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 /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */ 35 #include <sys/cdefs.h> 36 __FBSDID("$FreeBSD$"); 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 48 /* METHOD: 49 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 50 * At negative integers, return NaN and raise invalid. 51 * 52 * x < 6.5: 53 * Use argument reduction G(x+1) = xG(x) to reach the 54 * range [1.066124,2.066124]. Use a rational 55 * approximation centered at the minimum (x0+1) to 56 * ensure monotonicity. 57 * 58 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 59 * adjusted for equal-ripples: 60 * 61 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 62 * 63 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 64 * avoid premature round-off. 65 * 66 * Special values: 67 * -Inf: return NaN and raise invalid; 68 * negative integer: return NaN and raise invalid; 69 * other x ~< 177.79: return +-0 and raise underflow; 70 * +-0: return +-Inf and raise divide-by-zero; 71 * finite x ~> 171.63: return +Inf and raise overflow; 72 * +Inf: return +Inf; 73 * NaN: return NaN. 74 * 75 * Accuracy: tgamma(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 (x / 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 u.a = one - tiny; /* raise inexact */ 146 return (one/x); 147 } else if (!finite(x)) 148 return (x - x); /* x is NaN or -Inf */ 149 else 150 return (neg_gam(x)); 151 } 152 /* 153 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 154 */ 155 static struct Double 156 large_gam(x) 157 double x; 158 { 159 double z, p; 160 struct Double t, u, v; 161 162 z = one/(x*x); 163 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 164 p = p/x; 165 166 u = __log__D(x); 167 u.a -= one; 168 v.a = (x -= .5); 169 TRUNC(v.a); 170 v.b = x - v.a; 171 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 172 t.b = v.b*u.a + x*u.b; 173 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 174 t.b += lns2pi_lo; t.b += p; 175 u.a = lns2pi_hi + t.b; u.a += t.a; 176 u.b = t.a - u.a; 177 u.b += lns2pi_hi; u.b += t.b; 178 return (u); 179 } 180 /* 181 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 182 * It also has correct monotonicity. 183 */ 184 static double 185 small_gam(x) 186 double x; 187 { 188 double y, ym1, t; 189 struct Double yy, r; 190 y = x - one; 191 ym1 = y - one; 192 if (y <= 1.0 + (LEFT + x0)) { 193 yy = ratfun_gam(y - x0, 0); 194 return (yy.a + yy.b); 195 } 196 r.a = y; 197 TRUNC(r.a); 198 yy.a = r.a - one; 199 y = ym1; 200 yy.b = r.b = y - yy.a; 201 /* Argument reduction: G(x+1) = x*G(x) */ 202 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 203 t = r.a*yy.a; 204 r.b = r.a*yy.b + y*r.b; 205 r.a = t; 206 TRUNC(r.a); 207 r.b += (t - r.a); 208 } 209 /* Return r*tgamma(y). */ 210 yy = ratfun_gam(y - x0, 0); 211 y = r.b*(yy.a + yy.b) + r.a*yy.b; 212 y += yy.a*r.a; 213 return (y); 214 } 215 /* 216 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 217 */ 218 static double 219 smaller_gam(x) 220 double x; 221 { 222 double t, d; 223 struct Double r, xx; 224 if (x < x0 + LEFT) { 225 t = x, TRUNC(t); 226 d = (t+x)*(x-t); 227 t *= t; 228 xx.a = (t + x), TRUNC(xx.a); 229 xx.b = x - xx.a; xx.b += t; xx.b += d; 230 t = (one-x0); t += x; 231 d = (one-x0); d -= t; d += x; 232 x = xx.a + xx.b; 233 } else { 234 xx.a = x, TRUNC(xx.a); 235 xx.b = x - xx.a; 236 t = x - x0; 237 d = (-x0 -t); d += x; 238 } 239 r = ratfun_gam(t, d); 240 d = r.a/x, TRUNC(d); 241 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 242 return (d + r.a/x); 243 } 244 /* 245 * returns (z+c)^2 * P(z)/Q(z) + a0 246 */ 247 static struct Double 248 ratfun_gam(z, c) 249 double z, c; 250 { 251 double p, q; 252 struct Double r, t; 253 254 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 255 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 256 257 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 258 p = p/q; 259 t.a = z, TRUNC(t.a); /* t ~= z + c */ 260 t.b = (z - t.a) + c; 261 t.b *= (t.a + z); 262 q = (t.a *= t.a); /* t = (z+c)^2 */ 263 TRUNC(t.a); 264 t.b += (q - t.a); 265 r.a = p, TRUNC(r.a); /* r = P/Q */ 266 r.b = p - r.a; 267 t.b = t.b*p + t.a*r.b + a0_lo; 268 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 269 r.a = t.a + a0_hi, TRUNC(r.a); 270 r.b = ((a0_hi-r.a) + t.a) + t.b; 271 return (r); /* r = a0 + t */ 272 } 273 274 static double 275 neg_gam(x) 276 double x; 277 { 278 int sgn = 1; 279 struct Double lg, lsine; 280 double y, z; 281 282 y = ceil(x); 283 if (y == x) /* Negative integer. */ 284 return ((x - x) / zero); 285 z = y - x; 286 if (z > 0.5) 287 z = one - z; 288 y = 0.5 * y; 289 if (y == ceil(y)) 290 sgn = -1; 291 if (z < .25) 292 z = sin(M_PI*z); 293 else 294 z = cos(M_PI*(0.5-z)); 295 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 296 if (x < -170) { 297 if (x < -190) 298 return ((double)sgn*tiny*tiny); 299 y = one - x; /* exact: 128 < |x| < 255 */ 300 lg = large_gam(y); 301 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 302 lg.a -= lsine.a; /* exact (opposite signs) */ 303 lg.b -= lsine.b; 304 y = -(lg.a + lg.b); 305 z = (y + lg.a) + lg.b; 306 y = __exp__D(y, z); 307 if (sgn < 0) y = -y; 308 return (y); 309 } 310 y = one-x; 311 if (one-y == x) 312 y = tgamma(y); 313 else /* 1-x is inexact */ 314 y = -x*tgamma(-x); 315 if (sgn < 0) y = -y; 316 return (M_PI / (y*z)); 317 } 318