1 /* $OpenBSD: b_tgamma.c,v 1.3 2009/10/27 23:59:29 deraadt 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 /* 32 * This code by P. McIlroy, Oct 1992; 33 * 34 * The financial support of UUNET Communications Services is greatfully 35 * acknowledged. 36 */ 37 38 #include "math.h" 39 #include "math_private.h" 40 41 /* METHOD: 42 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) 43 * At negative integers, return NaN and raise invalid. 44 * 45 * x < 6.5: 46 * Use argument reduction G(x+1) = xG(x) to reach the 47 * range [1.066124,2.066124]. Use a rational 48 * approximation centered at the minimum (x0+1) to 49 * ensure monotonicity. 50 * 51 * x >= 6.5: Use the asymptotic approximation (Stirling's formula) 52 * adjusted for equal-ripples: 53 * 54 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x)) 55 * 56 * Keep extra precision in multiplying (x-.5)(log(x)-1), to 57 * avoid premature round-off. 58 * 59 * Special values: 60 * -Inf: return NaN and raise invalid; 61 * negative integer: return NaN and raise invalid; 62 * other x ~< -177.79: return +-0 and raise underflow; 63 * +-0: return +-Inf and raise divide-by-zero; 64 * finite x ~> 171.63: return +Inf and raise overflow; 65 * +Inf: return +Inf; 66 * NaN: return NaN. 67 * 68 * Accuracy: tgamma(x) is accurate to within 69 * x > 0: error provably < 0.9ulp. 70 * Maximum observed in 1,000,000 trials was .87ulp. 71 * x < 0: 72 * Maximum observed error < 4ulp in 1,000,000 trials. 73 */ 74 75 static double neg_gam(double); 76 static double small_gam(double); 77 static double smaller_gam(double); 78 static struct Double large_gam(double); 79 static struct Double ratfun_gam(double, double); 80 81 /* 82 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval 83 * [1.066.., 2.066..] accurate to 4.25e-19. 84 */ 85 #define LEFT -.3955078125 /* left boundary for rat. approx */ 86 #define x0 .461632144968362356785 /* xmin - 1 */ 87 88 #define a0_hi 0.88560319441088874992 89 #define a0_lo -.00000000000000004996427036469019695 90 #define P0 6.21389571821820863029017800727e-01 91 #define P1 2.65757198651533466104979197553e-01 92 #define P2 5.53859446429917461063308081748e-03 93 #define P3 1.38456698304096573887145282811e-03 94 #define P4 2.40659950032711365819348969808e-03 95 #define Q0 1.45019531250000000000000000000e+00 96 #define Q1 1.06258521948016171343454061571e+00 97 #define Q2 -2.07474561943859936441469926649e-01 98 #define Q3 -1.46734131782005422506287573015e-01 99 #define Q4 3.07878176156175520361557573779e-02 100 #define Q5 5.12449347980666221336054633184e-03 101 #define Q6 -1.76012741431666995019222898833e-03 102 #define Q7 9.35021023573788935372153030556e-05 103 #define Q8 6.13275507472443958924745652239e-06 104 /* 105 * Constants for large x approximation (x in [6, Inf]) 106 * (Accurate to 2.8*10^-19 absolute) 107 */ 108 #define lns2pi_hi 0.418945312500000 109 #define lns2pi_lo -.000006779295327258219670263595 110 #define Pa0 8.33333333333333148296162562474e-02 111 #define Pa1 -2.77777777774548123579378966497e-03 112 #define Pa2 7.93650778754435631476282786423e-04 113 #define Pa3 -5.95235082566672847950717262222e-04 114 #define Pa4 8.41428560346653702135821806252e-04 115 #define Pa5 -1.89773526463879200348872089421e-03 116 #define Pa6 5.69394463439411649408050664078e-03 117 #define Pa7 -1.44705562421428915453880392761e-02 118 119 static const double zero = 0., one = 1.0, tiny = 1e-300; 120 121 double 122 tgamma(double x) 123 { 124 struct Double u; 125 126 if (x >= 6) { 127 if(x > 171.63) 128 return(x/zero); 129 u = large_gam(x); 130 return(__exp__D(u.a, u.b)); 131 } else if (x >= 1.0 + LEFT + x0) 132 return (small_gam(x)); 133 else if (x > 1.e-17) 134 return (smaller_gam(x)); 135 else if (x > -1.e-17) { 136 if (x != 0.0) 137 u.a = one - tiny; /* raise inexact */ 138 return (one/x); 139 } else if (!finite(x)) { 140 return (x - x); /* x = NaN, -Inf */ 141 } else 142 return (neg_gam(x)); 143 } 144 145 /* 146 * We simply call tgamma() rather than bloating the math library 147 * with a float-optimized version of it. The reason is that tgammaf() 148 * is essentially useless, since