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 38 /* 39 * This code by P. McIlroy, Oct 1992; 40 * 41 * The financial support of UUNET Communications Services is greatfully 42 * acknowledged. 43 */ 44 45 #include <math.h> 46 #include "mathimpl.h" 47 #include <errno.h> 48 49 /* METHOD: 50 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 51 * At negative integers, return +Inf, and set errno. 52 * 53 * x < 6.5: 54 * Use argument reduction G(x+1) = xG(x) to reach the 55 * range [1.066124,2.066124]. Use a rational 56 * approximation centered at the minimum (x0+1) to 57 * ensure monotonicity. 58 * 59 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 60 * adjusted for equal-ripples: 61 * 62 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 63 * 64 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 65 * avoid premature round-off. 66 * 67 * Special values: 68 * non-positive integer: Set overflow trap; return +Inf; 69 * x > 171.63: Set overflow trap; return +Inf; 70 * NaN: Set invalid trap; return NaN 71 * 72 * Accuracy: Gamma(x) is accurate to within 73 * x > 0: error provably < 0.9ulp. 74 * Maximum observed in 1,000,000 trials was .87ulp. 75 * x < 0: 76 * Maximum observed error < 4ulp in 1,000,000 trials. 77 */ 78 79 static double neg_gam __P((double)); 80 static double small_gam __P((double)); 81 static double smaller_gam __P((double)); 82 static struct Double large_gam __P((double)); 83 static struct Double ratfun_gam __P((double, double)); 84 85 /* 86 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 87 * [1.066.., 2.066..] accurate to 4.25e-19. 88 */ 89 #define LEFT -.3955078125 /* left boundary for rat. approx */ 90 #define x0 .461632144968362356785 /* xmin - 1 */ 91 92 #define a0_hi 0.88560319441088874992 93 #define a0_lo -.00000000000000004996427036469019695 94 #define P0 6.21389571821820863029017800727e-01 95 #define P1 2.65757198651533466104979197553e-01 96 #define P2 5.53859446429917461063308081748e-03 97 #define P3 1.38456698304096573887145282811e-03 98 #define P4 2.40659950032711365819348969808e-03 99 #define Q0 1.45019531250000000000000000000e+00 100 #define Q1 1.06258521948016171343454061571e+00 101 #define Q2 -2.07474561943859936441469926649e-01 102 #define Q3 -1.46734131782005422506287573015e-01 103 #define Q4 3.07878176156175520361557573779e-02 104 #define Q5 5.12449347980666221336054633184e-03 105 #define Q6 -1.76012741431666995019222898833e-03 106 #define Q7 9.35021023573788935372153030556e-05 107 #define Q8 6.13275507472443958924745652239e-06 108 /* 109 * Constants for large x approximation (x in [6, Inf]) 110 * (Accurate to 2.8*10^-19 absolute) 111 */ 112 #define lns2pi_hi 0.418945312500000 113 #define lns2pi_lo -.000006779295327258219670263595 114 #define Pa0 8.33333333333333148296162562474e-02 115 #define Pa1 -2.77777777774548123579378966497e-03 116 #define Pa2 7.93650778754435631476282786423e-04 117 #define Pa3 -5.95235082566672847950717262222e-04 118 #define Pa4 8.41428560346653702135821806252e-04 119 #define Pa5 -1.89773526463879200348872089421e-03 120 #define Pa6 5.69394463439411649408050664078e-03 121 #define Pa7 -1.44705562421428915453880392761e-02 122 123 static const double zero = 0., one = 1.0, tiny = 1e-300; 124 static int endian; 125 /* 126 * TRUNC sets trailing bits in a floating-point number to zero. 127 * is a temporary variable. 