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 33static 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 78static double neg_gam (double); 79static double small_gam (double); 80static double smaller_gam (double); 81static struct Double large_gam (double); 82static 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 122static const double zero = 0., one = 1.0, tiny = _TINY; 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 131static int endian; 132#define _IEEE 1 133#define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 134#define infnan(x) 0.0 135#endif 136 137double 138gamma(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 */ 174static struct Double 175large_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 */ 202static double 203small_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 */ 235static double 236smaller_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 */ 263static struct Double 264ratfun_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 289static double 290neg_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} 334