1 /*- 2 * Copyright (c) 1992, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* 13 * This code by P. McIlroy, Oct 1992; 14 * 15 * The financial support of UUNET Communications Services is greatfully 16 * acknowledged. 17 */ 18 19 #include <math.h> 20 #include "mathimpl.h" 21 #include <errno.h> 22 23 /* METHOD: 24 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 25 * At negative integers, return +Inf, and set errno. 26 * 27 * x < 6.5: 28 * Use argument reduction G(x+1) = xG(x) to reach the 29 * range [1.066124,2.066124]. Use a rational 30 * approximation centered at the minimum (x0+1) to 31 * ensure monotonicity. 32 * 33 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 34 * adjusted for equal-ripples: 35 * 36 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 37 * 38 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 39 * avoid premature round-off. 40 * 41 * Special values: 42 * non-positive integer: Set overflow trap; return +Inf; 43 * x > 171.63: Set overflow trap; return +Inf; 44 * NaN: Set invalid trap; return NaN 45 * 46 * Accuracy: Gamma(x) is accurate to within 47 * x > 0: error provably < 0.9ulp. 48 * Maximum observed in 1,000,000 trials was .87ulp. 49 * x < 0: 50 * Maximum observed error < 4ulp in 1,000,000 trials. 51 */ 52 53 static double neg_gam __P((double)); 54 static double small_gam __P((double)); 55 static double smaller_gam __P((double)); 56 static struct Double large_gam __P((double)); 57 static struct Double ratfun_gam __P((double, double)); 58 59 /* 60 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 61 * [1.066.., 2.066..] accurate to 4.25e-19. 62 */ 63 #define LEFT -.3955078125 /* left boundary for rat. approx */ 64 #define x0 .461632144968362356785 /* xmin - 1 */ 65 66 #define a0_hi 0.88560319441088874992 67 #define a0_lo -.00000000000000004996427036469019695 68 #define P0 6.21389571821820863029017800727e-01 69 #define P1 2.65757198651533466104979197553e-01 70 #define P2 5.53859446429917461063308081748e-03 71 #define P3 1.38456698304096573887145282811e-03 72 #define P4 2.40659950032711365819348969808e-03 73 #define Q0 1.45019531250000000000000000000e+00 74 #define Q1 1.06258521948016171343454061571e+00 75 #define Q2 -2.07474561943859936441469926649e-01 76 #define Q3 -1.46734131782005422506287573015e-01 77 #define Q4 3.07878176156175520361557573779e-02 78 #define Q5 5.12449347980666221336054633184e-03 79 #define Q6 -1.76012741431666995019222898833e-03 80 #define Q7 9.35021023573788935372153030556e-05 81 #define Q8 6.13275507472443958924745652239e-06 82 /* 83 * Constants for large x approximation (x in [6, Inf]) 84 * (Accurate to 2.8*10^-19 absolute) 85 */ 86 #define lns2pi_hi 0.418945312500000 87 #define lns2pi_lo -.000006779295327258219670263595 88 #define Pa0 8.33333333333333148296162562474e-02 89 #define Pa1 -2.77777777774548123579378966497e-03 90 #define Pa2 7.93650778754435631476282786423e-04 91 #define Pa3 -5.95235082566672847950717262222e-04 92 #define Pa4 8.41428560346653702135821806252e-04 93 #define Pa5 -1.89773526463879200348872089421e-03 94 #define Pa6 5.69394463439411649408050664078e-03 95 #define Pa7 -1.44705562421428915453880392761e-02 96 97 static const double zero = 0., one = 1.0, tiny = 1e-300; 98 static int endian; 99 /* 100 * TRUNC sets trailing bits in a floating-point number to zero. 101 * is a temporary variable. 102 */ 103 #if defined(vax) || defined(tahoe) 104 #define _IEEE 0 105 #define TRUNC(x) x = (double) (float) (x) 106 #else 107 #define _IEEE 1 108 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 109 #define infnan(x) 0.0 110 #endif 111 112 double 113 gamma(x) 114 double x; 115 { 116 struct Double u; 117 endian = (*(int *) &one) ? 1 : 0; 118 119 if (x >= 6) { 120 if(x > 171.63) 121 return(one/zero); 122 u = large_gam(x); 123 return(__exp__D(u.a, u.b)); 124 } else if (x >= 1.0 + LEFT + x0) 125 return (small_gam(x)); 126 else if (x > 1.e-17) 127 return (smaller_gam(x)); 128 else if (x > -1.e-17) { 129 if (x == 0.0) 130 if (!