1 /*- 2 * SPDX-License-Identifier: BSD-3-Clause 3 * 4 * Copyright (c) 1992, 1993 5 * The Regents of the University of California. All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 3. Neither the name of the University nor the names of its contributors 16 * may be used to endorse or promote products derived from this software 17 * without specific prior written permission. 18 * 19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 29 * SUCH DAMAGE. 30 */ 31 32 /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */ 33 #include <sys/cdefs.h> 34 __FBSDID("$FreeBSD$"); 35 36 /* 37 * This code by P. McIlroy, Oct 1992; 38 * 39 * The financial support of UUNET Communications Services is greatfully 40 * acknowledged. 41 */ 42 43 #include <math.h> 44 #include "mathimpl.h" 45 46 /* METHOD: 47 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 48 * At negative integers, return NaN and raise invalid. 49 * 50 * x < 6.5: 51 * Use argument reduction G(x+1) = xG(x) to reach the 52 * range [1.066124,2.066124]. Use a rational 53 * approximation centered at the minimum (x0+1) to 54 * ensure monotonicity. 55 * 56 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 57 * adjusted for equal-ripples: 58 * 59 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 60 * 61 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 62 * avoid premature round-off. 63 * 64 * Special values: 65 * -Inf: return NaN and raise invalid; 66 * negative integer: return NaN and raise invalid; 67 * other x ~< 177.79: return +-0 and raise underflow; 68 * +-0: return +-Inf and raise divide-by-zero; 69 * finite x ~> 171.63: return +Inf and raise overflow; 70 * +Inf: return +Inf; 71 * NaN: return NaN. 72 * 73 * Accuracy: tgamma(x) is accurate to within 74 * x > 0: error provably < 0.9ulp. 75 * Maximum observed in 1,000,000 trials was .87ulp. 76 * x < 0: 77 * Maximum observed error < 4ulp in 1,000,000 trials. 78 */ 79 80 static double neg_gam(double); 81 static double small_gam(double); 82 static double smaller_gam(double); 83 static struct Double large_gam(double); 84 static struct Double ratfun_gam(double, double); 85 86 /* 87 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 88 * [1.066.., 2.066..] accurate to 4.25e-19. 89 */ 90 #define LEFT -.3955078125 /* left boundary for rat. approx */ 91 #define x0 .461632144968362356785 /* xmin - 1 */ 92 93 #define a0_hi 0.88560319441088874992 94 #define a0_lo -.00000000000000004996427036469019695 95 #define P0 6.21389571821820863029017800727e-01 96 #define P1 2.65757198651533466104979197553e-01 97 #define P2 5.53859446429917461063308081748e-03 98 #define P3 1.38456698304096573887145282811e-03 99 #define P4 2.40659950032711365819348969808e-03 100 #define Q0 1.45019531250000000000000000000e+00 101 #define Q1 1.06258521948016171343454061571e+00 102 #define Q2 -2.07474561943859936441469926649e-01 103 #define Q3 -1.46734131782005422506287573015e-01 104 #define Q4 3.07878176156175520361557573779e-02 105 #define Q5 5.12449347980666221336054633184e-03 106 #define Q6 -1.76012741431666995019222898833e-03 107 #define Q7 9.35021023573788935372153030556e-05 108 #define Q8 6.13275507472443958924745652239e-06 109 /* 110 * Constants for large x approximation (x in [6, Inf]) 111 * (Accurate to 2.8*10^-19 absolute) 112 */ 113 #define lns2pi_hi 0.418945312500000 114 #define lns2pi_lo -.000006779295327258219670263595 115 #define Pa0 8.33333333333333148296162562474e-02 116 #define Pa1 -2.77777777774548123579378966497e-03 117 #define Pa2 7.93650778754435631476282786423e-04 118 #define Pa3 -5.95235082566672847950717262222e-04 119 #define Pa4 8.41428560346653702135821806252e-04 120 #define Pa5 -1.89773526463879200348872089421e-03 121 #define Pa6 5.69394463439411649408050664078e-03 122 #define Pa7 -1.44705562421428915453880392761e-02 123 124 static const double zero = 0., one = 1.0, tiny = 1e-300; 125 126 double 127 tgamma(x) 128 double x; 129 { 130 struct Double u; 131 132 if (x >= 6) { 133 if(x > 171.63) 134 return (x / zero); 135 u = large_gam(x); 136 return(__exp__D(u.a, u.b)); 137 } else if (x >= 1.0 + LEFT + x0) 138 return (small_gam(x)); 139 else if (x > 1.e-17) 140 