1 /* $NetBSD: n_gamma.c,v 1.8 2012/06/08 11:13:33 abs 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 33 static 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 78 static double neg_gam (double); 79 static double small_gam (double); 80 static double smaller_gam (double); 81 static struct Double large_gam (double); 82 static 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-02 121 122 static 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 131 static int endian; 132 #define _IEEE 1 133 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 134 #define infnan(x) 0.0 135 #endif 136 137 double 138 gamma(double x) 139 { 140 double b; 141 struct Double u; 142 #if _IEEE 143 int endian = (*(int *) &one) ? 1 : 0; 144 #endif 145 146 if (x >= 6) { 147 if(x > 171.63) 148 return(one/zero); 149 u = large_gam(x); 150 return(__exp__D(u.a, u.b)); 151 } else if (x >= 1.0 + LEFT + x0) { 152 return (small_gam(x)); 153 } else if (x > 1.e-17) { 154 return (smaller_gam(x)); 155 } else if (x > -1.e-17) { 156 if (x == 0.0) { 157 if (!_IEEE) return (infnan(ERANGE)); 158 else return (one/x); 159 } 160 b =one+1e-20; /* Raise inexact flag. ??? -ragge */ 161 return (one/x); 162 } else if (!finite(x)) { 163 if (_IEEE) /* x = NaN, -Inf */ 164 return (x*x); 165 else 166 return (infnan(EDOM)); 167 } else 168 return (neg_gam(x)); 169 } 170 /* 171 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 172 */ 173 static struct Double 174 large_gam(double x) 175 { 176 double z, p; 177 struct Double t, u, v; 178 179 z = one/(x*x); 180 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 181 p = p/x; 182 183 u = __log__D(x); 184 u.a -= one; 185 v.a = (x -= .5); 186 TRUNC(v.a); 187 v.b = x - v.a; 188 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 189 t.b = v.b*u.a + x*u.b; 190 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 191 t.b += lns2pi_lo; t.b += p; 192 u.a = lns2pi_hi + t.b; u.a += t.a; 193 u.b = t.a - u.a; 194 u.b += lns2pi_hi; u.b += t.b; 195 return (u); 196 } 197 /* 198 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 199 * It also has correct monotonicity. 200 */ 201 static double 202 small_gam(double x) 203 { 204 double y, ym1, t; 205 struct Double yy, r; 206 y = x - one; 207 ym1 = y - one; 208 if (y <= 1.0 + (LEFT + x0)) { 209 yy = ratfun_gam(y - x0, 0); 210 return (yy.a + yy.b); 211 } 212 r.a = y; 213 TRUNC(r.a); 214 yy.a = r.a - one; 215 y = ym1; 216 yy.b = r.b = y - yy.a; 217 /* Argument reduction: G(x+1) = x*G(x) */ 218 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 219 t = r.a*yy.a; 220 r.b = r.a*yy.b + y*r.b; 221 r.a = t; 222 TRUNC(r.a); 223 r.b += (t - r.a); 224 } 225 /* Return r*gamma(y). */ 226 yy = ratfun_gam(y - x0, 0); 227 y = r.b*(yy.a + yy.b) + r.a*yy.b; 228 y += yy.a*r.a; 229 return (y); 230 } 231 /* 232 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 233 */ 234 static double 235 smaller_gam(double x) 236 { 237 double t, d; 238 struct Double r, xx; 239 if (x < x0 + LEFT) { 240 t = x, TRUNC(t); 241 d = (t+x)*(x-t); 242 t *= t; 243 xx.a = (t + x), TRUNC(xx.a); 244 xx.b = x - xx.a; xx.b += t; xx.b += d; 245 t = (one-x0); t += x; 246 d = (one-x0); d -= t; d += x; 247 x = xx.a + xx.b; 248 } else { 249 xx.a = x, TRUNC(xx.a); 250 xx.b = x - xx.a; 251 t = x - x0; 252 d = (-x0 -t); d += x; 253 } 254 r = ratfun_gam(t, d); 255 d = r.a/x, TRUNC(d); 256 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 257 return (d + r.a/x); 258 } 259 /* 260 * returns (z+c)^2 * P(z)/Q(z) + a0 261 */ 262 static struct Double 263 ratfun_gam(double z, double c) 264 { 265 double p, q; 266 struct Double r, t; 267 268 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 269 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 270 271 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 272 p = p/q; 273 t.a = z, TRUNC(t.a); /* t ~= z + c */ 274 t.b = (z - t.a) + c; 275 t.b *= (t.a + z); 276 q = (t.a *= t.a); /* t = (z+c)^2 */ 277 TRUNC(t.a); 278 t.b += (q - t.a); 279 r.a = p, TRUNC(r.a); /* r = P/Q */ 280 r.b = p - r.a; 281 t.b = t.b*p + t.a*r.b + a0_lo; 282 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 283 r.a = t.a + a0_hi, TRUNC(r.a); 284 r.b = ((a0_hi-r.a) + t.a) + t.b; 285 return (r); /* r = a0 + t */ 286 } 287 288 static double 289 neg_gam(double x) 290 { 291 int sgn = 1; 292 struct Double lg, lsine; 293 double y, z; 294 295 y = floor(x + .5); 296 if (y == x) { /* Negative integer. */ 297 if(!_IEEE) 298 return (infnan(ERANGE)); 299 else 300 return (one/zero); 301 } 302 z = fabs(x - y); 303 y = .5*ceil(x); 304 if (y == ceil(y)) 305 sgn = -1; 306 if (z < .25) 307 z = sin(M_PI*z); 308 else 309 z = cos(M_PI*(0.5-z)); 310 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 311 if (x < -170) { 312 if (x < -190) 313 return ((double)sgn*tiny*tiny); 314 y = one - x; /* exact: 128 < |x| < 255 */ 315 lg = large_gam(y); 316 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 317 lg.a -= lsine.a; /* exact (opposite signs) */ 318 lg.b -= lsine.b; 319 y = -(lg.a + lg.b); 320 z = (y + lg.a) + lg.b; 321 y = __exp__D(y, z); 322 if (sgn < 0) y = -y; 323 return (y); 324 } 325 y = one-x; 326 if (one-y == x) 327 y = gamma(y); 328 else /* 1-x is inexact */ 329 y = -x*gamma(-x); 330 if (sgn < 0) y = -y; 331 return (M_PI / (y*z)); 332 } 333