lib/libm/src/b_tgamma.c

   1 /*      $NetBSD: b_tgamma.c,v 1.1 2012/05/05 17:54:14 christos Exp $    */
   2
   3 /*-
   4  * Copyright (c) 1992, 1993
   5  *      The Regents of the University of California.  All rights reserved.
   6  *
   7  * Redistribution and use in source and binary forms, with or without
   8  * modification, are permitted provided that the following conditions
   9  * are met:
  10  * 1. Redistributions of source code must retain the above copyright
  11  *    notice, this list of conditions and the following disclaimer.
  12  * 2. Redistributions in binary form must reproduce the above copyright
  13  *    notice, this list of conditions and the following disclaimer in the
  14  *    documentation and/or other materials provided with the distribution.
  15  * 3. All advertising materials mentioning features or use of this software
  16  *    must display the following acknowledgement:
  17  *      This product includes software developed by the University of
  18  *      California, Berkeley and its contributors.
  19  * 4. Neither the name of the University nor the names of its contributors
  20  *    may be used to endorse or promote products derived from this software
  21  *    without specific prior written permission.
  22  *
  23  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  24  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  25  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  26  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  27  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  28  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  29  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  30  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  31  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  32  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  33  * SUCH DAMAGE.
  34  */
  35
  36 /* @(#)gamma.c  8.1 (Berkeley) 6/4/93 */
  37 #include <sys/cdefs.h>
  38 #if 0
  39 __FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_tgamma.c 176449 2008-02-22 02:26:51Z das $");
  40 #else
  41 __RCSID("$NetBSD: b_tgamma.c,v 1.1 2012/05/05 17:54:14 christos Exp $");
  42 #endif
  43
  44 /*
  45  * This code by P. McIlroy, Oct 1992;
  46  *
  47  * The financial support of UUNET Communications Services is greatfully
  48  * acknowledged.
  49  */
  50
  51 #include "math.h"
  52 #include "math_private.h"
  53
  54 /* METHOD:
  55  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
  56  *      At negative integers, return NaN and raise invalid.
  57  *
  58  * x < 6.5:
  59  *      Use argument reduction G(x+1) = xG(x) to reach the
  60  *      range [1.066124,2.066124].  Use a rational
  61  *      approximation centered at the minimum (x0+1) to
  62  *      ensure monotonicity.
  63  *
  64  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
  65  *      adjusted for equal-ripples:
  66  *
  67  *      log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
  68  *
  69  *      Keep extra precision in multiplying (x-.5)(log(x)-1), to
  70  *      avoid premature round-off.
  71  *
  72  * Special values:
  73  *      -Inf:                   return NaN and raise invalid;
  74  *      negative integer:       return NaN and raise invalid;
  75  *      other x ~< 177.79:      return +-0 and raise underflow;
  76  *      +-0:                    return +-Inf and raise divide-by-zero;
  77  *      finite x ~> 171.63:     return +Inf and raise overflow;
  78  *      +Inf:                   return +Inf;
  79  *      NaN:                    return NaN.
  80  *
  81  * Accuracy: tgamma(x) is accurate to within
  82  *      x > 0:  error provably < 0.9ulp.
  83  *      Maximum observed in 1,000,000 trials was .87ulp.
  84  *      x < 0:
  85  *      Maximum observed error < 4ulp in 1,000,000 trials.
  86  */
  87
  88 static double neg_gam(double);
  89 static double small_gam(double);
  90 static double smaller_gam(double);
  91 static struct Double large_gam(double);
  92 static struct Double ratfun_gam(double, double);
  93
  94 /*
  95  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
  96  * [1.066.., 2.066..] accurate to 4.25e-19.
