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: src/lib/msun/bsdsrc/b_tgamma.c,v 1.7 2005/09/19 11:28:19 bde Exp $"); */ 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 "../include/math.h" 49 #include "mathimpl.h" 50 #include <errno.h> 51 52 /* METHOD: 53 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 54 * At negative integers, return +Inf, and set errno. 55 * 56 * x < 6.5: 57 * Use argument reduction G(x+1) = xG(x) to reach the 58 * range [1.066124,2.066124]. Use a rational 59 * approximation centered at the minimum (x0+1) to 60 * ensure monotonicity. 61 * 62 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 63 * adjusted for equal-ripples: 64 * 65 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 66 * 67 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 68 * avoid premature round-off. 69 * 70 * Special values: 71 * non-positive integer: Set overflow trap; return +Inf; 72 * x > 171.63: Set overflow trap; return +Inf; 73 * NaN: Set invalid trap; return NaN 74 * 75 * Accuracy: Gamma(x) is accurate to within 76 * x > 0: error provably < 0.9ulp. 77 * Maximum observed in 1,000,000 trials was .87ulp. 78 * x < 0: 79 * Maximum observed error < 4ulp in 1,000,000 trials. 80 */ 81 82 static double neg_gam(double); 83 static double small_gam(double); 84 static double smaller_gam(double); 85 static struct Double large_gam(double); 86 static struct Double ratfun_gam(double, double); 87 88 /* 89 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 90 * [1.066.., 2.066..] accurate to 4.25e-19. 91 */ 92 #define LEFT -.3955078125 /* left boundary for rat. approx */ 93 #define x0 .461632144968362356785 /* xmin - 1 */ 94 95 #define a0_hi 0.88560319441088874992 96 #define a0_lo -.00000000000000004996427036469019695 97 #define P0 6.21389571821820863029017800727e-01 98 #define P1 2.65757198651533466104979197553e-01 99 #define P2 5.53859446429917461063308081748e-03 100 #define P3 1.38456698304096573887145282811e-03 101 #define P4 2.40659950032711365819348969808e-03 102 #define Q0 1.45019531250000000000000000000e+00 103 #define Q1 1.06258521948016171343454061571e+00 104 #define Q2 -2.07474561943859936441469926649e-01 105 #define Q3 -1.46734131782005422506287573015e-01 106 #define Q4 3.07878176156175520361557573779e-02 107 #define Q5 5.12449347980666221336054633184e-03 108 #define Q6 -1.76012741431666995019222898833e-03 109 #define Q7 9.35021023573788935372153030556e-05 110 #define Q8 6.13275507472443958924745652239e-06 111 /* 112 * Constants for large x approximation (x in [6, Inf]) 113 * (Accurate to 2.8*10^-19 absolute) 114 */ 115 #define lns2pi_hi 0.418945312500000 116 #define lns2pi_lo -.000006779295327258219670263595 117 #define Pa0 8.33333333333333148296162562474e-02 118 #define Pa1 -2.77777777774548123579378966497e-03 119 #define Pa2 7.93650778754435631476282786423e-04 120 #define Pa3 -5.95235082566672847950717262222e-04 121 #define Pa4 8.41428560346653702135821806252e-04 122 #define Pa5 -1.89773526463879200348872089421e-03 123 #define Pa6 5.69394463439411649408050664078e-03 124 #define Pa7 -1.44705562421428915453880392761e-02 125 126 static const double zero = 0., one = 1.0, tiny = 1e-300; 127 128 double 129 tgamma(x) 130 double x; 131 { 132 struct Double u; 133 134 if (x >= 6) { 135 if(x > 171.63) 136 return(one/zero); 137 u = large_gam(x); 138 return(__exp__D(u.a, u.b)); 139 } else if (x >= 1.0 + LEFT + x0) 140 return (small_gam(x)); 141 else if (x > 1.e-17) 142 return (smaller_gam(x)); 143 else if (x > -1.e-17) { 144 if (x == 0.0) 145 return (one/x); 146 one+1e-20; /* Raise inexact flag. */ 147 return (one/x); 148 } else if (!finite(x)) 149 return (x*x); /* x = NaN, -Inf */ 150 else 151 return (neg_gam(x)); 152 } 153 /* 154 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 155 */ 156 static