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. Neither the name of the University nor the names of its contributors 14 * may be used to endorse or promote products derived from this software 15 * without specific prior written permission. 16 * 17 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 18 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 21 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 22 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 23 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 24 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 25 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 26 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 27 * SUCH DAMAGE. 28 */ 29 30 /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */ 31 #include "cdefs-compat.h" 32 //__FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_tgamma.c,v 1.10 2008/02/22 02:26:51 das Exp $"); 33 34 /* 35 * This code by P. McIlroy, Oct 1992; 36 * 37 * The financial support of UUNET Communications Services is greatfully 38 * acknowledged. 39 */ 40 41 #include <openlibm_math.h> 42 43 #include "mathimpl.h" 44 45 /* METHOD: 46 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 47 * At negative integers, return NaN and raise invalid. 48 * 49 * x < 6.5: 50 * Use argument reduction G(x+1) = xG(x) to reach the 51 * range [1.066124,2.066124]. Use a rational 52 * approximation centered at the minimum (x0+1) to 53 * ensure monotonicity. 54 * 55 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 56 * adjusted for equal-ripples: 57 * 58 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 59 * 60 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 61 * avoid premature round-off. 62 * 63 * Special values: 64 * -Inf: return NaN and raise invalid; 65 * negative integer: return NaN and raise invalid; 66 * other x ~< 177.79: return +-0 and raise underflow; 67 * +-0: return +-Inf and raise divide-by-zero; 68 * finite x ~> 171.63: return +Inf and raise overflow; 69 * +Inf: return +Inf; 70 * NaN: return NaN. 71 * 72 * Accuracy: tgamma(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(double); 80 static double small_gam(double); 81 static double smaller_gam(double); 82 static struct Double large_gam(double); 83 static struct Double ratfun_gam(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; 124 125 OLM_DLLEXPORT double 126 tgamma(x) 127 double x; 128 { 129 struct Double u; 130 131 if (isgreaterequal(x, 6)) { 132 if(x > 171.63) 133 return (x / zero); 134 u = large_gam(x); 135 return(__exp__D(u.a, u.b)); 136 } else if (isgreaterequal(x, 1.0 + LEFT + x0)) 137 return (small_gam(x)); 138 else if (isgreater(x, 1.e-17)) 139 return (smaller_gam(x)); 140 else if (isgreater(x, -1.e-17)) { 141 if (x != 0.0) 142 u.a = one - tiny; /* raise inexact */ 143 return (one/x); 144 } else if (!isfinite(x)) 145 return (x - x); /* x is NaN or -Inf */ 146 else 147 return (neg_gam(x)); 148 } 149 /* 150 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 151 */ 152 static struct Double 153 large_gam(x) 154 double x; 155 { 156 double z, p; 157 struct Double t, u, v; 158 159 z = one/(x*x); 160 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 161 p = p/x; 162 163 u = __log__D(x); 164 u.a -= one; 165 v.a = (x -= .5); 166 TRUNC(v.a); 167 v.b = x - v.a; 168 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 169 t.b = v.b*u.a + x*u.b; 170 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 171 t.b += lns2pi_lo; t.b += p; 172 u.a = lns2pi_hi + t.b; u.a += t.a; 173 u.b = t.a - u.a; 174 u.b += lns2pi_hi; u.b += t.b; 175 return (u); 176 } 177 /* 178 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 179 * It also has correct monotonicity. 180 */ 181 static double 182 small_gam(x) 183 double x; 184 { 185 double y, ym1, t; 186 struct Double yy, r; 187 y = x - one; 188 ym1 = y - one; 189 if (y <= 1.0 + (LEFT + x0)) { 190 yy = ratfun_gam(y - x0, 0); 191 return (yy.a + yy.b); 192 } 193 r.a = y; 194 TRUNC(r.a); 195 yy.a = r.a - one; 196 y = ym1; 197 yy.b = r.b = y - yy.a; 198 /* Argument reduction: G(x+1) = x*G(x) */ 199 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 200 t = r.a*yy.a; 201 r.b = r.a*yy.b + y*r.b; 202 r.a = t; 203 TRUNC(r.a); 204 r.b += (t - r.a); 205 } 206 /* Return r*tgamma(y). */ 207 yy = ratfun_gam(y - x0, 0); 208 y = r.b*(yy.a + yy.b) + r.a*yy.b; 209 y += yy.a*r.a; 210 return (y); 211 } 212 /* 213 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 214 */ 215 static double 216 smaller_gam(x) 217 double x; 218 { 219 double t, d; 220 struct Double r, xx; 221 if (x < x0 + LEFT) { 222 t = x, TRUNC(t); 223 d = (t+x)*(x-t); 224 t *= t; 225 xx.a = (t + x), TRUNC(xx.a); 226 xx.b = x - xx.a; xx.b += t; xx.b += d; 227 t = (one-x0); t += x; 228 d = (one-x0); d -= t; d += x; 229 x = xx.a + xx.b; 230 } else { 231 xx.a = x, TRUNC(xx.a); 232 xx.b = x - xx.a; 233 t = x - x0; 234 d = (-x0 -t); d += x; 235 } 236 r = ratfun_gam(t, d); 237 d = r.a/x, TRUNC(d); 238 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 239 return (d + r.a/x); 240 } 241 /* 242 * returns (z+c)^2 * P(z)/Q(z) + a0 243 */ 244 static struct Double 245 ratfun_gam(z, c) 246 double z, c; 247 { 248 double p, q; 249 struct Double r, t; 250 251 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 252 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 253 254 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 255 p = p/q; 256 t.a = z, TRUNC(t.a); /* t ~= z + c */ 257 t.b = (z - t.a) + c; 258 t.b *= (t.a + z); 259 q = (t.a *= t.a); /* t = (z+c)^2 */ 260 TRUNC(t.a); 261 t.b += (q - t.a); 262 r.a = p, TRUNC(r.a); /* r = P/Q */ 263 r.b = p - r.a; 264 t.b = t.b*p + t.a*r.b + a0_lo; 265 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 266 r.a = t.a + a0_hi, TRUNC(r.a); 267 r.b = ((a0_hi-r.a) + t.a) + t.b; 268 return (r); /* r = a0 + t */ 269 } 270 271 static double 272 neg_gam(x) 273 double x; 274 { 275 int sgn = 1; 276 struct Double lg, lsine; 277 double y, z; 278 279 y = ceil(x); 280 if (y == x) /* Negative integer. */ 281 return ((x - x) / zero); 282 z = y - x; 283 if (z > 0.5) 284 z = one - z; 285 y = 0.5 * y; 286 if (y == ceil(y)) 287 sgn = -1; 288 if (z < .25) 289 z = sin(M_PI*z); 290 else 291 z = cos(M_PI*(0.5-z)); 292 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 293 if (x < -170) { 294 if (x < -190) 295 return ((double)sgn*tiny*tiny); 296 y = one - x; /* exact: 128 < |x| < 255 */ 297 lg = large_gam(y); 298 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 299 lg.a -= lsine.a; /* exact (opposite signs) */ 300 lg.b -= lsine.b; 301 y = -(lg.a + lg.b); 302 z = (y + lg.a) + lg.b; 303 y = __exp__D(y, z); 304 if (sgn < 0) y = -y; 305 return (y); 306 } 307 y = one-x; 308 if (one-y == x) 309 y = tgamma(y); 310 else /* 1-x is inexact */ 311 y = -x*tgamma(-x); 312 if (sgn < 0) y = -y; 313 return (M_PI / (y*z)); 314 } 315