1 /*- 2 * Copyright (c) 1992, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 3. All advertising materials mentioning features or use of this software 14 * must display the following acknowledgement: 15 * This product includes software developed by the University of 16 * California, Berkeley and its contributors. 17 * 4. Neither the name of the University nor the names of its contributors 18 * may be used to endorse or promote products derived from this software 19 * without specific prior written permission. 20 * 21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 31 * SUCH DAMAGE. 32 */ 33 34 #ifndef lint 35 static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93"; 36 #endif /* not lint */ 37 include <sys/cdefs.h> 38 __FBSDID("$FreeBSD$"); 39 40 /* 41 * This code by P. McIlroy, Oct 1992; 42 * 43 * The financial support of UUNET Communications Services is greatfully 44 * acknowledged. 45 */ 46 47 #include <math.h> 48 #include "mathimpl.h" 49 #include <errno.h> 50 51 /* METHOD: 52 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 53 * At negative integers, return +Inf, and set errno. 54 * 55 * x < 6.5: 56 * Use argument reduction G(x+1) = xG(x) to reach the 57 * range [1.066124,2.066124]. Use a rational 58 * approximation centered at the minimum (x0+1) to 59 * ensure monotonicity. 60 * 61 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 62 * adjusted for equal-ripples: 63 * 64 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 65 * 66 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 67 * avoid premature round-off. 68 * 69 * Special values: 70 * non-positive integer: Set overflow trap; return +Inf; 71 * x > 171.63: Set overflow trap; return +Inf; 72 * NaN: Set invalid trap; return NaN 73 * 74 * Accuracy: Gamma(x) is accurate to within 75 * x > 0: error provably < 0.9ulp. 76 * Maximum observed in 1,000,000 trials was .87ulp. 77 * x < 0: 78 * Maximum observed error < 4ulp in 1,000,000 trials. 79 */ 80 81 static double neg_gam(double); 82 static double small_gam(double); 83 static double smaller_gam(double); 84 static struct Double large_gam(double); 85 static struct Double ratfun_gam(double, double); 86 87 /* 88 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 89 * [1.066.., 2.066..] accurate to 4.25e-19. 90 */ 91 #define LEFT -.3955078125 /* left boundary for rat. approx */ 92 #define x0 .461632144968362356785 /* xmin - 1 */ 93 94 #define a0_hi 0.88560319441088874992 95 #define a0_lo -.00000000000000004996427036469019695 96 #define P0 6.21389571821820863029017800727e-01 97 #define P1 2.65757198651533466104979197553e-01 98 #define P2 5.53859446429917461063308081748e-03 99 #define P3 1.38456698304096573887145282811e-03 100 #define P4 2.40659950032711365819348969808e-03 101 #define Q0 1.45019531250000000000000000000e+00 102 #define Q1 1.06258521948016171343454061571e+00 103 #define Q2 -2.07474561943859936441469926649e-01 104 #define Q3 -1.46734131782005422506287573015e-01 105 #define Q4 3.07878176156175520361557573779e-02 106 #define Q5 5.12449347980666221336054633184e-03 107 #define Q6 -1.76012741431666995019222898833e-03 108 #define Q7 9.35021023573788935372153030556e-05 109 #define Q8 6.13275507472443958924745652239e-06 110 /* 111 * Constants for large x approximation (x in [6, Inf]) 112 * (Accurate to 2.8*10^-19 absolute) 113 */ 114 #define lns2pi_hi 0.418945312500000 115 #define lns2pi_lo -.000006779295327258219670263595 116 #define Pa0 8.33333333333333148296162562474e-02 117 #define Pa1 -2.77777777774548123579378966497e-03 118 #define Pa2 7.93650778754435631476282786423e-04 119 #define Pa3 -5.95235082566672847950717262222e-04 120 #define Pa4 8.41428560346653702135821806252e-04 121 #define Pa5 -1.89773526463879200348872089421e-03 122 #define Pa6 5.69394463439411649408050664078e-03 123 #define Pa7 -1.44705562421428915453880392761e-02 124 125 static const double zero = 0., one = 1.0, tiny = 1e-300; 126 static int endian; 127 /* 128 * TRUNC sets trailing bits in a floating-point number to zero. 129 * is a temporary variable. 