1 /* $NetBSD: n_lgamma.c,v 1.8 2024/06/09 14:09:27 riastradh 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[] = "@(#)lgamma.c 8.2 (Berkeley) 11/30/93"; 34 #endif 35 #endif /* not lint */ 36 37 /* 38 * Coded by Peter McIlroy, Nov 1992; 39 * 40 * The financial support of UUNET Communications Services is gratefully 41 * acknowledged. 42 */ 43 44 #include <math.h> 45 #include <errno.h> 46 47 #include "mathimpl.h" 48 49 /* Log gamma function. 50 * Error: x > 0 error < 1.3ulp. 51 * x > 4, error < 1ulp. 52 * x > 9, error < .6ulp. 53 * x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0) 54 * Method: 55 * x > 6: 56 * Use the asymptotic expansion (Stirling's Formula) 57 * 0 < x < 6: 58 * Use gamma(x+1) = x*gamma(x) for argument reduction. 59 * Use rational approximation in 60 * the range 1.2, 2.5 61 * Two approximations are used, one centered at the 62 * minimum to ensure monotonicity; one centered at 2 63 * to maintain small relative error. 64 * x < 0: 65 * Use the reflection formula, 66 * G(1-x)G(x) = PI/sin(PI*x) 67 * Special values: 68 * non-positive integer returns +Inf. 69 * NaN returns NaN 70 */ 71 #if defined(__vax__) || defined(tahoe) 72 #define _IEEE 0 73 /* double and float have same size exponent field */ 74 #define TRUNC(x) x = (double) (float) (x) 75 #else 76 static int endian; 77 #define _IEEE 1 78 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 79 #define infnan(x) 0.0 80 #endif 81 82 static double small_lgam(double); 83 static double large_lgam(double); 84 static double neg_lgam(double, int *); 85 static const double one = 1.0; 86 int signgam; 87 88 #define UNDERFL (1e-1020 * 1e-1020) 89 90 #define LEFT (1.0 - (x0 + .25)) 91 #define RIGHT (x0 - .218) 92 /* 93 * Constants for approximation in [1.244,1.712] 94 */ 95 #define x0 0.461632144968362356785 96 #define x0_lo -.000000000000000015522348162858676890521 97 #define a0_hi -0.12148629128932952880859 98 #define a0_lo .0000000007534799204229502 99 #define r0 -2.771227512955130520e-002 100 #define r1 -2.980729795228150847e-001 101 #define r2 -3.257411333183093394e-001 102 #define r3 -1.126814387531706041e-001 103 #define r4 -1.129130057170225562e-002 104 #define r5 -2.259650588213369095e-005 105 #define s0 1.714457160001714442e+000 106 #define s1 2.786469504618194648e+000 107 #define s2 1.564546365519179805e+000 108 #define s3 3.485846389981109850e-001 109 #define s4 2.467759345363656348e-002 110 /* 111 * Constants for approximation in [1.71, 2.5] 112 */ 113 #define a1_hi 4.227843350984671344505727574870e-01 114 #define a1_lo 4.670126436531227189e-18 115 #define p0 3.224670334241133695662995251041e-01 116 #define p1 3.569659696950364669021382724168e-01 117 #define p2 1.342918716072560025853732668111e-01 118 #define p3 1.950702176409779831089963408886e-02 119 #define p4 8.546740251667538090796227834289e-04 120 #define q0 1.000000000000000444089209850062e+00 121 #define q1 1.315850076960161985084596381057e+00 122 #define q2 6.274644311862156431658377186977e-01 123 #define q3 1.304706631926259297049597307705e-01 124 #define q4 1.102815279606722369265536798366e-02 125 #define q5 2.512690594856678929537585620579e-04 126 #define q6 -1.003597548112371003358107325598e-06 127 /* 128 * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf]. 129 */ 130 #define lns2pi .418938533204672741780329736405 131 #define pb0 8.33333333333333148296162562474e-02 132 #define pb1 -2.77777777774548123579378966497e-03 133 #define pb2 7.93650778754435631476282786423e-04 134 #define pb3 -5.95235082566672847950717262222e-04 135 #define pb4 8.41428560346653702135821806252e-04 136 #define pb5 -1.89773526463879200348872089421e-03 137 #define pb6 5.69394463439411649408050664078e-03 138 #define pb7 -1.44705562421428915453880392761e-02 139 140 __weak_alias(lgammal, lgamma) 141 __weak_alias(lgammal_r, lgamma_r) 142 143 double 144 lgamma(double x) 145 { 146 147 return lgamma_r(x, &signgam); 148 } 149 150 double 151 lgamma_r(double x, int *signgamp) 152 { 153 double r; 154 155 *signgamp = 1; 156 #if _IEEE 157 endian = ((*(int *) &one)) ? 