return (smaller_gam(x)); 141 else if (x > -1.e-17) { 142 if (x != 0.0) 143 u.a = one - tiny; /* raise inexact */ 144 return (one/x); 145 } else if (!finite(x)) 146 return (x - x); /* x is NaN or -Inf */ 147 else 148 return (neg_gam(x)); 149 } 150 /* 151 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 152 */ 153 static struct Double 154 large_gam(x) 155 double x; 156 { 157 double z, p; 158 struct Double t, u, v; 159 160 z = one/(x*x); 161 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 162 p = p/x; 163 164 u = __log__D(x); 165 u.a -= one; 166 v.a = (x -= .5); 167 TRUNC(v.a); 168 v.b = x - v.a; 169 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 170 t.b = v.b*u.a + x*u.b; 171 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 172 t.b += lns2pi_lo; t.b += p; 173 u.a = lns2pi_hi + t.b; u.a += t.a; 174 u.b = t.a - u.a; 175 u.b += lns2pi_hi; u.b += t.b; 176 return (u); 177 } 178 /* 179 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 180 * It also has correct monotonicity. 181 */ 182 static double 183 small_gam(x) 184 double x; 185 { 186 double y, ym1, t; 187 struct Double yy, r; 188 y = x - one; 189 ym1 = y - one; 190 if (y <= 1.0 + (LEFT + x0)) { 191 yy = ratfun_gam(y - x0, 0); 192 return (yy.a + yy.b); 193 } 194 r.a = y; 195 TRUNC(r.a); 196 yy.a = r.a - one; 197 y = ym1; 198 yy.b = r.b = y - yy.a; 199 /* Argument reduction: G(x+1) = x*G(x) */ 200 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 201 t = r.a*yy.a; 202 r.b = r.a*yy.b + y*r.b; 203 r.a = t; 204 TRUNC(r.a); 205 r.b += (t - r.a); 206 } 207 /* Return r*tgamma(y). */ 208 yy = ratfun_gam(y - x0, 0); 209 y = r.b*(yy.a + yy.b) + r.a*yy.b; 210 y += yy.a*r.a; 211 return (y); 212 } 213 /* 214 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 215 */ 216 static double 217 smaller_gam(x) 218 double x; 219 { 220 double t, d; 221 struct Double r, xx; 222 if (x < x0 + LEFT) { 223 t = x, TRUNC(t); 224 d = (t+x)*(x-t); 225 t *= t; 226 xx.a = (t + x), TRUNC(xx.a); 227 xx.b = x - xx.a; xx.b += t; xx.b += d; 228 t = (one-x0); t += x; 229 d = (one-x0); d -= t; d += x; 230 x = xx.a + xx.b; 231 } else { 232 xx.a = x, TRUNC(xx.a); 233 xx.b = x - xx.a; 234 t = x - x0; 235 d = (-x0 -t); d += x; 236 } 237 r = ratfun_gam(t, d); 238 d = r.a/x, TRUNC(d); 239 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 240 return (d + r.a/x); 241 } 242 /* 243 * returns (z+c)^2 * P(z)/Q(z) + a0 244 */ 245 static struct Double 246 ratfun_gam(z, c) 247 double z, c; 248 { 249 double p, q; 250 struct Double r, t; 251 252 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 253 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 254 255 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 256 p = p/q; 257 t.a = z, TRUNC(t.a); /* t ~= z + c */ 258 t.b = (z - t.a) + c; 259 t.b *= (t.a + z); 260 q = (t.a *= t.a); /* t = (z+c)^2 */ 261 TRUNC(t.a); 262 t.b += (q - t.a); 263 r.a = p, TRUNC(r.a); /* r = P/Q */ 264 r.b = p - r.a; 265 t.b = t.b*p + t.a*r.b + a0_lo; 266 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 267 r.a = t.a + a0_hi, TRUNC(r.a); 268 r.b = ((a0_hi-r.a) + t.a) + t.b; 269 return (r); /* r = a0 + t */ 270 } 271 272 static double 273 neg_gam(x) 274 double x; 275 { 276 int sgn = 1; 277 struct Double lg, lsine; 278 double y, z; 279 280 y = ceil(x); 281 if (y == x) /* Negative integer. */ 282 return ((x - x) / zero); 283 z = y - x; 284 if (z > 0.5) 285 z = one - z; 286 y = 0.5 * y; 287 if (y == ceil(y)) 288 sgn = -1; 289 if (z < .25) 290 z = sin(M_PI*z); 291 else 292 z = cos(M_PI*(0.5-z)); 293 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 294 if (x < -170) { 295 if (x < -190) 296 return ((double)sgn*tiny*tiny); 297 y = one - x; /* exact: 128 < |x| < 255 */ 298 lg = large_gam(y); 299 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 300 lg.a -= lsine.a; /* exact (opposite signs) */ 301 lg.b -= lsine.b; 302 y = -(lg.a + lg.b); 303 z = (y + lg.a) + lg.b; 304 y = __exp__D(y, z); 305 if (sgn < 0) y = -y; 306 return (y); 307 } 308 y = one-x; 309 if (one-y == x) 310 y = tgamma(y); 311 else /* 1-x is inexact */ 312 y = -x*tgamma(-x); 313 if (sgn < 0) y = -y; 314 return (M_PI / (y*z)); 315 } 316