z+c)^2 */
 260         TRUNC(t.a);
 261         t.b += (q - t.a);
 262         r.a = p, TRUNC(r.a);            /* r = P/Q */
 263         r.b = p - r.a;
 264         t.b = t.b*p + t.a*r.b + a0_lo;
 265         t.a *= r.a;                     /* t = (z+c)^2*(P/Q) */
 266         r.a = t.a + a0_hi, TRUNC(r.a);
 267         r.b = ((a0_hi-r.a) + t.a) + t.b;
 268         return (r);                     /* r = a0 + t */
 269 }
 270
 271 static double
 272 neg_gam(x)
 273         double x;
 274 {
 275         int sgn = 1;
 276         struct Double lg, lsine;
 277         double y, z;
 278
 279         y = ceil(x);
 280         if (y == x)             /* Negative integer. */
 281                 return ((x - x) / zero);
 282         z = y - x;
 283         if (z > 0.5)
 284                 z = one - z;
 285         y = 0.5 * y;
 286         if (y == ceil(y))
 287                 sgn = -1;
 288         if (z < .25)
 289                 z = sin(M_PI*z);
 290         else
 291                 z = cos(M_PI*(0.5-z));
 292         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
 293         if (x < -170) {
 294                 if (x < -190)
 295                         return ((double)sgn*tiny*tiny);
 296                 y = one - x;            /* exact: 128 < |x| < 255 */
 297                 lg = large_gam(y);
 298                 lsine = __log__D(M_PI/z);       /* = TRUNC(log(u)) + small */
 299                 lg.a -= lsine.a;                /* exact (opposite signs) */
 300                 lg.b -= lsine.b;
 301                 y = -(lg.a + lg.b);
 302                 z = (y + lg.a) + lg.b;
 303                 y = __exp__D(y, z);
 304                 if (sgn < 0) y = -y;
 305                 return (y);
 306         }
 307         y = one-x;
 308         if (one-y == x)
 309                 y = tgamma(y);
 310         else            /* 1-x is inexact */
 311                 y = -x*tgamma(-x);
 312         if (sgn < 0) y = -y;
 313         return (M_PI / (y*z));
 314 }
 315
 316 #if (LDBL_MANT_DIG == DBL_MANT_DIG)
 317 AROS_MAKE_ASM_SYM(typeof(tgammal), tgammal, AROS_CSYM_FROM_ASM_NAME(tgammal), AROS_CSYM_FROM_ASM_NAME(tgamma));
 318 AROS_EXPORT_ASM_SYM(AROS_CSYM_FROM_ASM_NAME(tgammal));
 319 #endif