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 50 /* METHOD: 51 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 52 * At negative integers, return NaN and raise invalid. 53 * 54 * x < 6.5: 55 * Use argument reduction G(x+1) = xG(x) to reach the 56 * range [1.066124,2.066124]. Use a rational 57 * approximation centered at the minimum (x0+1) to 58 * ensure monotonicity. 59 * 60 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 61 * adjusted for equal-ripples: 62 * 63 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 64 * 65 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 66 * avoid premature round-off. 67 * 68 * Special values: 69 * -Inf: return NaN and raise invalid; 70 * negative integer: return NaN and raise invalid; 71 * other x ~< 177.79: return +-0 and raise underflow; 72 * +-0: return +-Inf and raise divide-by-zero; 73 * finite x ~> 171.63: return +Inf and raise overflow; 74 * +Inf: return +Inf; 75 * NaN: return NaN. 76 * 77 * Accuracy: tgamma(x) is accurate to within 78 * x > 0: error provably < 0.9ulp. 79 * Maximum observed in 1,000,000 trials was .87ulp. 80 * x < 0: 81 * Maximum observed error < 4ulp in 1,000,000 trials. 82 */ 83 84 static double neg_gam(double); 85 static double small_gam(double); 86 static double smaller_gam(double); 87 static struct Double large_gam(double); 88 static struct Double ratfun_gam(double, double); 89 90 /* 91 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 92 * [1.066.., 2.066..] accurate to 4.25e-19. 93 */ 94 #define LEFT -.3955078125 /* left boundary for rat. approx */ 95 #define x0 .461632144968362356785 /* xmin - 1 */ 96 97 #define a0_hi 0.88560319441088874992 98 #define a0_lo -.00000000000000004996427036469019695 99 #define P0 6.21389571821820863029017800727e-01 100 #define P1 2.65757198651533466104979197553e-01 101 #define P2 5.53859446429917461063308081748e-03 102 #define P3 1.38456698304096573887145282811e-03 103 #define P4 2.40659950032711365819348969808e-03 104 #define Q0 1.45019531250000000000000000000e+00 105 #define Q1 1.06258521948016171343454061571e+00 106 #define Q2 -2.07474561943859936441469926649e-01 107 #define Q3 -1.46734131782005422506287573015e-01 108 #define Q4 3.07878176156175520361557573779e-02 109 #define Q5 5.12449347980666221336054633184e-03 110 #define Q6 -1.76012741431666995019222898833e-03 111 #define Q7 9.35021023573788935372153030556e-05 112 #define Q8 6.13275507472443958924745652239e-06 113 /* 114 * Constants for large x approximation (x in [6, Inf]) 115 * (Accurate to 2.8*10^-19 absolute) 116 */ 117 #define lns2pi_hi 0.418945312500000 118 #define lns2pi_lo -.000006779295327258219670263595 119 #define Pa0 8.33333333333333148296162562474e-02 120 #define Pa1 -2.77777777774548123579378966497e-03 121 #define Pa2 7.93650778754435631476282786423e-04 122 #define Pa3 -5.95235082566672847950717262222e-04 123 #define Pa4 8.41428560346653702135821806252e-04 124 #define Pa5 -1.89773526463879200348872089421e-03 125 #define Pa6 5.69394463439411649408050664078e-03 126 #define Pa7 -1.44705562421428915453880392761e-02 127 128 static const double zero = 0., one = 1.0, tiny = 1e-300; 129 130 double 131 tgamma(x) 132 double x; 133 { 134 struct Double u; 135 136 if (x >= 6) { 137 if(x > 171.63) 138 return (x / zero); 139 u = large_gam(x); 140 return(__exp__D(u.a, u.b)); 141 } else if (x >= 1.0 + LEFT + x0) 142 return (small_gam(x)); 143 else if (x > 1.e-17) 144 return (smaller_gam(x)); 145 else if (x > -1.e-17) { 146 if (x != 0.0) 147 u.a = one - tiny; /* raise inexact */ 148 return (one/x); 149 } else if (!finite(x)) 150 return (x - x); /* x is NaN or -Inf */ 151 else 152 return (neg_gam(x)); 153 } 154 /* 155 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 156 */ 157 static struct Double 158 large_gam(x) 159 double