the function is superexponential 149 * and floats have very limited range. -- das@freebsd.org 150 */ 151 152 float 153 tgammaf(float x) 154 { 155 return tgamma(x); 156 } 157 158 /* 159 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 160 */ 161 162 static struct Double 163 large_gam(double x) 164 { 165 double z, p; 166 struct Double t, u, v; 167 168 z = one/(x*x); 169 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7)))))); 170 p = p/x; 171 172 u = __log__D(x); 173 u.a -= one; 174 v.a = (x -= .5); 175 TRUNC(v.a); 176 v.b = x - v.a; 177 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ 178 t.b = v.b*u.a + x*u.b; 179 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ 180 t.b += lns2pi_lo; t.b += p; 181 u.a = lns2pi_hi + t.b; u.a += t.a; 182 u.b = t.a - u.a; 183 u.b += lns2pi_hi; u.b += t.b; 184 return (u); 185 } 186 187 /* 188 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 189 * It also has correct monotonicity. 190 */ 191 192 static double 193 small_gam(double x) 194 { 195 double y, ym1, t; 196 struct Double yy, r; 197 y = x - one; 198 ym1 = y - one; 199 if (y <= 1.0 + (LEFT + x0)) { 200 yy = ratfun_gam(y - x0, 0); 201 return (yy.a + yy.b); 202 } 203 r.a = y; 204 TRUNC(r.a); 205 yy.a = r.a - one; 206 y = ym1; 207 yy.b = r.b = y - yy.a; 208 /* Argument reduction: G(x+1) = x*G(x) */ 209 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { 210 t = r.a*yy.a; 211 r.b = r.a*yy.b + y*r.b; 212 r.a = t; 213 TRUNC(r.a); 214 r.b += (t - r.a); 215 } 216 /* Return r*tgamma(y). */ 217 yy = ratfun_gam(y - x0, 0); 218 y = r.b*(yy.a + yy.b) + r.a*yy.b; 219 y += yy.a*r.a; 220 return (y); 221 } 222 223 /* 224 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp. 225 */ 226 227 static double 228 smaller_gam(double x) 229 { 230 double t, d; 231 struct Double r, xx; 232 if (x < x0 + LEFT) { 233 t = x; 234 TRUNC(t); 235 d = (t+x)*(x-t); 236 t *= t; 237 xx.a = (t + x); 238 TRUNC(xx.a); 239 xx.b = x - xx.a; xx.b += t; xx.b += d; 240 t = (one-x0); t += x; 241 d = (one-x0); d -= t; d += x; 242 x = xx.a + xx.b; 243 } else { 244 xx.a = x; 245 TRUNC(xx.a); 246 xx.b = x - xx.a; 247 t = x - x0; 248 d = (-x0 -t); d += x; 249 } 250 r = ratfun_gam(t, d); 251 d = r.a/x; 252 TRUNC(d); 253 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; 254 return (d + r.a/x); 255 } 256 257 /* 258 * returns (z+c)^2 * P(z)/Q(z) + a0 259 */ 260 261 static struct Double 262 ratfun_gam(double z, double c) 263 { 264 double p, q; 265 struct Double r, t; 266 267 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8))))))); 268 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4))); 269 270 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ 271 p = p/q; 272 t.a = z; 273 TRUNC(t.a); /* t ~= z + c */ 274 t.b = (z - t.a) + c; 275 t.b *= (t.a + z); 276 q = (t.a *= t.a); /* t = (z+c)^2 */ 277 TRUNC(t.a); 278 t.b += (q - t.a); 279 r.a = p; 280 TRUNC(r.a); /* r = P/Q */ 281 r.b = p - r.a; 282 t.b = t.b*p + t.a*r.b + a0_lo; 283 t.a *= r.a; /* t = (z+c)^2*(P/Q) */ 284 r.a = t.a + a0_hi; 285 TRUNC(r.a); 286 r.b = ((a0_hi-r.a) + t.a) + t.b; 287 return (r); /* r = a0 + t */ 288 } 289 290 static double 291 neg_gam(double x) 292 { 293 int sgn = 1; 294 struct Double lg, lsine; 295 double y, z; 296 297 y = ceil(x); 298 if (y == x) /* Negative integer. */ 299 return ((x - x) / zero); 300 z = y - x; 301 if (z > 0.5) 302 z = one - z; 303 y = 0.5 * y; 304 if (y == ceil(y)) 305 sgn = -1; 306 if (z < .25) 307 z = sin(M_PI*z); 308 else 309 z = cos(M_PI*(0.5-z)); 310 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 311 if (x < -170) { 312 if (x < -190) 313 return ((double)sgn*tiny*tiny); 314 y = one - x; /* exact: 128 < |x| < 255 */ 315 lg = large_gam(y); 316 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */ 317 lg.a -= lsine.a; /* exact (opposite signs) */ 318 lg.b -= lsine.b; 319 y = -(lg.a + lg.b); 320 z = (y + lg.a) + lg.b; 321 y = __exp__D(y, z); 322 if (sgn < 0) y = -y; 323 return (y); 324 } 325 y = one-x; 326 if (one-y == x) 327 y = tgamma(y); 328 else /* 1-x is inexact */ 329 y = -x*tgamma(-x); 330 if (sgn < 0) y = -y; 331 return (M_PI / (y*z)); 332 } 333