struct Double 157 large_gam(x) 158 double x; 159 { 160 double z, p; 161 struct Double t, u, v; 162 163 z = one/(x*x); 164 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 165 p = p/x; 166 167 u = __log__D(x); 168 u.a -= one; 169 v.a = (x -= .5); 170 TRUNC(v.a); 171 v.b = x - v.a; 172 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 173 t.b = v.b*u.a + x*u.b; 174 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 175 t.b += lns2pi_lo; t.b += p; 176 u.a = lns2pi_hi + t.b; u.a += t.a; 177 u.b = t.a - u.a; 178 u.b += lns2pi_hi; u.b += t.b; 179 return (u); 180 } 181 /* 182 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 183 * It also has correct monotonicity. 184 */ 185 static double 186 small_gam(x) 187 double x; 188 { 189 double y, ym1, t; 190 struct Double yy, r; 191 y = x - one; 192 ym1 = y - one; 193 if (y <= 1.0 + (LEFT + x0)) { 194 yy = ratfun_gam(y - x0, 0); 195 return (yy.a + yy.b); 196 } 197 r.a = y; 198 TRUNC(r.a); 199 yy.a = r.a - one; 200 y = ym1; 201 yy.b = r.b = y - yy.a; 202 /* Argument reduction: G(x+1) = x*G(x) */ 203 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 204 t = r.a*yy.a; 205 r.b = r.a*yy.b + y*r.b; 206 r.a = t; 207 TRUNC(r.a); 208 r.b += (t - r.a); 209 } 210 /* Return r*tgamma(y). */ 211 yy = ratfun_gam(y - x0, 0); 212 y = r.b*(yy.a + yy.b) + r.a*yy.b; 213 y += yy.a*r.a; 214 return (y); 215 } 216 /* 217 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 218 */ 219 static double 220 smaller_gam(x) 221 double x; 222 { 223 double t, d; 224 struct Double r, xx; 225 if (x < x0 + LEFT) { 226 t = x, TRUNC(t); 227 d = (t+x)*(x-t); 228 t *= t; 229 xx.a = (t + x), TRUNC(xx.a); 230 xx.b = x - xx.a; xx.b += t; xx.b += d; 231 t = (one-x0); t += x; 232 d = (one-x0); d -= t; d += x; 233 x = xx.a + xx.b; 234 } else { 235 xx.a = x, TRUNC(xx.a); 236 xx.b = x - xx.a; 237 t = x - x0; 238 d = (-x0 -t); d += x; 239 } 240 r = ratfun_gam(t, d); 241 d = r.a/x, TRUNC(d); 242 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 243 return (d + r.a/x); 244 } 245 /* 246 * returns (z+c)^2 * P(z)/Q(z) + a0 247 */ 248 static struct Double 249 ratfun_gam(z, c) 250 double z, c; 251 { 252 double p, q; 253 struct Double r, t; 254 255 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 256 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 257 258 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 259 p = p/q; 260 t.a = z, TRUNC(t.a); /* t ~= z + c */ 261 t.b = (z - t.a) + c; 262 t.b *= (t.a + z); 263 q = (t.a *= t.a); /* t = (z+c)^2 */ 264 TRUNC(t.a); 265 t.b += (q - t.a); 266 r.a = p, TRUNC(r.a); /* r = P/Q */ 267 r.b = p - r.a; 268 t.b = t.b*p + t.a*r.b + a0_lo; 269 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 270 r.a = t.a + a0_hi, TRUNC(r.a); 271 r.b = ((a0_hi-r.a) + t.a) + t.b; 272 return (r); /* r = a0 + t */ 273 } 274 275 static double 276 neg_gam(x) 277 double x; 278 { 279 int sgn = 1; 280 struct Double lg, lsine; 281 double y, z; 282 283 y = floor(x + .5); 284 if (y == x) /* Negative integer. */ 285 return (one/zero); 286 z = fabs(x - y); 287 y = .5*ceil(x); 288 if (y == ceil(y)) 289 sgn = -1; 290 if (z < .25) 291 z = sin(M_PI*z); 292 else 293 z = cos(M_PI*(0.5-z)); 294 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 295 if (x < -170) { 296 if (x < -190) 297 return ((double)sgn*tiny*tiny); 298 y = one - x; /* exact: 128 < |x| < 255 */ 299 lg = large_gam(y); 300 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 301 lg.a -= lsine.a; /* exact (opposite signs) */ 302 lg.b -= lsine.b; 303 y = -(lg.a + lg.b); 304 z = (y + lg.a) + lg.b; 305 y = __exp__D(y, z); 306 if (sgn < 0) y = -y; 307 return (y); 308 } 309 y = one-x; 310 if (one-y == x) 311 y = tgamma(y); 312 else /* 1-x is inexact */ 313 y = -x*tgamma(-x); 314 if (sgn < 0) y = -y; 315 return (M_PI / (y*z)); 316 } 317
