1 /* $OpenBSD: b_tgamma.c,v 1.4 2011/07/06 00:02:42 martynas Exp $ */ 2 /*- 3 * Copyright (c) 1992, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. Neither the name of the University nor the names of its contributors 15 * may be used to endorse or promote products derived from this software 16 * without specific prior written permission. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28 * SUCH DAMAGE. 29 */ 30 31 /* LINTLIBRARY */ 32 33 /* 34 * This code by P. McIlroy, Oct 1992; 35 * 36 * The financial support of UUNET Communications Services is greatfully 37 * acknowledged. 38 */ 39 40 #include <sys/cdefs.h> 41 #include <float.h> 42 #include <math.h> 43 44 #include "math_private.h" 45 46 /* METHOD: 47 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 48 * At negative integers, return NaN and raise invalid. 49 * 50 * x < 6.5: 51 * Use argument reduction G(x+1) = xG(x) to reach the 52 * range [1.066124,2.066124]. Use a rational 53 * approximation centered at the minimum (x0+1) to 54 * ensure monotonicity. 55 * 56 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 57 * adjusted for equal-ripples: 58 * 59 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 60 * 61 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 62 * avoid premature round-off. 63 * 64 * Special values: 65 * -Inf: return NaN and raise invalid; 66 * negative integer: return NaN and raise invalid; 67 * other x ~< -177.79: return +-0 and raise underflow; 68 * +-0: return +-Inf and raise divide-by-zero; 69 * finite x ~> 171.63: return +Inf and raise overflow; 70 * +Inf: return +Inf; 71 * NaN: return NaN. 72 * 73 * Accuracy: tgamma(x) is accurate to within 74 * x > 0: error provably < 0.9ulp. 75 * Maximum observed in 1,000,000 trials was .87ulp. 76 * x < 0: 77 * Maximum observed error < 4ulp in 1,000,000 trials. 78 */ 79 80 static double neg_gam(double); 81 static double small_gam(double); 82 static double smaller_gam(double); 83 static struct Double large_gam(double); 84 static struct Double ratfun_gam(double, double); 85 86 /* 87 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 88 * [1.066.., 2.066..] accurate to 4.25e-19. 89 */ 90 #define LEFT -.3955078125 /* left boundary for rat. approx */ 91 #define x0 .461632144968362356785 /* xmin - 1 */ 92 93 #define a0_hi 0.88560319441088874992 94 #define a0_lo -.00000000000000004996427036469019695 95 #define P0 6.21389571821820863029017800727e-01 96 #define P1 2.65757198651533466104979197553e-01 97 #define P2 5.53859446429917461063308081748e-03 98 #define P3 1.38456698304096573887145282811e-03 99 #define P4 2.40659950032711365819348969808e-03 100 #define Q0 1.45019531250000000000000000000e+00 101 #define Q1 1.06258521948016171343454061571e+00 102 #define Q2 -2.07474561943859936441469926649e-01 103 #define Q3 -1.46734131782005422506287573015e-01 104 #define Q4 3.07878176156175520361557573779e-02 105 #define Q5 5.12449347980666221336054633184e-03 106 #define Q6 -1.76012741431666995019222898833e-03 107 #define Q7 9.35021023573788935372153030556e-05 108 #define Q8 6.13275507472443958924745652239e-06 109 /* 110 * Constants for large x approximation (x in [6, Inf]) 111 * (Accurate to 2.8*10^-19 absolute) 112 */ 113 #define lns2pi_hi 0.418945312500000 114 #define lns2pi_lo -.000006779295327258219670263595 115 #define Pa0 8.33333333333333148296162562474e-02 116 #define Pa1 -2.77777777774548123579378966497e-03 117 #define Pa2 7.93650778754435631476282786423e-04 118 #define Pa3 -5.95235082566672847950717262222e-04 119 #define Pa4 8.41428560346653702135821806252e-04 120 #define Pa5 -1.89773526463879200348872089421e-03 121 #define Pa6 5.69394463439411649408050664078e-03 122 #define Pa7 -1.44705562421428915453880392761e-02 123 124 static const double zero = 0., one = 1.0, tiny = 1e-300; 125 126 double 127 tgamma(double x) 128 { 129 struct Double u; 130 131 if (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 (x >= 1.0 + LEFT + x0) 137 return (small_gam(x)); 138 else if (x > 1.e-17) 139 return (smaller_gam(x)); 140 else if (x > -1.e-17) { 141 if (x != 0.0) 142 u.a = one - tiny; /* raise inexact */ 143 return (one/x); 144 } else if (!finite(x)) { 145 return (x - x); /* x = NaN, -Inf */ 146 } else 147 return (neg_gam(x)); 148 } 149 150 /* 151 * We simply call tgamma() rather than bloating the math library 152 * with a float-optimized version of it. The reason is