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 __FBSDID("$FreeBSD$"); 48 49 #include <float.h> 50 51 #include "math.h" 52 #include "math_private.h" 53 54 /* Used in b_log.c and below. */ 55 struct Double { 56 double a; 57 double b; 58 }; 59 60 #include "b_log.c" 61 #include "b_exp.c" 62 63 /* 64 * The range is broken into several subranges. Each is handled by its 65 * helper functions. 66 * 67 * x >= 6.0: large_gam(x) 68 * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0. 69 * xleft > x > iota: smaller_gam(x) where iota = 1e-17. 70 * iota > x > -itoa: Handle x near 0. 71 * -iota > x : neg_gam 72 * 73 * Special values: 74 * -Inf: return NaN and raise invalid; 75 * negative integer: return NaN and raise invalid; 76 * other x ~< 177.79: return +-0 and raise underflow; 77 * +-0: return +-Inf and raise divide-by-zero; 78 * finite x ~> 171.63: return +Inf and raise overflow; 79 * +Inf: return +Inf; 80 * NaN: return NaN. 81 * 82 * Accuracy: tgamma(x) is accurate to within 83 * x > 0: error provably < 0.9ulp. 84 * Maximum observed in 1,000,000 trials was .87ulp. 85 * x < 0: 86 * Maximum observed error < 4ulp in 1,000,000 trials. 87 */ 88 89 /* 90 * Constants for large x approximation (x in [6, Inf]) 91 * (Accurate to 2.8*10^-19 absolute) 92 */ 93 94 static const double zero = 0.; 95 static const volatile double tiny = 1e-300; 96 /* 97 * x >= 6 98 * 99 * Use the asymptotic approximation (Stirling's formula) adjusted fof 100 * equal-ripples: 101 * 102 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) 103 * 104 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid 105 * premature round-off. 106 * 107 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 108 */ 109 static const double 110 ln2pi_hi = 0.41894531250000000, 111 ln2pi_lo = -6.7792953272582197e-6, 112 Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ 113 Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */ 114 Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */ 115 Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */ 116 Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */ 117 Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */ 118 Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */ 119 Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */ 120 121 static struct Double 122 large_gam(double x) 123 { 124 double p, z, thi, tlo, xhi, xlo; 125 struct Double u; 126 127 z = 1 / (x * x); 128 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + 129 z * (Pa6 + z * Pa7)))))); 130 p = p / x; 131 132 u = __log__D(x); 133 u.a -= 1; 134 135 /* Split (x - 0.5) in high and low parts. */ 136 x -= 0.5; 137 xhi = (float)x; 138 xlo = x - xhi; 139 140 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ 141 thi = xhi * u.a; 142 tlo = xlo * u.a + x * u.b; 143 144 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ 145 tlo += ln2pi_lo; 146 tlo += p; 147 u.a = ln2pi_hi + tlo; 148 u.a += thi; 149 u.b = thi - u.a; 150 u.b += ln2pi_hi; 151 u.b += tlo; 152 return (u); 153 } 154 /* 155 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval 156 * [1.066.., 2.066..] accurate to 4.25e-19. 157 * 158 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. 159 */ 160 static const double 161 #if 0 162 a0_hi = 8.8560319441088875e-1, 163 a0_lo = -4.9964270364690197e-17, 164 #else 165 a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */ 166 a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */ 167 #endif 168 P0 = 6.2138957182182086e-1, 169 P1 = 2.6575719865153347e-1, 170 P2 = 5.5385944642991746e-3, 171 P3 = 1.3845669830409657e-3, 172 P4 = 2.4065995003271137e-3, 173 Q0 = 1.4501953125000000e+0, 174 Q1 = 1.0625852194801617e+0, 175 Q2 = -2.0747456194385994e-1, 176 Q3 = -1.4673413178200542e-1, 177 Q4 = 3.0787817615617552e-2, 178 Q5 = 5.1244934798066622e-3, 179 Q6 = -1.7601274143166700e-3, 180 Q7 = 9.3502102357378894e-5, 181 Q8 = 6.1327550747244396e-6; 182 183 static struct Double 184 ratfun_gam(double z, double c) 185 { 186 double p, q, thi, tlo; 187 struct Double r; 188 189 q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + 190 z * (Q6 + z * (Q7 + z * Q8))))))); 191 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4))); 192 p = p / q; 193 194 /* Split z into high and low parts. */ 195 thi = (float)z; 196 tlo = (z - thi) + c; 197 tlo *= (thi + z); 198 199 /* Split (z+c)^2 into high and low parts. */ 200 thi *= thi; 201 q = thi; 202 thi = (float)thi; 203 tlo += (q - thi); 204 205 /* Split p/q into high and low parts. */ 206 r.a = (float)p; 207 r.b = p - r.a; 208 209 tlo = tlo * p + thi * r.b + a0_lo; 210 thi *= r.a; /* t = (z+c)^2*(P/Q) */ 211 r.a = (float)(thi + a0_hi); 212 r.b = ((a0_hi - r.a) + thi) + tlo; 213 return (r); /* r = a0 + t */ 214 } 215 /* 216 * x < 6 217 * 218 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, 219 * 2.066124]. Use a rational approximation centered at the minimum 220 * (x0+1) to ensure monotonicity. 