1 /*- 2 * SPDX-License-Identifier: BSD-3-Clause 3 * 4 * Copyright (c) 1992, 1993 5 * The Regents of the University of California. All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 3. Neither the name of the University nor the names of its contributors 16 * may be used to endorse or promote products derived from this software 17 * without specific prior written permission. 18 * 19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 29 * SUCH DAMAGE. 30 */ 31 32 /* 33 * The original code, FreeBSD's old svn r93211, contained the following 34 * attribution: 35 * 36 * This code by P. McIlroy, Oct 1992; 37 * 38 * The financial support of UUNET Communications Services is greatfully 39 * acknowledged. 40 * 41 * The algorithm remains, but the code has been re-arranged to facilitate 42 * porting to other precisions. 43 */ 44 45 /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */ 46 #include <sys/cdefs.h> 47 #include <float.h> 48 49 #include "math.h" 50 #include "math_private.h" 51 52 /* Used in b_log.c and below. */ 53 struct Double { 54 double a; 55 double b; 56 }; 57 58 #include "b_log.c" 59 #include "b_exp.c" 60 61 /* 62 * The range is broken into several subranges. Each is handled by its 63 * helper functions. 64 * 65 * x >= 6.0: large_gam(x) 66 * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0. 67 * xleft > x > iota: smaller_gam(x) where iota = 1e-17. 68 * iota > x > -itoa: Handle x near 0. 69 * -iota > x : neg_gam 70 * 71 * Special values: 72 * -Inf: return NaN and raise invalid; 73 * negative integer: return NaN and raise invalid; 74 * other x ~< 177.79: return +-0 and raise underflow; 75 * +-0: return +-Inf and raise divide-by-zero; 76 * finite x ~> 171.63: return +Inf and raise overflow; 77 * +Inf: return +Inf; 78 * NaN: return NaN. 79 * 80 * Accuracy: tgamma(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 /* 88 * Constants for large x approximation (x in [6, Inf]) 89 * (Accurate to 2.8*10^-19 absolute) 90 */ 91 92 static const double zero = 0.; 93 static const volatile double tiny = 1e-300; 94 /* 95 * x >= 6 96 * 97 * Use the asymptotic approximation (Stirling's formula) adjusted fof 98 * equal-ripples: 99 * 100 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) 101 * 102 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid 103 * premature round-off. 104 * 105 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 106 */ 107 static const double 108 ln2pi_hi = 0.41894531250000000, 109 ln2pi_lo = -6.7792953272582197e-6, 110 Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ 111 Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */ 112 Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */ 113 Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */ 114 Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */ 115 Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */ 116 Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */ 117 Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */ 118 119 static struct Double 120 large_gam(double x) 121 { 122 double p, z, thi, tlo, xhi, xlo; 123 struct Double u; 124 125 z = 1 / (x * x); 126 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + 127 z * (Pa6 + z * Pa7)))))); 128 p = p / x; 129 130 u = __log__D(x); 131 u.a -= 1; 132 133 /* Split (x - 0.5) in high and low parts. */ 134 x -= 0.5; 135 xhi = (float)x; 136 xlo = x - xhi; 137 138 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ 139 thi = xhi * u.a; 140 tlo = xlo * u.a + x * u.b; 141 142 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ 143 tlo += ln2pi_lo; 144 tlo += p; 145 u.a = ln2pi_hi + tlo; 146 u.a += thi; 147 u.b = thi - u.a; 148 u.b += ln2pi_hi; 149 u.b += tlo; 150 return (u); 151 } 152 /* 153 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval 154 * [1.066.., 2.066..] accurate to 4.25e-19. 155 * 156 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. 157 */ 158 static const double 159 #if 0 160 a0_hi = 8.8560319441088875e-1, 161 a0_lo = -4.9964270364690197e-17, 162 #else 163 a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */ 164 a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */ 165 #endif 166 P0 = 6.2138957182182086e-1, 167 P1 = 2.6575719865153347e-1, 168 P2 = 5.5385944642991746e-3, 169 P3 = 1.3845669830409657e-3, 170 P4 = 2.4065995003271137e-3, 171 Q0 = 1.4501953125000000e+0, 172 Q1 = 1.0625852194801617e+0, 173 Q2 = -2.0747456194385994e-1, 174 Q3 = -1.4673413178200542e-1, 175 Q4 = 3.0787817615617552e-2, 176 Q5 = 5.1244934798066622e-3, 177 Q6 = -1.7601274143166700e-3, 178 Q7 = 9.3502102357378894e-5, 179 Q8 = 6.1327550747244396e-6; 180 181 static struct Double 182 ratfun_gam(double z, double c) 183 { 184 double p, q, thi, tlo; 185 