1 /* $NetBSD: n_lgamma.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 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); 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 __pure double 141 lgamma(double x) 142 { 143 double r; 144 145 signgam = 1; 146 #if _IEEE 147 endian = ((*(int *) &one)) ? 1 : 0; 148 #endif 149 150 if (!finite(x)) { 151 if (_IEEE) 152 return (x+x); 153 else return (infnan(EDOM)); 154 } 155 156 if (x > 6 + RIGHT) { 157 r = large_lgam(x); 158 return (r); 159 } else if (x > 1e-16) 160 return (small_lgam(x)); 161 else if (x > -1e-16) { 162 if (x < 0) 163 signgam = -1, x = -x; 164 return (-log(x)); 165 } else 166 return (neg_lgam(x)); 167 } 168 169 static double 170 large_lgam(double x) 171 { 172 double z, p, x1; 173 struct Double t, u, v; 174 u = __log__D(x); 175 u.a -= 1.0; 176 if (x > 1e15) { 177 v.a = x - 0.5; 178 TRUNC(v.a); 179 v.b = (x - v.a) - 0.5; 180 t.a = u.a*v.a; 181 t.b = x*u.b + v.b*u.a; 182 if (_IEEE == 0 && !finite(t.a)) 183 return(infnan(ERANGE)); 184 return(t.a + t.b); 185 } 186 x1 = 1./x; 187 z = x1*x1; 188 p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7)))))); 189 /* error in approximation = 2.8e-19 */ 190 191 p = p*x1; /* error < 2.3e-18 absolute */ 192 /* 0 < p < 1/64 (at x = 5.5) */ 193 v.a = x = x - 0.5; 194 TRUNC(v.a); /* truncate v.a to 26 bits. */ 195 v.b = x - v.a; 196 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 197 t.b = v.b*u.a + x*u.b; 198 t.b += p; t.b += lns2pi; /* return t + lns2pi + p */ 199 return (t.a + t.b); 200 } 201 202 static double 203 small_lgam(double x) 204 { 205 int x_int; 206 double y, z, t, r = 0, p, q, hi, lo; 207 struct Double rr; 208 x_int = (x + .5); 209 y = x - x_int; 210 if (x_int <= 2 && y > RIGHT) { 211 t = y - x0; 212 y--; x_int++; 213 goto CONTINUE; 214 } else if (y < -LEFT) { 215 t = y +(1.0-x0); 216 CONTINUE: 217 z = t - x0_lo; 218 p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5)))); 219 q = s0+z*(s1+z*(s2+z*(s3+z*s4))); 220 r = t*(z*(p/q) - x0_lo); 221 t = .5*t*t; 222 z = 1.0; 223 switch (x_int) { 224 case 6: z = (y + 5); 225 case 5: z *= (y + 4); 226 case 4: z *= (y + 3); 227 case 3: z *= (y + 2); 228 rr = __log__D(z); 229 rr.b += a0_lo; rr.a += a0_hi; 230 return(((r+rr.b)+t+rr.a)); 231 case 2: return(((r+a0_lo)+t)+a0_hi); 232 case 0: r -= log1p(x); 233 default: rr = __log__D(x); 234 rr.a -= a0_hi; rr.b -= a0_lo; 235 return(((r - rr.b) + t) - rr.a); 236 } 237 } else { 238 p = p0+y*(p1+y*(p2+y*(p3+y*p4))); 239 q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6))))); 240 p = p*(y/q); 241 t = (double)(float) y; 242 z = y-t; 243 hi = (double)(float) (p+a1_hi); 244 lo = a1_hi - hi; lo += p; lo += a1_lo; 245 r = lo*y + z*hi; /* q + r = y*(a0+p/q) */ 246 q = hi*t; 247 z = 1.0; 248 switch (x_int) { 249 case 6: z = (y + 5); 250 case 5: z *= (y + 4); 251 case 4: z *= (y + 3); 252 case 3: z *= (y + 2); 253 rr = __log__D(z); 254 r += rr.b; r += q; 255 return(rr.a + r); 256 case 2: return (q+ r); 257 case 0: rr = __log__D(x); 258 r -= rr.b; r -= log1p(x); 259 r += q; r-= rr.a; 260 return(r); 261 default: rr = __log__D(x); 262 r -= rr.b; 263 q -= rr.a; 264 return (r+q); 265 } 266 } 267 } 268 269 static double 270 neg_lgam(double x) 271 { 272 int xi; 273 double y, z, zero = 0.0; 274 275 /* avoid destructive cancellation as much as possible */ 276 if (x > -170) { 277 xi = x; 278 if (xi == x) { 279 if (_IEEE) 280 return(one/zero); 281 else 282 return(infnan(ERANGE)); 283 } 284 y = gamma(x); 285 if (y < 0) 286 y = -y, signgam = -1; 287 return (log(y)); 288 } 289 z = floor(x + .5); 290 if (z == x) { /* convention: G(-(integer)) -> +Inf */ 291 if (_IEEE) 292 return (one/zero); 293 else 294 return (infnan(ERANGE)); 295 } 296 y = .5*ceil(x); 297 if (y == ceil(y)) 298 signgam = -1; 299 x = -x; 300 z = fabs(x + z); /* 0 < z <= .5 */ 301 if (z < .25) 302 z = sin(M_PI*z); 303 else 304 z = cos(M_PI*(0.5-z)); 305 z = log(M_PI/(z*x)); 306 y = large_lgam(x); 307 return (z - y); 308 } 309