_IEEE) return (infnan(ERANGE)); 131 else return (one/x); 132 one+1e-20; /* Raise inexact flag. */ 133 return (one/x); 134 } else if (!finite(x)) { 135 if (_IEEE) /* x = NaN, -Inf */ 136 return (x*x); 137 else 138 return (infnan(EDOM)); 139 } else 140 return (neg_gam(x)); 141 } 142 /* 143 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 144 */ 145 static struct Double 146 large_gam(x) 147 double x; 148 { 149 double z, p; 150 int i; 151 struct Double t, u, v; 152 153 z = one/(x*x); 154 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 155 p = p/x; 156 157 u = __log__D(x); 158 u.a -= one; 159 v.a = (x -= .5); 160 TRUNC(v.a); 161 v.b = x - v.a; 162 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 163 t.b = v.b*u.a + x*u.b; 164 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 165 t.b += lns2pi_lo; t.b += p; 166 u.a = lns2pi_hi + t.b; u.a += t.a; 167 u.b = t.a - u.a; 168 u.b += lns2pi_hi; u.b += t.b; 169 return (u); 170 } 171 /* 172 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 173 * It also has correct monotonicity. 174 */ 175 static double 176 small_gam(x) 177 double x; 178 { 179 double y, ym1, t, x1; 180 struct Double yy, r; 181 y = x - one; 182 ym1 = y - one; 183 if (y <= 1.0 + (LEFT + x0)) { 184 yy = ratfun_gam(y - x0, 0); 185 return (yy.a + yy.b); 186 } 187 r.a = y; 188 TRUNC(r.a); 189 yy.a = r.a - one; 190 y = ym1; 191 yy.b = r.b = y - yy.a; 192 /* Argument reduction: G(x+1) = x*G(x) */ 193 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 194 t = r.a*yy.a; 195 r.b = r.a*yy.b + y*r.b; 196 r.a = t; 197 TRUNC(r.a); 198 r.b += (t - r.a); 199 } 200 /* Return r*gamma(y). */ 201 yy = ratfun_gam(y - x0, 0); 202 y = r.b*(yy.a + yy.b) + r.a*yy.b; 203 y += yy.a*r.a; 204 return (y); 205 } 206 /* 207 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 208 */ 209 static double 210 smaller_gam(x) 211 double x; 212 { 213 double t, d; 214 struct Double r, xx; 215 if (x < x0 + LEFT) { 216 t = x, TRUNC(t); 217 d = (t+x)*(x-t); 218 t *= t; 219 xx.a = (t + x), TRUNC(xx.a); 220 xx.b = x - xx.a; xx.b += t; xx.b += d; 221 t = (one-x0); t += x; 222 d = (one-x0); d -= t; d += x; 223 x = xx.a + xx.b; 224 } else { 225 xx.a = x, TRUNC(xx.a); 226 xx.b = x - xx.a; 227 t = x - x0; 228 d = (-x0 -t); d += x; 229 } 230 r = ratfun_gam(t, d); 231 d = r.a/x, TRUNC(d); 232 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 233 return (d + r.a/x); 234 } 235 /* 236 * returns (z+c)^2 * P(z)/Q(z) + a0 237 */ 238 static struct Double 239 ratfun_gam(z, c) 240 double z, c; 241 { 242 int i; 243 double p, q; 244 struct Double r, t; 245 246 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 247 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 248 249 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 250 p = p/q; 251 t.a = z, TRUNC(t.a); /* t ~= z + c */ 252 t.b = (z - t.a) + c; 253 t.b *= (t.a + z); 254 q = (t.a *= t.a); /* t = (z+c)^2 */ 255 TRUNC(t.a); 256 t.b += (q - t.a); 257 r.a = p, TRUNC(r.a); /* r = P/Q */ 258 r.b = p - r.a; 259 t.b = t.b*p + t.a*r.b + a0_lo; 260 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 261 r.a = t.a + a0_hi, TRUNC(r.a); 262 r.b = ((a0_hi-r.a) + t.a) + t.b; 263 return (r); /* r = a0 + t */ 264 } 265 266 static double 267 neg_gam(x) 268 double x; 269 { 270 int sgn = 1; 271 struct Double lg, lsine; 272 double y, z; 273 274 y = floor(x + .5); 275 if (y == x) /* Negative integer. */ 276 if(!_IEEE) 277 return (infnan(ERANGE)); 278 else 279 return (one/zero); 280 z = fabs(x - y); 281 y = .5*ceil(x); 282 if (y == ceil(y)) 283 sgn = -1; 284 if (z < .25) 285 z = sin(M_PI*z); 286 else 287 z = cos(M_PI*(0.5-z)); 288 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 289 if (x < -170) { 290 if (x < -190) 291 return ((double)sgn*tiny*tiny); 292 y = one - x; /* exact: 128 < |x| < 255 */ 293 lg = large_gam(y); 294 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 295 lg.a -= lsine.a; /* exact (opposite signs) */ 296 lg.b -= lsine.b; 297 y = -(lg.a + lg.b); 298 z = (y + lg.a) + lg.b; 299 y = __exp__D(y, z); 300 if (sgn < 0) y = -y; 301 return (y); 302 } 303 y = one-x; 304 if (one-y == x) 305 y = gamma(y); 306 else /* 1-x is inexact */ 307 y = -x*gamma(-x); 308 if (sgn < 0) y = -y; 309 return (M_PI / (y*z)); 310 } 311