struct Double r; 186 187 q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + 188 z * (Q6 + z * (Q7 + z * Q8))))))); 189 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4))); 190 p = p / q; 191 192 /* Split z into high and low parts. */ 193 thi = (float)z; 194 tlo = (z - thi) + c; 195 tlo *= (thi + z); 196 197 /* Split (z+c)^2 into high and low parts. */ 198 thi *= thi; 199 q = thi; 200 thi = (float)thi; 201 tlo += (q - thi); 202 203 /* Split p/q into high and low parts. */ 204 r.a = (float)p; 205 r.b = p - r.a; 206 207 tlo = tlo * p + thi * r.b + a0_lo; 208 thi *= r.a; /* t = (z+c)^2*(P/Q) */ 209 r.a = (float)(thi + a0_hi); 210 r.b = ((a0_hi - r.a) + thi) + tlo; 211 return (r); /* r = a0 + t */ 212 } 213 /* 214 * x < 6 215 * 216 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, 217 * 2.066124]. Use a rational approximation centered at the minimum 218 * (x0+1) to ensure monotonicity. 219 * 220 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 221 * It also has correct monotonicity. 222 */ 223 static const double 224 left = -0.3955078125, /* left boundary for rat. approx */ 225 x0 = 4.6163214496836236e-1; /* xmin - 1 */ 226 227 static double 228 small_gam(double x) 229 { 230 double t, y, ym1; 231 struct Double yy, r; 232 233 y = x - 1; 234 if (y <= 1 + (left + x0)) { 235 yy = ratfun_gam(y - x0, 0); 236 return (yy.a + yy.b); 237 } 238 239 r.a = (float)y; 240 yy.a = r.a - 1; 241 y = y - 1 ; 242 r.b = yy.b = y - yy.a; 243 244 /* Argument reduction: G(x+1) = x*G(x) */ 245 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { 246 t = r.a * yy.a; 247 r.b = r.a * yy.b + y * r.b; 248 r.a = (float)t; 249 r.b += (t - r.a); 250 } 251 252 /* Return r*tgamma(y). */ 253 yy = ratfun_gam(y - x0, 0); 254 y = r.b * (yy.a + yy.b) + r.a * yy.b; 255 y += yy.a * r.a; 256 return (y); 257 } 258 /* 259 * Good on (0, 1+x0+left]. Accurate to 1 ulp. 260 */ 261 static double 262 smaller_gam(double x) 263 { 264 double d, rhi, rlo, t, xhi, xlo; 265 struct Double r; 266 267 if (x < x0 + left) { 268 t = (float)x; 269 d = (t + x) * (x - t); 270 t *= t; 271 xhi = (float)(t + x); 272 xlo = x - xhi; 273 xlo += t; 274 xlo += d; 275 t = 1 - x0; 276 t += x; 277 d = 1 - x0; 278 d -= t; 279 d += x; 280 x = xhi + xlo; 281 } else { 282 xhi = (float)x; 283 xlo = x - xhi; 284 t = x - x0; 285 d = - x0 - t; 286 d += x; 287 } 288 289 r = ratfun_gam(t, d); 290 d = (float)(r.a / x); 291 r.a -= d * xhi; 292 r.a -= d * xlo; 293 r.a += r.b; 294 295 return (d + r.a / x); 296 } 297 /* 298 * x < 0 299 * 300 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). 301 * At negative integers, return NaN and raise invalid. 302 */ 303 static double 304 neg_gam(double x) 305 { 306 int sgn = 1; 307 struct Double lg, lsine; 308 double y, z; 309 310 y = ceil(x); 311 if (y == x) /* Negative integer. */ 312 return ((x - x) / zero); 313 314 z = y - x; 315 if (z > 0.5) 316 z = 1 - z; 317 318 y = y / 2; 319 if (y == ceil(y)) 320 sgn = -1; 321 322 if (z < 0.25) 323 z = sinpi(z); 324 else 325 z = cospi(0.5 - z); 326 327 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 328 if (x < -170) { 329 330 if (x < -190) 331 return (sgn * tiny * tiny); 332 333 y = 1 - x; /* exact: 128 < |x| < 255 */ 334 lg = large_gam(y); 335 lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */ 336 lg.a -= lsine.a; /* exact (opposite signs) */ 337 lg.b -= lsine.b; 338 y = -(lg.a + lg.b); 339 z = (y + lg.a) + lg.b; 340 y = __exp__D(y, z); 341 if (sgn < 0) y = -y; 342 return (y); 343 } 344 345 y = 1 - x; 346 if (1 - y == x) 347 y = tgamma(y); 348 else /* 1-x is inexact */ 349 y = - x * tgamma(-x); 350 351 if (sgn < 0) y = -y; 352 return (M_PI / (y * z)); 353 } 354 /* 355 * xmax comes from lgamma(xmax) - emax * log(2) = 0. 356 * static const float xmax = 35.040095f 357 * static const double xmax = 171.624376956302725; 358 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), 359 * ld128: 1.75554834290446291700388921607020320e+03L, 360 * 361 * iota is a sloppy threshold to isolate x = 0. 362 */ 363 static const double xmax = 171.624376956302725; 364 static const double iota = 0x1p-56; 365 366 double 367 tgamma(double x) 368 { 369 struct Double u; 370 371 if (x >= 6) { 372 if (x > xmax) 373 return (x / zero); 374 u = large_gam(x); 375 return (__exp__D(u.a, u.b)); 376 } 377 378 if (x >= 1 + left + x0) 379 return (small_gam(x)); 380 381 if (x > iota) 382 return (smaller_gam(x)); 383 384 if (x > -iota) { 385 if (x != 0.) 386 u.a = 1 - tiny; /* raise inexact */ 387 return (1 / x); 388 } 389 390 if (!isfinite(x)) 391 return (x - x); /* x is NaN or -Inf */ 392 393 return (neg_gam(x)); 394 } 395 396 #if (LDBL_MANT_DIG == 53) 397 __weak_reference(tgamma, tgammal); 398 #endif 399