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, contain 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 * bsdrc/b_tgamma.c converted to long double by Steven G. Kargl. 42 */ 43 44 /* 45 * See bsdsrc/t_tgamma.c for implementation details. 46 */ 47 48 #include <float.h> 49 50 #if LDBL_MAX_EXP != 0x4000 51 #error "Unsupported long double format" 52 #endif 53 54 #ifdef __i386__ 55 #include <ieeefp.h> 56 #endif 57 58 #include "fpmath.h" 59 #include "math.h" 60 #include "math_private.h" 61 62 /* Used in b_log.c and below. */ 63 struct Double { 64 long double a; 65 long double b; 66 }; 67 68 #include "b_logl.c" 69 #include "b_expl.c" 70 71 static const double zero = 0.; 72 static const volatile double tiny = 1e-300; 73 /* 74 * x >= 6 75 * 76 * Use the asymptotic approximation (Stirling's formula) adjusted for 77 * equal-ripples: 78 * 79 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) 80 * 81 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid 82 * premature round-off. 83 * 84 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 85 */ 86 87 /* 88 * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the 89 * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The 90 * variables are clearly misnamed. 91 */ 92 static const union IEEEl2bits 93 ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L), 94 ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L); 95 #define ln2pi_hi (ln2pi_hiu.e) 96 #define ln2pi_lo (ln2pi_lou.e) 97 98 static const union IEEEl2bits 99 Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L), 100 Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L), 101 Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L), 102 Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L), 103 Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L), 104 Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L), 105 Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L), 106 Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L), 107 Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L), 108 Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L); 109 #define Pa0 (Pa0u.e) 110 #define Pa1 (Pa1u.e) 111 #define Pa2 (Pa2u.e) 112 #define Pa3 (Pa3u.e) 113 #define Pa4 (Pa4u.e) 114 #define Pa5 (Pa5u.e) 115 #define Pa6 (Pa6u.e) 116 #define Pa7 (Pa7u.e) 117 #define Pa8 (Pa8u.e) 118 #define Pa9 (Pa9u.e) 119 120 static struct Double 121 large_gam(long double x) 122 { 123 long double p, z, thi, tlo, xhi, xlo; 124 long double logx; 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 + z * (Pa8 + z * Pa9)))))))); 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 -= 0.5L; 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 union IEEEl2bits 161 a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L), 162 a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L); 163 #define a0_hi (a0_hiu.e) 164 #define a0_lo (a0_lou.e) 165 166 static const union IEEEl2bits 167 P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L), 168 P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L), 169 P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L), 170 P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L), 171 P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L), 172 P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L), 173 P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L), 174 P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L), 175 P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L), 176 Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L), 177 Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L), 178 Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L), 179 Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L), 180 Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L), 181 Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L), 182 Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L), 183 Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L); 184 #define P0 (P0u.e) 185 #define P1 (P1u.e) 186 #define P2 (P2u.e) 187 #define P3 (P3u.e) 188 #define P4 (P4u.e) 189 #define P5 (P5u.e) 190 #define P6 (P6u.e) 191 #define P7 (P7u.e) 192 #define P8 (P8u.e) 193 #define Q1 (Q1u.e) 194 #define Q2 (Q2u.e) 195 #define Q3 (Q3u.e) 196 #define Q4 (Q4u.e) 197 #define Q5 (Q5u.e) 198 #define Q6 (Q6u.e) 199 #define Q7 (Q7u.e) 200 #define Q8 (Q8u.e) 201 202 static struct Double 203 ratfun_gam(long double z, long double c) 204 { 205 long double p, q, thi, tlo; 206 struct Double r; 207 208 q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + 209 z * (Q6 + z * (Q7 + z * Q8))))))); 210 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + 211 z * (P6 + z * (P7 + z * P8))))))); 212 p = p / q; 213 214 /* Split z into high and low parts. */ 215 thi = (float)z; 216 tlo = (z - thi) + c; 217 tlo *= (thi + z); 218 219 /* Split (z+c)^2 into high and low parts. */ 220 thi *= thi; 221 q = thi; 222 thi = (float)thi; 223 tlo += (q - thi); 224 225 /* Split p/q into high and low parts. */ 226 r.a = (float)p; 227 r.b = p - r.a; 228 229 tlo = tlo * p + thi * r.b + a0_lo; 230 thi *= r.a; /* t = (z+c)^2*(P/Q) */ 231 r.a = (float)(thi + a0_hi); 232 r.b = ((a0_hi - r.a) + thi) + tlo; 233 return (r); /* r = a0 + t */ 234 } 235 /* 236 * x < 6 237 * 238 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, 239 * 2.066124]. Use a rational approximation centered at the minimum 240 * (x0+1) to ensure monotonicity. 241 * 242 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 243 * It also has correct monotonicity. 244 */ 245 static const union IEEEl2bits 246 xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L); 247 #define x0 (xm1u.e) 248 249 static const double 250 left = -0.3955078125; /* left boundary for rat. approx */ 251 252 static long double 253 small_gam(long double x) 254 { 255 long double t, y, ym1; 256 struct Double yy, r; 257 258 y = x - 1; 259 260 if (y <= 1 + (left + x0)) { 261 yy = ratfun_gam(y - x0, 0); 262 return (yy.a + yy.b); 263 } 264 265 r.a = (float)y; 266 yy.a = r.a - 1; 267 y = y - 1 ; 268 r.b = yy.b = y - yy.a; 269 270 /* Argument reduction: G(x+1) = x*G(x) */ 271 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { 272 t = r.a * yy.a; 273 r.b = r.a * yy.b + y * r.b; 274 r.a = (float)t; 275 r.b += (t - r.a); 276 } 277 278 /* Return r*tgamma(y). */ 279 yy = ratfun_gam(y - x0, 0); 280 y = r.b * (yy.a + yy.b) + r.a * yy.b; 281 y += yy.a * r.a; 282 return (y); 283 } 284 /* 285 * Good on (0, 1+x0+left]. Accurate to 1 ulp. 286 */ 287 static long double 288 smaller_gam(long double x) 289 { 290 long double d, rhi, rlo, t, xhi, xlo; 291 struct Double r; 292 293 if (x < x0 + left) { 294 t = (float)x; 295 d = (t + x) * (x - t); 296 t *= t; 297 xhi = (float)(t + x); 298 xlo = x - xhi; 299 xlo += t; 300 xlo += d; 301 t = 1 - x0; 302 t += x; 303 d = 1 - x0; 304 d -= t; 305 d += x; 306 x = xhi + xlo; 307 } else { 308 xhi = (float)x; 309 xlo = x - xhi; 310 t = x - x0; 311 d = - x0 - t; 312 d += x; 313 } 314 315 r = ratfun_gam(t, d); 316 d = (float)(r.a / x); 317 r.a -= d * xhi; 318 r.a -= d * xlo; 319 r.a += r.b; 320 321 return (d + r.a / x); 322 } 323 /* 324 * x < 0 325 * 326 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). 327 * At negative integers, return NaN and raise invalid. 328 */ 329 static const union IEEEl2bits 330 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L); 331 #define pi (piu.e) 332 333 static long double 334 neg_gam(long double x) 335 { 336 int sgn = 1; 337 struct Double lg, lsine; 338 long double y, z; 339 340 y = ceill(x); 341 if (y == x) /* Negative integer. */ 342 return ((x - x) / zero); 343 344 z = y - x; 345 if (z > 0.5) 346 z = 1 - z; 347 348 y = y / 2; 349 if (y == ceill(y)) 350 sgn = -1; 351 352 if (z < 0.25) 353 z = sinpil(z); 354 else 355 z = cospil(0.5 - z); 356 357 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 358 if (x < -1753) { 359 360 if (x < -1760) 361 return (sgn * tiny * tiny); 362 y = expl(lgammal(x) / 2); 363 y *= y; 364 return (sgn < 0 ? -y : y); 365 } 366 367 368 y = 1 - x; 369 if (1 - y == x) 370 y = tgammal(y); 371 else /* 1-x is inexact */ 372 y = - x * tgammal(-x); 373 374 if (sgn < 0) y = -y; 375 return (pi / (y * z)); 376 } 377 /* 378 * xmax comes from lgamma(xmax) - emax * log(2) = 0. 379 * static const float xmax = 35.040095f 380 * static const double xmax = 171.624376956302725; 381 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), 382 * ld128: 1.75554834290446291700388921607020320e+03L, 383 * 384 * iota is a sloppy threshold to isolate x = 0. 385 */ 386 static const double xmax = 1755.54834290446291689; 387 static const double iota = 0x1p-116; 388 389 long double 390 tgammal(long double x) 391 { 392 struct Double u; 393 394 ENTERI(); 395 396 if (x >= 6) { 397 if (x > xmax) 398 RETURNI(x / zero); 399 u = large_gam(x); 400 RETURNI(__exp__D(u.a, u.b)); 401 } 402 403 if (x >= 1 + left + x0) 404 RETURNI(small_gam(x)); 405 406 if (x > iota) 407 RETURNI(smaller_gam(x)); 408 409 if (x > -iota) { 410 if (x != 0) 411 u.a = 1 - tiny; /* raise inexact */ 412 RETURNI(1 / x); 413 } 414 415 if (!isfinite(x)) 416 RETURNI(x - x); /* x is NaN or -Inf */ 417 418 RETURNI(neg_gam(x)); 419 } 420