221 * 222 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 223 * It also has correct monotonicity. 224 */ 225 static const double 226 left = -0.3955078125, /* left boundary for rat. approx */ 227 x0 = 4.6163214496836236e-1; /* xmin - 1 */ 228 229 static double 230 small_gam(double x) 231 { 232 double t, y, ym1; 233 struct Double yy, r; 234 235 y = x - 1; 236 if (y <= 1 + (left + x0)) { 237 yy = ratfun_gam(y - x0, 0); 238 return (yy.a + yy.b); 239 } 240 241 r.a = (float)y; 242 yy.a = r.a - 1; 243 y = y - 1 ; 244 r.b = yy.b = y - yy.a; 245 246 /* Argument reduction: G(x+1) = x*G(x) */ 247 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { 248 t = r.a * yy.a; 249 r.b = r.a * yy.b + y * r.b; 250 r.a = (float)t; 251 r.b += (t - r.a); 252 } 253 254 /* Return r*tgamma(y). */ 255 yy = ratfun_gam(y - x0, 0); 256 y = r.b * (yy.a + yy.b) + r.a * yy.b; 257 y += yy.a * r.a; 258 return (y); 259 } 260 /* 261 * Good on (0, 1+x0+left]. Accurate to 1 ulp. 262 */ 263 static double 264 smaller_gam(double x) 265 { 266 double d, rhi, rlo, t, xhi, xlo; 267 struct Double r; 268 269 if (x < x0 + left) { 270 t = (float)x; 271 d = (t + x) * (x - t); 272 t *= t; 273 xhi = (float)(t + x); 274 xlo = x - xhi; 275 xlo += t; 276 xlo += d; 277 t = 1 - x0; 278 t += x; 279 d = 1 - x0; 280 d -= t; 281 d += x; 282 x = xhi + xlo; 283 } else { 284 xhi = (float)x; 285 xlo = x - xhi; 286 t = x - x0; 287 d = - x0 - t; 288 d += x; 289 } 290 291 r = ratfun_gam(t, d); 292 d = (float)(r.a / x); 293 r.a -= d * xhi; 294 r.a -= d * xlo; 295 r.a += r.b; 296 297 return (d + r.a / x); 298 } 299 /* 300 * x < 0 301 * 302 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). 303 * At negative integers, return NaN and raise invalid. 304 */ 305 static double 306 neg_gam(double x) 307 { 308 int sgn = 1; 309 struct Double lg, lsine; 310 double y, z; 311 312 y = ceil(x); 313 if (y == x) /* Negative integer. */ 314 return ((x - x) / zero); 315 316 z = y - x; 317 if (z > 0.5) 318 z = 1 - z; 319 320 y = y / 2; 321 if (y == ceil(y)) 322 sgn = -1; 323 324 if (z < 0.25) 325 z = sinpi(z); 326 else 327 z = cospi(0.5 - z); 328 329 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 330 if (x < -170) { 331 332 if (x < -190) 333 return (sgn * tiny * tiny); 334 335 y = 1 - x; /* exact: 128 < |x| < 255 */ 336 lg = large_gam(y); 337 lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */ 338 lg.a -= lsine.a; /* exact (opposite signs) */ 339 lg.b -= lsine.b; 340 y = -(lg.a + lg.b); 341 z = (y + lg.a) + lg.b; 342 y = __exp__D(y, z); 343 if (sgn < 0) y = -y; 344 return (y); 345 } 346 347 y = 1 - x; 348 if (1 - y == x) 349 y = tgamma(y); 350 else /* 1-x is inexact */ 351 y = - x * tgamma(-x); 352 353 if (sgn < 0) y = -y; 354 return (M_PI / (y * z)); 355 } 356 /* 357 * xmax comes from lgamma(xmax) - emax * log(2) = 0. 358 * static const float xmax = 35.040095f 359 * static const double xmax = 171.624376956302725; 360 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), 361 * ld128: 1.75554834290446291700388921607020320e+03L, 362 * 363 * iota is a sloppy threshold to isolate x = 0. 364 */ 365 static const double xmax = 171.624376956302725; 366 static const double iota = 0x1p-56; 367 368 double 369 tgamma(double x) 370 { 371 struct Double u; 372 373 if (x >= 6) { 374 if (x > xmax) 375 return (x / zero); 376 u = large_gam(x); 377 return (__exp__D(u.a, u.b)); 378 } 379 380 if (x >= 1 + left + x0) 381 return (small_gam(x)); 382 383 if (x > iota) 384 return (smaller_gam(x)); 385 386 if (x > -iota) { 387 if (x != 0.) 388 u.a = 1 - tiny; /* raise inexact */ 389 return (1 / x); 390 } 391 392 if (!isfinite(x)) 393 return (x - x); /* x is NaN or -Inf */ 394 395 return (neg_gam(x)); 396 } 397 398 #if (LDBL_MANT_DIG == 53) 399 __weak_reference(tgamma, tgammal); 400 #endif 401