lib/libm/src/b_tgamma.c

   1 /*      $OpenBSD: b_tgamma.c,v 1.7 2013/03/28 18:09:38 martynas 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 /*
  32  * This code by P. McIlroy, Oct 1992;
  33  *
  34  * The financial support of UUNET Communications Services is greatfully
  35  * acknowledged.
  36  */
  37
  38 #include <float.h>
  39 #include <math.h>
  40
  41 #include "math_private.h"
  42
  43 /* METHOD:
  44  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
  45  *      At negative integers, return NaN and raise invalid.
  46  *
  47  * x < 6.5:
  48  *      Use argument reduction G(x+1) = xG(x) to reach the
  49  *      range [1.066124,2.066124].  Use a rational
  50  *      approximation centered at the minimum (x0+1) to
  51  *      ensure monotonicity.
  52  *
  53  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
  54  *      adjusted for equal-ripples:
  55  *
  56  *      log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
  57  *
  58  *      Keep extra precision in multiplying (x-.5)(log(x)-1), to
  59  *      avoid premature round-off.
  60  *
  61  * Special values:
  62  *      -Inf:                   return NaN and raise invalid;
  63  *      negative integer:       return NaN and raise invalid;
  64  *      other x ~< -177.79:     return +-0 and raise underflow;
  65  *      +-0:                    return +-Inf and raise divide-by-zero;
  66  *      finite x ~> 171.63:     return +Inf and raise overflow;
  67  *      +Inf:                   return +Inf;
  68  *      NaN:                    return NaN.
  69  *
  70  * Accuracy: tgamma(x) is accurate to within
  71  *      x > 0:  error provably < 0.9ulp.
  72  *      Maximum observed in 1,000,000 trials was .87ulp.
  73  *      x < 0:
  74  *      Maximum observed error < 4ulp in 1,000,000 trials.
  75  */
  76
  77 static double neg_gam(double);
  78 static double small_gam(double);
  79 static double smaller_gam(double);
  80 static struct Double large_gam(double);
  81 static struct Double ratfun_gam(double, double);
  82
  83 /*
  84  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
  85  * [1.066.., 2.066..] accurate to 4.25e-19.
  86  */
  87 #define LEFT -.3955078125       /* left boundary for rat. approx */
  88 #define x0 .461632144968362356785       /* xmin - 1 */
  89
  90 #define a0_hi 0.88560319441088874992
  91 #define a0_lo -.00000000000000004996427036469019695
  92 #define P0       6.21389571821820863029017800727e-01
  93 #define P1       2.65757198651533466104979197553e-01
  94 #define P2       5.53859446429917461063308081748e-03
  95 #define P3       1.38456698304096573887145282811e-03
  96 #define P4       2.40659950032711365819348969808e-03
  97 #define Q0       1.45019531250000000000000000000e+00
  98 #define Q1       1.06258521948016171343454061571e+00
  99 #define Q2      -2.07474561943859936441469926649e-01
 100 #define Q3      -1.46734131782005422506287573015e-01
 101 #define Q4       3.07878176156175520361557573779e-02
 102 #define Q5       5.12449347980666221336054633184e-03
 103 #define Q6      -1.76012741431666995019222898833e-03
 104 #define Q7       9.35021023573788935372153030556e-05
 105 #define Q8       6.13275507472443958924745652239e-06
 106 /*
 107  * Constants for large x approximation (x in [6, Inf])
 108  * (Accurate to 2.8*10^-19 absolute)
 109  */
 110 #define lns2pi_hi 0.418945312500000
 111 #define lns2pi_lo -.000006779295327258219670263595
 112 #define Pa0      8.33333333333333148296162562474e-02
 113 #define Pa1     -2.77777777774548123579378966497e-03
 114 #define Pa2      7.93650778754435631476282786423e-04
 115 #define Pa3     -5.95235082566672847950717262222e-04
 116 #define Pa4      8.41428560346653702135821806252e-04
 117 #define Pa5     -1.89773526463879200348872089421e-03
 118 #define Pa6      5.69394463439411649408050664078e-03
 119 #define Pa7     -1.44705562421428915453880392761e-02
 120
 121 static const double zero = 0., one = 1.0, tiny = 1e-300;
 122
 123 double
 124 tgamma(double x)
 125 {
 126         struct Double u;
 127
 128         if (x >= 6) {
 129                 if(x > 171.63)
 130                         return(x/zero);
 131                 u = large_gam(x);
 132                 return(__exp__D(u.a, u.b));
 133         } else if (x >= 1.0 + LEFT + x0)
 134                 return (small_gam(x));
 135         else if (x > 1.e-17)
 136                 return (smaller_gam(x));
 137         else if (x > -1.e-17) {
 138                 if (x != 0.0)
 139                         u.a = one - tiny;       /* raise inexact */
 140                 return (one/x);
 141         } else if (!finite(x)) {
 142                 return (x - x);                 /* x = NaN, -Inf */
 143          } else
 144                 return (neg_gam(x));
 145 }
 146
 147 /*
 148  * We simply call tgamma() rather than bloating the math library
 149  * with a float-optimized version of it.  The reason is that tgammaf()
 150  * is essentially useless, since the function is superexponential
 151  * and floats have very limited range.  -- das@freebsd.org
 152  */
 153
 154 float
 155 tgammaf(float x)
 156 {
 157         return tgamma(x);
 158 }
 159
 160 /*
 161  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
 162  */
 163
 164 static struct Double
 165 large_gam(double x)
 166 {
 167         double z, p;
 168         struct Double t, u, v;
 169
 170         z = one/(x*x);
 171         p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
 172         p = p/x;
 173
 174         u = __log__D(x);
 175         u.a -= one;
 176         v.a = (x -= .5);
 177         TRUNC(v.a);
 178         v.b = x - v.a;
 179         t.a = v.a*u.a;                  /* t = (x-.5)*(log(x)-1) */
 180         t.b = v.b*u.a + x*u.b;
 181         /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
 182         t.b += lns2pi_lo; t.b += p;
 183         u.a = lns2pi_hi + t.b; u.a += t.a;
 184         u.b = t.a - u.a;
 185         u.b += lns2pi_hi; u.b += t.b;
 186         return (u);
 187 }
 188
 189 /*
 190  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
 191  * It also has correct monotonicity.
