1 #include "FEATURE/uwin" 2 3 #if !_UWIN || _lib_gamma 4 5 void _STUB_gamma(){} 6 7 #else 8 9 /*- 10 * Copyright (c) 1992, 1993 11 * The Regents of the University of California. All rights reserved. 12 * 13 * Redistribution and use in source and binary forms, with or without 14 * modification, are permitted provided that the following conditions 15 * are met: 16 * 1. Redistributions of source code must retain the above copyright 17 * notice, this list of conditions and the following disclaimer. 18 * 2. Redistributions in binary form must reproduce the above copyright 19 * notice, this list of conditions and the following disclaimer in the 20 * documentation and/or other materials provided with the distribution. 21 * 3. Neither the name of the University nor the names of its contributors 22 * may be used to endorse or promote products derived from this software 23 * without specific prior written permission. 24 * 25 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 26 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 27 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 28 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 29 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 30 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 31 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 32 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 33 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 34 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 35 * SUCH DAMAGE. 36 */ 37 38 #ifndef lint 39 static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93"; 40 #endif /* not lint */ 41 42 /* 43 * This code by P. McIlroy, Oct 1992; 44 * 45 * The financial support of UUNET Communications Services is greatfully 46 * acknowledged. 47 */ 48 49 #define gamma ______gamma 50 51 #include <math.h> 52 #include <errno.h> 53 #include "mathimpl.h" 54 55 #undef gamma 56 57 /* METHOD: 58 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 59 * At negative integers, return +Inf, and set errno. 60 * 61 * x < 6.5: 62 * Use argument reduction G(x+1) = xG(x) to reach the 63 * range [1.066124,2.066124]. Use a rational 64 * approximation centered at the minimum (x0+1) to 65 * ensure monotonicity. 66 * 67 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 68 * adjusted for equal-ripples: 69 * 70 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 71 * 72 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 73 * avoid premature round-off. 74 * 75 * Special values: 76 * non-positive integer: Set overflow trap; return +Inf; 77 * x > 171.63: Set overflow trap; return +Inf; 78 * NaN: Set invalid trap; return NaN 79 * 80 * Accuracy: Gamma(x) is accurate to within 81 * x > 0: error provably < 0.9ulp. 82 * Maximum observed in 1,000,000 trials was .87ulp. 83 * x < 0: 84 * Maximum observed error < 4ulp in 1,000,000 trials. 85 */ 86 87 static double neg_gam __P((double)); 88 static double small_gam __P((double)); 89 static double smaller_gam __P((double)); 90 static struct Double large_gam __P((double)); 91 static struct Double ratfun_gam __P((double, double)); 92 93 /* 94 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 95 * [1.066.., 2.066..] accurate to 4.25e-19. 96 */ 97 #define LEFT -.3955078125 /* left boundary for rat. approx */ 98 #define x0 .461632144968362356785 /* xmin - 1 */ 99 100 #define a0_hi 0.88560319441088874992 101 #define a0_lo -.00000000000000004996427036469019695 102 #define P0 6.21389571821820863029017800727e-01 103 #define P1 2.65757198651533466104979197553e-01 104 #define P2 5.53859446429917461063308081748e-03 105 #define P3 1.38456698304096573887145282811e-03 106 #define P4 2.40659950032711365819348969808e-03 107 #define Q0 1.45019531250000000000000000000e+00 108 #define Q1 1.06258521948016171343454061571e+00 109 #define Q2 -2.07474561943859936441469926649e-01 110 #define Q3 -1.46734131782005422506287573015e-01 111 #define Q4 3.07878176156175520361557573779e-02 112 #define Q5 5.12449347980666221336054633184e-03 113 #define Q6 -1.76012741431666995019222898833e-03 114 #define Q7 9.35021023573788935372153030556e-05 115 #define Q8 6.13275507472443958924745652239e-06 116 /* 117 * Constants for large x approximation (x in [6, Inf]) 118 * (Accurate to 2.8*10^-19 absolute) 119 */ 120 #define lns2pi_hi 0.418945312500000 121 #define lns2pi_lo -.000006779295327258219670263595 122 #define Pa0 8.33333333333333148296162562474e-02 123 #define Pa1 -2.77777777774548123579378966497e-03 124 #define Pa2 7.93650778754435631476282786423e-04 125 #define Pa3 -5.95235082566672847950717262222e-04 126 #define Pa4 8.41428560346653702135821806252e-04 127 #define Pa5 -1.89773526463879200348872089421e-03 128 #define Pa6 5.69394463439411649408050664078e-03 129 #define Pa7 -1.44705562421428915453880392761e-02 130 131 static const double zero = 0., one = 1.0, tiny = 1e-300; 132 static int endian; 133 /* 134 * TRUNC sets trailing bits in a floating-point number to zero. 135 * is a temporary variable. 