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 #include <float.h> 46 47 #include "math.h" 48 #include "math_private.h" 49 50 /* Used in b_log.c and below. */ 51 struct Double { 52 double a; 53 double b; 54 }; 55 56 #include "b_log.c" 57 #include "b_exp.c" 58 59 /* 60 * The range is broken into several subranges. Each is handled by its 61 * helper functions. 62 * 63 * x >= 6.0: large_gam(x) 64 * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0. 65 * xleft > x > iota: smaller_gam(x) where iota = 1e-17. 66 * iota > x > -itoa: Handle x near 0. 67 * -iota > x : neg_gam 68 * 69 * Special values: 70 * -Inf: return NaN and raise invalid; 71 * negative integer: return NaN and raise invalid; 72 * other x ~< 177.79: return +-0 and raise underflow; 73 * +-0: return +-Inf and raise divide-by-zero; 74 * finite x ~> 171.63: return +Inf and raise overflow; 75 * +Inf: return +Inf; 76 * NaN: return NaN. 77 * 78 * Accuracy: tgamma(x) is accurate to within 79 * x > 0: error provably < 0.9ulp. 80 * Maximum observed in 1,000,000 trials was .87ulp. 81 * x < 0: 82 * Maximum observed error < 4ulp in 1,000,000 trials. 83 */ 84 85 /* 86 * Constants for large x approximation (x in [6, Inf]) 87 * (Accurate to 2.8*10^-19 absolute) 88 */ 89 90 static const double zero = 0.; 91 static const volatile double tiny = 1e-300; 92 /* 93 * x >= 6 94 * 95 * Use the asymptotic approximation (Stirling's formula) adjusted fof 96 * equal-ripples: 97 * 98 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) 99 * 100 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid 101 * premature round-off. 102 * 103 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 104 */ 105 static const double 106 ln2pi_hi = 0.41894531250000000, 107 ln2pi_lo = -6.7792953272582197e-6, 108 Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ 109 Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */ 110 Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */ 111 Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */ 112 Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */ 113 Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */ 114 Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */ 115 Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */ 116 117 static struct Double 118 large_gam(double x) 119 { 120 double p, z, thi, tlo, xhi, xlo; 121 struct Double u; 122 123 z = 1 / (x * x); 124 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + 125 z * (Pa6 + z * Pa7)))))); 126 p = p / x; 127 128 u = __log__D(x); 129 u.a -= 1; 130 131 /* Split (x - 0.5) in high and low parts. */ 132 x -= 0.5; 133 xhi = (float)x; 134 xlo = x - xhi; 135 136 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ 137 thi = xhi * u.a; 138 tlo = xlo * u.a + x * u.b; 139 140 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ 141 tlo += ln2pi_lo; 142 tlo += p; 143 u.a = ln2pi_hi + tlo; 144 u.a += thi; 145 u.b = thi - u.a; 146 u.b += ln2pi_hi; 147 u.b += tlo; 148 return (u); 149 } 150 /* 151 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval 152 * [1.066.., 2.066..] accurate to 4.25e-19. 153 * 154 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. 155 */ 156 static const double 157 #if 0 158 a0_hi = 8.8560319441088875e-1, 159 a0_lo = -4.9964270364690197e-17, 160 #else 161 a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */ 162 a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */ 163 #endif 164 P0 = 6.2138957182182086e-1, 165 P1 = 2.6575719865153347e-1, 166 P2 = 5.5385944642991746e-3, 167 P3 = 1.3845669830409657e-3, 168 P4 = 2.4065995003271137e-3, 169 Q0 = 1.4501953125000000e+0, 170 Q1 = 1.0625852194801617e+0, 171 Q2 = -2.0747456194385994e-1, 172 Q3 = -1.4673413178200542e-1, 173 Q4 = 3.0787817615617552e-2, 174 Q5 = 5.1244934798066622e-3, 175 Q6 = -1.7601274143166700e-3, 176 Q7 = 9.3502102357378894e-5, 177 Q8 = 6.1327550747244396e-6; 178 179 static struct Double 180 ratfun_gam(double z, double c) 181 { 182 double p, q, thi, tlo; 183 struct Double r; 184 185 q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + 186 z * (Q6 + z * (Q7 + z * Q8))))))); 187 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4))); 188 p = p / q; 189 190 /* Split z into high and low parts. */ 191 thi = (float)z; 192 tlo = (z - thi) + c; 193 tlo *= (thi + z); 194 195 /* Split (z+c)^2 into high and low parts. */ 196 thi *= thi; 197 q = thi; 198 thi = (float)thi; 199 tlo += (q - thi); 200 201 /* Split p/q into high and low parts. */ 202 r.a = (float)p; 203 r.b = p - r.a; 204 205 tlo = tlo * p + thi * r.b + a0_lo; 206 thi *= r.a; /* t = (z+c)^2*(P/Q) */ 207 r.a = (float)(thi + a0_hi); 208 r.b = ((a0_hi - r.a) + thi) + tlo; 209 return (r); /* r = a0 + t */ 210 } 211 /* 212 * x < 6 213 * 214 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, 215 * 2.066124]. Use a rational approximation centered at the minimum 216 * (x0+1) to ensure monotonicity. 