1 : 0; 158 #endif 159 160 if (!finite(x)) { 161 if (_IEEE) 162 return (x+x); 163 else return (infnan(EDOM)); 164 } 165 166 if (x > 6 + RIGHT) { 167 r = large_lgam(x); 168 return (r); 169 } else if (x > 1e-16) 170 return (small_lgam(x)); 171 else if (x > -1e-16) { 172 if (x < 0) 173 *signgamp = -1, x = -x; 174 return (-log(x)); 175 } else 176 return (neg_lgam(x, signgamp)); 177 } 178 179 static double 180 large_lgam(double x) 181 { 182 double z, p, x1; 183 struct Double t, u, v; 184 u = __log__D(x); 185 u.a -= 1.0; 186 if (x > 1e15) { 187 v.a = x - 0.5; 188 TRUNC(v.a); 189 v.b = (x - v.a) - 0.5; 190 t.a = u.a*v.a; 191 t.b = x*u.b + v.b*u.a; 192 if (_IEEE == 0 && !finite(t.a)) 193 return(infnan(ERANGE)); 194 return(t.a + t.b); 195 } 196 x1 = 1./x; 197 z = x1*x1; 198 p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7)))))); 199 /* error in approximation = 2.8e-19 */ 200 201 p = p*x1; /* error < 2.3e-18 absolute */ 202 /* 0 < p < 1/64 (at x = 5.5) */ 203 v.a = x = x - 0.5; 204 TRUNC(v.a); /* truncate v.a to 26 bits. */ 205 v.b = x - v.a; 206 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 207 t.b = v.b*u.a + x*u.b; 208 t.b += p; t.b += lns2pi; /* return t + lns2pi + p */ 209 return (t.a + t.b); 210 } 211 212 static double 213 small_lgam(double x) 214 { 215 int x_int; 216 double y, z, t, r = 0, p, q, hi, lo; 217 struct Double rr; 218 x_int = (x + .5); 219 y = x - x_int; 220 if (x_int <= 2 && y > RIGHT) { 221 t = y - x0; 222 y--; x_int++; 223 goto CONTINUE; 224 } else if (y < -LEFT) { 225 t = y +(1.0-x0); 226 CONTINUE: 227 z = t - x0_lo; 228 p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5)))); 229 q = s0+z*(s1+z*(s2+z*(s3+z*s4))); 230 r = t*(z*(p/q) - x0_lo); 231 t = .5*t*t; 232 z = 1.0; 233 switch (x_int) { 234 case 6: z = (y + 5); /* FALLTHROUGH */ 235 case 5: z *= (y + 4); /* FALLTHROUGH */ 236 case 4: z *= (y + 3); /* FALLTHROUGH */ 237 case 3: z *= (y + 2); 238 rr = __log__D(z); 239 rr.b += a0_lo; rr.a += a0_hi; 240 return(((r+rr.b)+t+rr.a)); 241 case 2: return(((r+a0_lo)+t)+a0_hi); 242 case 0: r -= log1p(x); /* FALLTHROUGH */ 243 default: rr = __log__D(x); 244 rr.a -= a0_hi; rr.b -= a0_lo; 245 return(((r - rr.b) + t) - rr.a); 246 } 247 } else { 248 p = p0+y*(p1+y*(p2+y*(p3+y*p4))); 249 q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6))))); 250 p = p*(y/q); 251 t = (double)(float) y; 252 z = y-t; 253 hi = (double)(float) (p+a1_hi); 254 lo = a1_hi - hi; lo += p; lo += a1_lo; 255 r = lo*y + z*hi; /* q + r = y*(a0+p/q) */ 256 q = hi*t; 257 z = 1.0; 258 switch (x_int) { 259 case 6: z = (y + 5); /* FALLTHROUGH */ 260 case 5: z *= (y + 4); /* FALLTHROUGH */ 261 case 4: z *= (y + 3); /* FALLTHROUGH */ 262 case 3: z *= (y + 2); 263 rr = __log__D(z); 264 r += rr.b; r += q; 265 return(rr.a + r); 266 case 2: return (q+ r); 267 case 0: rr = __log__D(x); 268 r -= rr.b; r -= log1p(x); 269 r += q; r-= rr.a; 270 return(r); 271 default: rr = __log__D(x); 272 r -= rr.b; 273 q -= rr.a; 274 return (r+q); 275 } 276 } 277 } 278 279 static double 280 neg_lgam(double x, int *signgamp) 281 { 282 int xi; 283 double y, z, zero = 0.0; 284 285 /* avoid destructive cancellation as much as possible */ 286 if (x > -170) { 287 xi = x; 288 if (xi == x) { 289 if (_IEEE) 290 return(one/zero); 291 else 292 return(infnan(ERANGE)); 293 } 294 y = gamma(x); 295 if (y < 0) 296 y = -y, *signgamp = -1; 297 return (log(y)); 298 } 299 z = floor(x + .5); 300 if (z == x) { /* convention: G(-(integer)) -> +Inf */ 301 if (_IEEE) 302 return (one/zero); 303 else 304 return (infnan(ERANGE)); 305 } 306 y = .5*ceil(x); 307 if (y == ceil(y)) 308 *signgamp = -1; 309 x = -x; 310 z = fabs(x + z); /* 0 < z <= .5 */ 311 if (z < .25) 312 z = sin(M_PI*z); 313 else 314 z = cos(M_PI*(0.5-z)); 315 z = log(M_PI/(z*x)); 316 y = large_lgam(x); 317 return (z - y); 318 } 319