  97  */
  98 #define LEFT -.3955078125       /* left boundary for rat. approx */
  99 #define x0 .461632144968362356785       /* xmin - 1 */
 100
 101 #define a0_hi 0.88560319441088874992
 102 #define a0_lo -.00000000000000004996427036469019695
 103 #define P0       6.21389571821820863029017800727e-01
 104 #define P1       2.65757198651533466104979197553e-01
 105 #define P2       5.53859446429917461063308081748e-03
 106 #define P3       1.38456698304096573887145282811e-03
 107 #define P4       2.40659950032711365819348969808e-03
 108 #define Q0       1.45019531250000000000000000000e+00
 109 #define Q1       1.06258521948016171343454061571e+00
 110 #define Q2      -2.07474561943859936441469926649e-01
 111 #define Q3      -1.46734131782005422506287573015e-01
 112 #define Q4       3.07878176156175520361557573779e-02
 113 #define Q5       5.12449347980666221336054633184e-03
 114 #define Q6      -1.76012741431666995019222898833e-03
 115 #define Q7       9.35021023573788935372153030556e-05
 116 #define Q8       6.13275507472443958924745652239e-06
 117 /*
 118  * Constants for large x approximation (x in [6, Inf])
 119  * (Accurate to 2.8*10^-19 absolute)
 120  */
 121 #define lns2pi_hi 0.418945312500000
 122 #define lns2pi_lo -.000006779295327258219670263595
 123 #define Pa0      8.33333333333333148296162562474e-02
 124 #define Pa1     -2.77777777774548123579378966497e-03
 125 #define Pa2      7.93650778754435631476282786423e-04
 126 #define Pa3     -5.95235082566672847950717262222e-04
 127 #define Pa4      8.41428560346653702135821806252e-04
 128 #define Pa5     -1.89773526463879200348872089421e-03
 129 #define Pa6      5.69394463439411649408050664078e-03
 130 #define Pa7     -1.44705562421428915453880392761e-02
 131
 132 static const double zero = 0., one = 1.0, tiny = 1e-300;
 133
 134 double
 135 tgamma(double x)
 136 {
 137         struct Double u;
 138
 139         if (x >= 6) {
 140                 if(x > 171.63)
 141                         return (x / zero);
 142                 u = large_gam(x);
 143                 return(__exp__D(u.a, u.b));
 144         } else if (x >= 1.0 + LEFT + x0)
 145                 return (small_gam(x));
 146         else if (x > 1.e-17)
 147                 return (smaller_gam(x));
 148         else if (x > -1.e-17) {
 149                 if (x != 0.0)
 150                         u.a = one - tiny;       /* raise inexact */
 151                 return (one/x);
 152         } else if (!finite(x))
 153                 return (x - x);         /* x is NaN or -Inf */
 154         else
 155                 return (neg_gam(x));
 156 }
 157 /*
 158  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
 159  */
 160 static struct Double
 161 large_gam(double x)
 162 {
 163         double z, p;
 164         struct Double t, u, v;
 165
 166         z = one/(x*x);
 167         p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
 168         p = p/x;
 169
 170         u = __log__D(x);
 171         u.a -= one;
 172         v.a = (x -= .5);
 173         TRUNC(v.a);
 174         v.b = x - v.a;
 175         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 176         t.b = v.b*u.a + x*u.b;
 177         /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
 178         t.b += lns2pi_lo; t.b += p;
 179         u.a = lns2pi_hi + t.b; u.a += t.a;
 180         u.b = t.a - u.a;
 181         u.b += lns2pi_hi; u.b += t.b;
 182         return (u);
 183 }
 184 /*
 185  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
 186  * It also has correct monotonicity.