217 * 218 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 219 * It also has correct monotonicity. 220 */ 221 static const double 222 left = -0.3955078125, /* left boundary for rat. approx */ 223 x0 = 4.6163214496836236e-1; /* xmin - 1 */ 224 225 static double 226 small_gam(double x) 227 { 228 double t, y, ym1; 229 struct Double yy, r; 230 231 y = x - 1; 232 if (y <= 1 + (left + x0)) { 233 yy = ratfun_gam(y - x0, 0); 234 return (yy.a + yy.b); 235 } 236 237 r.a = (float)y; 238 yy.a = r.a - 1; 239 y = y - 1 ; 240 r.b = yy.b = y - yy.a; 241 242 /* Argument reduction: G(x+1) = x*G(x) */ 243 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { 244 t = r.a * yy.a; 245 r.b = r.a * yy.b + y * r.b; 246 r.a = (float)t; 247 r.b += (t - r.a); 248 } 249 250 /* Return r*tgamma(y). */ 251 yy = ratfun_gam(y - x0, 0); 252 y = r.b * (yy.a + yy.b) + r.a * yy.b; 253 y += yy.a * r.a; 254 return (y); 255 } 256 /* 257 * Good on (0, 1+x0+left]. Accurate to 1 ulp. 258 */ 259 static double 260 smaller_gam(double x) 261 { 262 double d, rhi, rlo, t, xhi, xlo; 263 struct Double r; 264 265 if (x < x0 + left) { 266 t = (float)x; 267 d = (t + x) * (x - t); 268 t *= t; 269 xhi = (float)(t + x); 270 xlo = x - xhi; 271 xlo += t; 272 xlo += d; 273 t = 1 - x0; 274 t += x; 275 d = 1 - x0; 276 d -= t; 277 d += x; 278 x = xhi + xlo; 279 } else { 280 xhi = (float)x; 281 xlo = x - xhi; 282 t = x - x0; 283 d = - x0 - t; 284 d += x; 285 } 286 287 r = ratfun_gam(t, d); 288 d = (float)(r.a / x); 289 r.a -= d * xhi; 290 r.a -= d * xlo; 291 r.a += r.b; 292 293 return (d + r.a / x); 294 } 295 /* 296 * x < 0 297 * 298 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). 299 * At negative integers, return NaN and raise invalid. 300 */ 301 static double 302 neg_gam(double x) 303 { 304 int sgn = 1; 305 struct Double lg, lsine; 306 double y, z; 307 308 y = ceil(x); 309 if (y == x) /* Negative integer. */ 310 return ((x - x) / zero); 311 312 z = y - x; 313 if (z > 0.5) 314 z = 1 - z; 315 316 y = y / 2; 317 if (y == ceil(y)) 318 sgn = -1; 319 320 if (z < 0.25) 321 z = sinpi(z); 322 else 323 z = cospi(0.5 - z); 324 325 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 326 if (x < -170) { 327 328 if (x < -190) 329 return (sgn * tiny * tiny); 330 331 y = 1 - x; /* exact: 128 < |x| < 255 */ 332 lg = large_gam(y); 333 lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */ 334 lg.a -= lsine.a; /* exact (opposite signs) */ 335 lg.b -= lsine.b; 336 y = -(lg.a + lg.b); 337 z = (y + lg.a) + lg.b; 338 y = __exp__D(y, z); 339 if (sgn < 0) y = -y; 340 return (y); 341 } 342 343 y = 1 - x; 344 if (1 - y == x) 345 y = tgamma(y); 346 else /* 1-x is inexact */ 347 y = - x * tgamma(-x); 348 349 if (sgn < 0) y = -y; 350 return (M_PI / (y * z)); 351 } 352 /* 353 * xmax comes from lgamma(xmax) - emax * log(2) = 0. 354 * static const float xmax = 35.040095f 355 * static const double xmax = 171.624376956302725; 356 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), 357 * ld128: 1.75554834290446291700388921607020320e+03L, 358 * 359 * iota is a sloppy threshold to isolate x = 0. 360 */ 361 static const double xmax = 171.624376956302725; 362 static const double iota = 0x1p-56; 363 364 double 365 tgamma(double x) 366 { 367 struct Double u; 368 369 if (x >= 6) { 370 if (x > xmax) 371 return (x / zero); 372 u = large_gam(x); 373 return (__exp__D(u.a, u.b)); 374 } 375 376 if (x >= 1 + left + x0) 377 return (small_gam(x)); 378 379 if (x > iota) 380 return (smaller_gam(x)); 381 382 if (x > -iota) { 383 if (x != 0.) 384 u.a = 1 - tiny; /* raise inexact */ 385 return (1 / x); 386 } 387 388 if (!isfinite(x)) 389 return (x - x); /* x is NaN or -Inf */ 390 391 return (neg_gam(x)); 392 } 393 394 #if (LDBL_MANT_DIG == 53) 395 __weak_reference(tgamma, tgammal); 396 #endif 397