130 */ 131 #if defined(vax) || defined(tahoe) 132 #define _IEEE 0 133 #define TRUNC(x) x = (double) (float) (x) 134 #else 135 #define _IEEE 1 136 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 137 #define infnan(x) 0.0 138 #endif 139 140 double 141 gamma(x) 142 double x; 143 { 144 struct Double u; 145 endian = (*(int *) &one) ? 1 : 0; 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 one+1e-20; /* Raise inexact flag. */ 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(x) 175 double x; 176 { 177 double z, p; 178 int i; 179 struct Double t, u, v; 180 181 z = one/(x*x); 182 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 183 p = p/x; 184 185 u = __log__D(x); 186 u.a -= one; 187 v.a = (x -= .5); 188 TRUNC(v.a); 189 v.b = x - v.a; 190 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 191 t.b = v.b*u.a + x*u.b; 192 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 193 t.b += lns2pi_lo; t.b += p; 194 u.a = lns2pi_hi + t.b; u.a += t.a; 195 u.b = t.a - u.a; 196 u.b += lns2pi_hi; u.b += t.b; 197 return (u); 198 } 199 /* 200 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 201 * It also has correct monotonicity. 202 */ 203 static double 204 small_gam(x) 205 double x; 206 { 207 double y, ym1, t, x1; 208 struct Double yy, r; 209 y = x - one; 210 ym1 = y - one; 211 if (y <= 1.0 + (LEFT + x0)) { 212 yy = ratfun_gam(y - x0, 0); 213 return (yy.a + yy.b); 214 } 215 r.a = y; 216 TRUNC(r.a); 217 yy.a = r.a - one; 218 y = ym1; 219 yy.b = r.b = y - yy.a; 220 /* Argument reduction: G(x+1) = x*G(x) */ 221 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 222 t = r.a*yy.a; 223 r.b = r.a*yy.b + y*r.b; 224 r.a = t; 225 TRUNC(r.a); 226 r.b += (t - r.a); 227 } 228 /* Return r*gamma(y). */ 229 yy = ratfun_gam(y - x0, 0); 230 y = r.b*(yy.a + yy.b) + r.a*yy.b; 231 y += yy.a*r.a; 232 return (y); 233 } 234 /* 235 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 236 */ 237 static double 238 smaller_gam(x) 239 double x; 240 { 241 double t, d; 242 struct Double r, xx; 243 if (x < x0 + LEFT) { 244 t = x, TRUNC(t); 245 d = (t+x)*(x-t); 246 t *= t; 247 xx.a = (t + x), TRUNC(xx.a); 248 xx.b = x - xx.a; xx.b += t; xx.b += d; 249 t = (one-x0); t += x; 250 d = (one-x0); d -= t; d += x; 251 x = xx.a + xx.b; 252 } else { 253 xx.a = x, TRUNC(xx.a); 254 xx.b = x - xx.a; 255 t = x - x0; 256 d = (-x0 -t); d += x; 257 } 258 r = ratfun_gam(t, d); 259 d = r.a/x, TRUNC(d); 260 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 261 return (d + r.a/x); 262 } 263 /* 264 * returns (z+c)^2 * P(z)/Q(z) + a0 265 */ 266 static struct Double 267 ratfun_gam(z, c) 268 double z, c; 269 { 270 int i; 271 double p, q; 272 struct Double r, t; 273 274 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 275 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 276 277 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 278 p = p/q; 279 t.a = z, TRUNC(t.a); /* t ~= z + c */ 280 t.b = (z - t.a) + c; 281 t.b *= (t.a + z); 282 q = (t.a *= t.a); /* t = (z+c)^2 */ 283 TRUNC(t.a); 284 t.b += (q - t.a); 285 r.a = p, TRUNC(r.a); /* r = P/Q */ 286 r.b = p - r.a; 287 t.b = t.b*p + t.a*r.b + a0_lo; 288 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 289 r.a = t.a + a0_hi, TRUNC(r.a); 290 r.b = ((a0_hi-r.a) + t.a) + t.b; 291 return (r); /* r = a0 + t */ 292 } 293 294 static double 295 neg_gam(x) 296 double x; 297 { 298 int sgn = 1; 299 struct Double lg, lsine; 300 double y, z; 301 302 y = floor(x + .5); 303 if (y == x) /* Negative integer. */ 304 if(!_IEEE) 305 return (infnan(ERANGE)); 306 else 307 return (one/zero); 308 z = fabs(x - y); 309 y = .5*ceil(x); 310 if (y == ceil(y)) 311 sgn = -1; 312 if (z < .25) 313 z = sin(M_PI*z); 314 else 315 z = cos(M_PI*(0.5-z)); 316 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 317 if (x < -170) { 318 if (x < -190) 319 return ((double)sgn*tiny*tiny); 320 y = one - x; /* exact: 128 < |x| < 255 */ 321 lg = large_gam(y); 322 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 323 lg.a -= lsine.a; /* exact (opposite signs) */ 324 lg.b -= lsine.b; 325 y = -(lg.a + lg.b); 326 z = (y + lg.a) + lg.b; 327 y = __exp__D(y, z); 328 if (sgn < 0) y = -y; 329 return (y); 330 } 331 y = one-x; 332 if (one-y == x) 333 y = gamma(y); 334 else /* 1-x is inexact */ 335 y = -x*gamma(-x); 336 if (sgn < 0) y = -y; 337 return (M_PI / (y*z)); 338 } 339