 187  */
 188 static double
 189 small_gam(double x)
 190 {
 191         double y, ym1, t;
 192         struct Double yy, r;
 193         y = x - one;
 194         ym1 = y - one;
 195         if (y <= 1.0 + (LEFT + x0)) {
 196                 yy = ratfun_gam(y - x0, 0);
 197                 return (yy.a + yy.b);
 198         }
 199         r.a = y;
 200         TRUNC(r.a);
 201         yy.a = r.a - one;
 202         y = ym1;
 203         yy.b = r.b = y - yy.a;
 204         /* Argument reduction: G(x+1) = x*G(x) */
 205         for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
 206                 t = r.a*yy.a;
 207                 r.b = r.a*yy.b + y*r.b;
 208                 r.a = t;
 209                 TRUNC(r.a);
 210                 r.b += (t - r.a);
 211         }
 212         /* Return r*tgamma(y). */
 213         yy = ratfun_gam(y - x0, 0);
 214         y = r.b*(yy.a + yy.b) + r.a*yy.b;
 215         y += yy.a*r.a;
 216         return (y);
 217 }
 218 /*
 219  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
 220  */
 221 static double
 222 smaller_gam(double x)
 223 {
 224         double t, d;
 225         struct Double r, xx;
 226         if (x < x0 + LEFT) {
 227                 t = x, TRUNC(t);
 228                 d = (t+x)*(x-t);
 229                 t *= t;
 230                 xx.a = (t + x), TRUNC(xx.a);
 231                 xx.b = x - xx.a; xx.b += t; xx.b += d;
 232                 t = (one-x0); t += x;
 233                 d = (one-x0); d -= t; d += x;
 234                 x = xx.a + xx.b;
 235         } else {
 236                 xx.a =  x, TRUNC(xx.a);
 237                 xx.b = x - xx.a;
 238                 t = x - x0;
 239                 d = (-x0 -t); d += x;
 240         }
 241         r = ratfun_gam(t, d);
 242         d = r.a/x, TRUNC(d);
 243         r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
 244         return (d + r.a/x);
 245 }
 246 /*
 247  * returns (z+c)^2 * P(z)/Q(z) + a0
 248  */
 249 static struct Double
 250 ratfun_gam(double z, double c)
 251 {
 252         double p, q;
 253         struct Double r, t;
 254
 255         q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
 256         p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
 257
 258         /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
 259         p = p/q;
 260         t.a = z, TRUNC(t.a);            /* t ~= z + c */
 261         t.b = (z - t.a) + c;
 262         t.b *= (t.a + z);
 263         q = (t.a *= t.a);               /* t = (z+c)^2 */
 264         TRUNC(t.a);
 265         t.b += (q - t.a);
 266         r.a = p, TRUNC(r.a);            /* r = P/Q */
 267         r.b = p - r.a;
 268         t.b = t.b*p + t.a*r.b + a0_lo;
 269         t.a *= r.a;                     /* t = (z+c)^2*(P/Q) */
 270         r.a = t.a + a0_hi, TRUNC(r.a);
 271         r.b = ((a0_hi-r.a) + t.a) + t.b;
 272         return (r);                     /* r = a0 + t */
 273 }
 274
 275 static double
 276 neg_gam(double x)
 277 {
 278         int sgn = 1;
 279         struct Double lg, lsine;
 280         double y, z;
 281
 282         y = ceil(x);
 283         if (y == x)             /* Negative integer. */
 284                 return ((x - x) / zero);
 285         z = y - x;
 286         if (z > 0.5)
 287                 z = one - z;
 288         y = 0.5 * y;
 289         if (y == ceil(y))
 290                 sgn = -1;
 291         if (z < .25)
 292                 z = sin(M_PI*z);
 293         else
 294                 z = cos(M_PI*(0.5-z));
 295         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
 296         if (x < -170) {
 297                 if (x < -190)
 298                         return ((double)sgn*tiny*tiny);
 299                 y = one - x;            /* exact: 128 < |x| < 255 */
 300                 lg = large_gam(y);
 301                 lsine = __log__D(M_PI/z);       /* = TRUNC(log(u)) + small */
 302                 lg.a -= lsine.a;                /* exact (opposite signs) */
 303                 lg.b -= lsine.b;
 304                 y = -(lg.a + lg.b);
 305                 z = (y + lg.a) + lg.b;
 306                 y = __exp__D(y, z);
 307                 if (sgn < 0) y = -y;
 308                 return (y);
 309         }
 310         y = one-x;
 311         if (one-y == x)
 312                 y = tgamma(y);
 313         else            /* 1-x is inexact */
 314                 y = -x*tgamma(-x);
 315         if (sgn < 0) y = -y;
 316         return (M_PI / (y*z));
 317 }