136 */ 137 #if defined(vax) || defined(tahoe) 138 #define _IEEE 0 139 #define TRUNC(x) x = (double) (float) (x) 140 #else 141 #define _IEEE 1 142 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 143 #define infnan(x) 0.0 144 #endif 145 146 extern double gamma(x) 147 double x; 148 { 149 struct Double u; 150 endian = (*(int *) &one) ? 1 : 0; 151 152 if (x >= 6) { 153 if(x > 171.63) 154 return(one/zero); 155 u = large_gam(x); 156 return(__exp__D(u.a, u.b)); 157 } else if (x >= 1.0 + LEFT + x0) 158 return (small_gam(x)); 159 else if (x > 1.e-17) 160 return (smaller_gam(x)); 161 else if (x > -1.e-17) { 162 if (x == 0.0) 163 if (!_IEEE) return (infnan(ERANGE)); 164 else return (one/x); 165 one+1e-20; /* Raise inexact flag. */ 166 return (one/x); 167 } else if (!finite(x)) { 168 if (_IEEE) /* x = NaN, -Inf */ 169 return (x*x); 170 else 171 return (infnan(EDOM)); 172 } else 173 return (neg_gam(x)); 174 } 175 /* 176 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 177 */ 178 static struct Double 179 large_gam(x) 180 double x; 181 { 182 double z, p; 183 struct Double t, u, v; 184 185 z = one/(x*x); 186 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 187 p = p/x; 188 189 u = __log__D(x); 190 u.a -= one; 191 v.a = (x -= .5); 192 TRUNC(v.a); 193 v.b = x - v.a; 194 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 195 t.b = v.b*u.a + x*u.b; 196 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 197 t.b += lns2pi_lo; t.b += p; 198 u.a = lns2pi_hi + t.b; u.a += t.a; 199 u.b = t.a - u.a; 200 u.b += lns2pi_hi; u.b += t.b; 201 return (u); 202 } 203 /* 204 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 205 * It also has correct monotonicity. 206 */ 207 static double 208 small_gam(x) 209 double x; 210 { 211 double y, ym1, t; 212 struct Double yy, r; 213 y = x - one; 214 ym1 = y - one; 215 if (y <= 1.0 + (LEFT + x0)) { 216 yy = ratfun_gam(y - x0, 0); 217 return (yy.a + yy.b); 218 } 219 r.a = y; 220 TRUNC(r.a); 221 yy.a = r.a - one; 222 y = ym1; 223 yy.b = r.b = y - yy.a; 224 /* Argument reduction: G(x+1) = x*G(x) */ 225 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 226 t = r.a*yy.a; 227 r.b = r.a*yy.b + y*r.b; 228 r.a = t; 229 TRUNC(r.a); 230 r.b += (t - r.a); 231 } 232 /* Return r*gamma(y). */ 233 yy = ratfun_gam(y - x0, 0); 234 y = r.b*(yy.a + yy.b) + r.a*yy.b; 235 y += yy.a*r.a; 236 return (y); 237 } 238 /* 239 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 240 */ 241 static double 242 smaller_gam(x) 243 double x; 244 { 245 double t, d; 246 struct Double r, xx; 247 if (x < x0 + LEFT) { 248 t = x, TRUNC(t); 249 d = (t+x)*(x-t); 250 t *= t; 251 xx.a = (t + x), TRUNC(xx.a); 252 xx.b = x - xx.a; xx.b += t; xx.b += d; 253 t = (one-x0); t += x; 254 d = (one-x0); d -= t; d += x; 255 x = xx.a + xx.b; 256 } else { 257 xx.a = x, TRUNC(xx.a); 258 xx.b = x - xx.a; 259 t = x - x0; 260 d = (-x0 -t); d += x; 261 } 262 r = ratfun_gam(t, d); 263 d = r.a/x, TRUNC(d); 264 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 265 return (d + r.a/x); 266 } 267 /* 268 * returns (z+c)^2 * P(z)/Q(z) + a0 269 */ 270 static struct Double 271 ratfun_gam(z, c) 272 double z, c; 273 { 274 double p, q; 275 struct Double r, t; 276 277 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 278 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 279 280 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 281 p = p/q; 282 t.a = z, TRUNC(t.a); /* t ~= z + c */ 283 t.b = (z - t.a) + c; 284 t.b *= (t.a + z); 285 q = (t.a *= t.a); /* t = (z+c)^2 */ 286 TRUNC(t.a); 287 t.b += (q - t.a); 288 r.a = p, TRUNC(r.a); /* r = P/Q */ 289 r.b = p - r.a; 290 t.b = t.b*p + t.a*r.b + a0_lo; 291 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 292 r.a = t.a + a0_hi, TRUNC(r.a); 293 r.b = ((a0_hi-r.a) + t.a) + t.b; 294 return (r); /* r = a0 + t */ 295 } 296 297 static double 298 neg_gam(x) 299 double x; 300 { 301 int sgn = 1; 302 struct Double lg, lsine; 303 double y, z; 304 305 y = floor(x + .5); 306 if (y == x) /* Negative integer. */ 307 if(!_IEEE) 308 return (infnan(ERANGE)); 309 else 310 return (one/zero); 311 z = fabs(x - y); 312 y = .5*ceil(x); 313 if (y == ceil(y)) 314 sgn = -1; 315 if (z < .25) 316 z = sin(M_PI*z); 317 else 318 z = cos(M_PI*(0.5-z)); 319 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 320 if (x < -170) { 321 if (x < -190) 322 return ((double)sgn*tiny*tiny); 323 y = one - x; /* exact: 128 < |x| < 255 */ 324 lg = large_gam(y); 325 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 326 lg.a -= lsine.a; /* exact (opposite signs) */ 327 lg.b -= lsine.b; 328 y = -(lg.a + lg.b); 329 z = (y + lg.a) + lg.b; 330 y = __exp__D(y, z); 331 if (sgn < 0) y = -y; 332 return (y); 333 } 334 y = one-x; 335 if (one-y == x) 336 y = gamma(y); 337 else /* 1-x is inexact */ 338 y = -x*gamma(-x); 339 if (sgn < 0) y = -y; 340 return (M_PI / (y*z)); 341 } 342 343 #endif 344