 192  */
 193
 194 static double
 195 small_gam(double x)
 196 {
 197         double y, ym1, t;
 198         struct Double yy, r;
 199         y = x - one;
 200         ym1 = y - one;
 201         if (y <= 1.0 + (LEFT + x0)) {
 202                 yy = ratfun_gam(y - x0, 0);
 203                 return (yy.a + yy.b);
 204         }
 205         r.a = y;
 206         TRUNC(r.a);
 207         yy.a = r.a - one;
 208         y = ym1;
 209         yy.b = r.b = y - yy.a;
 210         /* Argument reduction: G(x+1) = x*G(x) */
 211         for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
 212                 t = r.a*yy.a;
 213                 r.b = r.a*yy.b + y*r.b;
 214                 r.a = t;
 215                 TRUNC(r.a);
 216                 r.b += (t - r.a);
 217         }
 218         /* Return r*tgamma(y). */
 219         yy = ratfun_gam(y - x0, 0);
 220         y = r.b*(yy.a + yy.b) + r.a*yy.b;
 221         y += yy.a*r.a;
 222         return (y);
 223 }
 224
 225 /*
 226  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
 227  */
 228
 229 static double
 230 smaller_gam(double x)
 231 {
 232         double t, d;
 233         struct Double r, xx;
 234         if (x < x0 + LEFT) {
 235                 t = x;
 236                 TRUNC(t);
 237                 d = (t+x)*(x-t);
 238                 t *= t;
 239                 xx.a = (t + x);
 240                 TRUNC(xx.a);
 241                 xx.b = x - xx.a; xx.b += t; xx.b += d;
 242                 t = (one-x0); t += x;
 243                 d = (one-x0); d -= t; d += x;
 244                 x = xx.a + xx.b;
 245         } else {
 246                 xx.a =  x;
 247                 TRUNC(xx.a);
 248                 xx.b = x - xx.a;
 249                 t = x - x0;
 250                 d = (-x0 -t); d += x;
 251         }
 252         r = ratfun_gam(t, d);
 253         d = r.a/x;
 254         TRUNC(d);
 255         r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
 256         return (d + r.a/x);
 257 }
 258
 259 /*
 260  * returns (z+c)^2 * P(z)/Q(z) + a0
 261  */
 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;
 275         TRUNC(t.a);                     /* t ~= z + c */
 276         t.b = (z - t.a) + c;
 277         t.b *= (t.a + z);
 278         q = (t.a *= t.a);               /* t = (z+c)^2 */
 279         TRUNC(t.a);
 280         t.b += (q - t.a);
 281         r.a = p;
 282         TRUNC(r.a);                     /* r = P/Q */
 283         r.b = p - r.a;
 284         t.b = t.b*p + t.a*r.b + a0_lo;
 285         t.a *= r.a;                     /* t = (z+c)^2*(P/Q) */
 286         r.a = t.a + a0_hi;
 287         TRUNC(r.a);
 288         r.b = ((a0_hi-r.a) + t.a) + t.b;
 289         return (r);                     /* r = a0 + t */
 290 }
 291
 292 static double
 293 neg_gam(double x)
 294 {
 295         int sgn = 1;
 296         struct Double lg, lsine;
 297         double y, z;
 298
 299         y = ceil(x);
 300         if (y == x)             /* Negative integer. */
 301                 return ((x - x) / zero);
 302         z = y - x;
 303         if (z > 0.5)
 304                 z = one - z;
 305         y = 0.5 * y;
 306         if (y == ceil(y))
 307                 sgn = -1;
 308         if (z < .25)
 309                 z = sin(M_PI*z);
 310         else
 311                 z = cos(M_PI*(0.5-z));
 312         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
 313         if (x < -170) {
 314                 if (x < -190)
 315                         return ((double)sgn*tiny*tiny);
 316                 y = one - x;            /* exact: 128 < |x| < 255 */
 317                 lg = large_gam(y);
 318                 lsine = __log__D(M_PI/z);       /* = TRUNC(log(u)) + small */
 319                 lg.a -= lsine.a;                /* exact (opposite signs) */
 320                 lg.b -= lsine.b;
 321                 y = -(lg.a + lg.b);
 322                 z = (y + lg.a) + lg.b;
 323                 y = __exp__D(y, z);
 324                 if (sgn < 0) y = -y;
 325                 return (y);
 326         }
 327         y = one-x;
 328         if (one-y == x)
 329                 y = tgamma(y);
 330         else            /* 1-x is inexact */
 331                 y = -x*tgamma(-x);
 332         if (sgn < 0) y = -y;
 333         return (M_PI / (y*z));
 334 }
 335
 336 #if     LDBL_MANT_DIG == 53
 337 __strong_alias(tgammal, tgamma);
 338 #endif  /* LDBL_MANT_DIG == 53 */