128 */ 129 #if defined(vax) || defined(tahoe) 130 #define _IEEE 0 131 #define TRUNC(x) x = (double) (float) (x) 132 #else 133 #define _IEEE 1 134 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 135 #define infnan(x) 0.0 136 #endif 137 138 double 139 gamma(x) 140 double x; 141 { 142 struct Double u; 143 endian = (*(int *) &one) ? 1 : 0; 144 145 if (x >= 6) { 146 if(x > 171.63) 147 return(one/zero); 148 u = large_gam(x); 149 return(__exp__D(u.a, u.b)); 150 } else if (x >= 1.0 + LEFT + x0) 151 return (small_gam(x)); 152 else if (x > 1.e-17) 153 return (smaller_gam(x)); 154 else if (x > -1.e-17) { 155 if (x == 0.0) 156 if (!_IEEE) return (infnan(ERANGE)); 157 else return (one/x); 158 one+1e-20; /* Raise inexact flag. */ 159 return (one/x); 160 } else if (!finite(x)) { 161 if (_IEEE) /* x = NaN, -Inf */ 162 return (x*x); 163 else 164 return (infnan(EDOM)); 165 } else 166 return (neg_gam(x)); 167 } 168 /* 169 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 170 */ 171 static struct Double 172 large_gam(x) 173 double x; 174 { 175 double z, p; 176 int i; 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(x) 203 double x; 204 { 205 double y, ym1, t, x1; 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 */ 235 static double 236 smaller_gam(x) 237 double x; 238 { 239 double t, d; 240 struct Double r, xx; 241 if (x < x0 + LEFT) { 242 t = x, TRUNC(t); 243 d = (t+x)*(x-t); 244 t *= t; 245 xx.a = (t + x), TRUNC(xx.a); 246 xx.b = x - xx.a; xx.b += t; xx.b += d; 247 t = (one-x0); t += x; 248 d = (one-x0); d -= t; d += x; 249 x = xx.a + xx.b; 250 } else { 251 xx.a = x, TRUNC(xx.a); 252 xx.b = x - xx.a; 253 t = x - x0; 254 d = (-x0 -t); d += x; 255 } 256 r = ratfun_gam(t, d); 257 d = r.a/x, TRUNC(d); 258 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 259 return (d + r.a/x); 260 } 261 /* 262 * returns (z+c)^2 * P(z)/Q(z) + a0 263 */ 264 static struct Double 265 ratfun_gam(z, c) 266 double z, c; 267 { 268 int i; 269 double p, q; 270 struct Double r, t; 271 272 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 273 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 274 275 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 276 p = p/q; 277 t.a = z, TRUNC(t.a); /* t ~= z + c */ 278 t.b = (z - t.a) + c; 279 t.b *= (t.a + z); 280 q = (t.a *= t.a); /* t = (z+c)^2 */ 281 TRUNC(t.a); 282 t.b += (q - t.a); 283 r.a = p, TRUNC(r.a); /* r = P/Q */ 284 r.b = p - r.a; 285 t.b = t.b*p + t.a*r.b + a0_lo; 286 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 287 r.a = t.a + a0_hi, TRUNC(r.a); 288 r.b = ((a0_hi-r.a) + t.a) + t.b; 289 return (r); /* r = a0 + t */ 290 } 291 292 static double 293 neg_gam(x) 294 double x; 295 { 296 int sgn = 1; 297 struct Double lg, lsine; 298 double y, z; 299 300 y = floor(x + .5); 301 if (y == x) /* Negative integer. */ 302 if(!_IEEE) 303 return (infnan(ERANGE)); 304 else 305 return (one/zero); 306 z = fabs(x - y); 307 y = .5*ceil(x); 308 if (y == ceil(y)) 309 sgn = -1; 310 if (z < .25) 311 z = sin(M_PI*z); 312 else 313 z = cos(M_PI*(0.5-z)); 314 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 315 if (x < -170) { 316 if (x < -190) 317 return ((double)sgn*tiny*tiny); 318 y = one - x; /* exact: 128 < |x| < 255 */ 319 lg = large_gam(y); 320 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 321 lg.a -= lsine.a; /* exact (opposite signs) */ 322 lg.b -= lsine.b; 323 y = -(lg.a + lg.b); 324 z = (y + lg.a) + lg.b; 325 y = __exp__D(y, z); 326 if (sgn < 0) y = -y; 327 return (y); 328 } 329 y = one-x; 330 if (one-y == x) 331 y = gamma(y); 332 else /* 1-x is inexact */ 333 y = -x*gamma(-x); 334 if (sgn < 0) y = -y; 335 return (M_PI / (y*z)); 336 } 337