that tgammaf() 153 * is essentially useless, since the function is superexponential 154 * and floats have very limited range. -- das@freebsd.org 155 */ 156 157 float 158 tgammaf(float x) 159 { 160 return tgamma(x); 161 } 162 163 /* 164 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 165 */ 166 167 static struct Double 168 large_gam(double x) 169 { 170 double z, p; 171 struct Double t, u, v; 172 173 z = one/(x*x); 174 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 175 p = p/x; 176 177 u = __log__D(x); 178 u.a -= one; 179 v.a = (x -= .5); 180 TRUNC(v.a); 181 v.b = x - v.a; 182 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 183 t.b = v.b*u.a + x*u.b; 184 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 185 t.b += lns2pi_lo; t.b += p; 186 u.a = lns2pi_hi + t.b; u.a += t.a; 187 u.b = t.a - u.a; 188 u.b += lns2pi_hi; u.b += t.b; 189 return (u); 190 } 191 192 /* 193 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 194 * It also has correct monotonicity. 195 */ 196 197 static double 198 small_gam(double x) 199 { 200 double y, ym1, t; 201 struct Double yy, r; 202 y = x - one; 203 ym1 = y - one; 204 if (y <= 1.0 + (LEFT + x0)) { 205 yy = ratfun_gam(y - x0, 0); 206 return (yy.a + yy.b); 207 } 208 r.a = y; 209 TRUNC(r.a); 210 yy.a = r.a - one; 211 y = ym1; 212 yy.b = r.b = y - yy.a; 213 /* Argument reduction: G(x+1) = x*G(x) */ 214 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 215 t = r.a*yy.a; 216 r.b = r.a*yy.b + y*r.b; 217 r.a = t; 218 TRUNC(r.a); 219 r.b += (t - r.a); 220 } 221 /* Return r*tgamma(y). */ 222 yy = ratfun_gam(y - x0, 0); 223 y = r.b*(yy.a + yy.b) + r.a*yy.b; 224 y += yy.a*r.a; 225 return (y); 226 } 227 228 /* 229 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 230 */ 231 232 static double 233 smaller_gam(double x) 234 { 235 double t, d; 236 struct Double r, xx; 237 if (x < x0 + LEFT) { 238 t = x; 239 TRUNC(t); 240 d = (t+x)*(x-t); 241 t *= t; 242 xx.a = (t + x); 243 TRUNC(xx.a); 244 xx.b = x - xx.a; xx.b += t; xx.b += d; 245 t = (one-x0); t += x; 246 d = (one-x0); d -= t; d += x; 247 x = xx.a + xx.b; 248 } else { 249 xx.a = x; 250 TRUNC(xx.a); 251 xx.b = x - xx.a; 252 t = x - x0; 253 d = (-x0 -t); d += x; 254 } 255 r = ratfun_gam(t, d); 256 d = r.a/x; 257 TRUNC(d); 258 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 259 return (d + r.a/x); 260 } 261 262 /* 263 * returns (z+c)^2 * P(z)/Q(z) + a0 264 */ 265 266 static struct Double 267 ratfun_gam(double z, double c) 268 { 269 double p, q; 270 struct Double r, t; 271 272 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 273 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 274 275 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 276 p = p/q; 277 t.a = z; 278 TRUNC(t.a); /* t ~= z + c */ 279 t.b = (z - t.a) + c; 280 t.b *= (t.a + z); 281 q = (t.a *= t.a); /* t = (z+c)^2 */ 282 TRUNC(t.a); 283 t.b += (q - t.a); 284 r.a = p; 285 TRUNC(r.a); /* r = P/Q */ 286 r.b = p - r.a; 287 t.b = t.b*p + t.a*r.b + a0_lo; 288 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 289 r.a = t.a + a0_hi; 290 TRUNC(r.a); 291 r.b = ((a0_hi-r.a) + t.a) + t.b; 292 return (r); /* r = a0 + t */ 293 } 294 295 static double 296 neg_gam(double x) 297 { 298 int sgn = 1; 299 struct Double lg, lsine; 300 double y, z; 301 302 y = ceil(x); 303 if (y == x) /* Negative integer. */ 304 return ((x - x) / zero); 305 z = y - x; 306 if (z > 0.5) 307 z = one - z; 308 y = 0.5 * y; 309 if (y == ceil(y)) 310 sgn = -1; 311 if (z < .25) 312 z = sin(M_PI*z); 313 else 314 z = cos(M_PI*(0.5-z)); 315 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 316 if (x < -170) { 317 if (x < -190) 318 return ((double)sgn*tiny*tiny); 319 y = one - x; /* exact: 128 < |x| < 255 */ 320 lg = large_gam(y); 321 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 322 lg.a -= lsine.a; /* exact (opposite signs) */ 323 lg.b -= lsine.b; 324 y = -(lg.a + lg.b); 325 z = (y + lg.a) + lg.b; 326 y = __exp__D(y, z); 327 if (sgn < 0) y = -y; 328 return (y); 329 } 330 y = one-x; 331 if (one-y == x) 332 y = tgamma(y); 333 else /* 1-x is inexact */ 334 y = -x*tgamma(-x); 335 if (sgn < 0) y = -y; 336 return (M_PI / (y*z)); 337 } 338 339 #if LDBL_MANT_DIG == 53 340 #ifdef lint 341 /* PROTOLIB1 */ 342 long double tgammal(long double); 343 #else /* lint */ 344 __weak_alias(tgammal, tgamma); 345 #endif /* lint */ 346 #endif /* LDBL_MANT_DIG == 53 */ 347