x; 160 { 161 double z, p; 162 struct Double t, u, v; 163 164 z = one/(x*x); 165 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 166 p = p/x; 167 168 u = __log__D(x); 169 u.a -= one; 170 v.a = (x -= .5); 171 TRUNC(v.a); 172 v.b = x - v.a; 173 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 174 t.b = v.b*u.a + x*u.b; 175 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 176 t.b += lns2pi_lo; t.b += p; 177 u.a = lns2pi_hi + t.b; u.a += t.a; 178 u.b = t.a - u.a; 179 u.b += lns2pi_hi; u.b += t.b; 180 return (u); 181 } 182 /* 183 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 184 * It also has correct monotonicity. 185 */ 186 static double 187 small_gam(x) 188 double x; 189 { 190 double y, ym1, t; 191 struct Double yy, r; 192 y = x - one; 193 ym1 = y - one; 194 if (y <= 1.0 + (LEFT + x0)) { 195 yy = ratfun_gam(y - x0, 0); 196 return (yy.a + yy.b); 197 } 198 r.a = y; 199 TRUNC(r.a); 200 yy.a = r.a - one; 201 y = ym1; 202 yy.b = r.b = y - yy.a; 203 /* Argument reduction: G(x+1) = x*G(x) */ 204 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 205 t = r.a*yy.a; 206 r.b = r.a*yy.b + y*r.b; 207 r.a = t; 208 TRUNC(r.a); 209 r.b += (t - r.a); 210 } 211 /* Return r*tgamma(y). */ 212 yy = ratfun_gam(y - x0, 0); 213 y = r.b*(yy.a + yy.b) + r.a*yy.b; 214 y += yy.a*r.a; 215 return (y); 216 } 217 /* 218 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 219 */ 220 static double 221 smaller_gam(x) 222 double x; 223 { 224 double t, d; 225 struct Double r, xx; 226 if (x < x0 + LEFT) { 227 t = x, TRUNC(t); 228 d = (t+x)*(x-t); 229 t *= t; 230 xx.a = (t + x), TRUNC(xx.a); 231 xx.b = x - xx.a; xx.b += t; xx.b += d; 232 t = (one-x0); t += x; 233 d = (one-x0); d -= t; d += x; 234 x = xx.a + xx.b; 235 } else { 236 xx.a = x, TRUNC(xx.a); 237 xx.b = x - xx.a; 238 t = x - x0; 239 d = (-x0 -t); d += x; 240 } 241 r = ratfun_gam(t, d); 242 d = r.a/x, TRUNC(d); 243 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 244 return (d + r.a/x); 245 } 246 /* 247 * returns (z+c)^2 * P(z)/Q(z) + a0 248 */ 249 static struct Double 250 ratfun_gam(z, c) 251 double z, c; 252 { 253 double p, q; 254 struct Double r, t; 255 256 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 257 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 258 259 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 260 p = p/q; 261 t.a = z, TRUNC(t.a); /* t ~= z + c */ 262 t.b = (z - t.a) + c; 263 t.b *= (t.a + z); 264 q = (t.a *= t.a); /* t = (z+c)^2 */ 265 TRUNC(t.a); 266 t.b += (q - t.a); 267 r.a = p, TRUNC(r.a); /* r = P/Q */ 268 r.b = p - r.a; 269 t.b = t.b*p + t.a*r.b + a0_lo; 270 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 271 r.a = t.a + a0_hi, TRUNC(r.a); 272 r.b = ((a0_hi-r.a) + t.a) + t.b; 273 return (r); /* r = a0 + t */ 274 } 275 276 static double 277 neg_gam(x) 278 double x; 279 { 280 int sgn = 1; 281 struct Double lg, lsine; 282 double y, z; 283 284 y = ceil(x); 285 if (y == x) /* Negative integer. */ 286 return ((x - x) / zero); 287 z = y - x; 288 if (z > 0.5) 289 z = one - z; 290 y = 0.5 * y; 291 if (y == ceil(y)) 292 sgn = -1; 293 if (z < .25) 294 z = sin(M_PI*z); 295 else 296 z = cos(M_PI*(0.5-z)); 297 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 298 if (x < -170) { 299 if (x < -190) 300 return ((double)sgn*tiny*tiny); 301 y = one - x; /* exact: 128 < |x| < 255 */ 302 lg = large_gam(y); 303 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 304 lg.a -= lsine.a; /* exact (opposite signs) */ 305 lg.b -= lsine.b; 306 y = -(lg.a + lg.b); 307 z = (y + lg.a) + lg.b; 308 y = __exp__D(y, z); 309 if (sgn < 0) y = -y; 310 return (y); 311 } 312 y = one-x; 313 if (one-y == x) 314 y = tgamma(y); 315 else /* 1-x is inexact */ 316 y = -x*tgamma(-x); 317 if (sgn < 0) y = -y; 318 return (M_PI / (y*z)); 319 } 320