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 #include <sys/cdefs.h> 45 46 /* 47 * See bsdsrc/t_tgamma.c for implementation details. 48 */ 49 50 #include <float.h> 51 52 #if LDBL_MAX_EXP != 0x4000 53 #error "Unsupported long double format" 54 #endif 55 56 #include "math.h" 57 #include "math_private.h" 58 59 /* Used in b_log.c and below. */ 60 struct LDouble { 61 long double a; 62 long double b; 63 }; 64 65 #include "b_logl.c" 66 #include "b_expl.c" 67 68 static const double zero = 0.; 69 static const volatile double tiny = 1e-300; 70 /* 71 * x >= 6 72 * 73 * Use the asymptotic approximation (Stirling's formula) adjusted for 74 * equal-ripples: 75 * 76 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) 77 * 78 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid 79 * premature round-off. 80 * 81 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. 82 */ 83 84 /* 85 * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the 86 * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The 87 * variables are clearly misnamed. 88 */ 89 static const union ieee_ext_u 90 ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L), 91 ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L); 92 #define ln2pi_hi (ln2pi_hiu.extu_ld) 93 #define ln2pi_lo (ln2pi_lou.extu_ld) 94 95 static const union ieee_ext_u 96 Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L), 97 Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L), 98 Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L), 99 Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L), 100 Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L), 101 Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L), 102 Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L), 103 Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L), 104 Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L), 105 Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L); 106 #define Pa0 (Pa0u.extu_ld) 107 #define Pa1 (Pa1u.extu_ld) 108 #define Pa2 (Pa2u.extu_ld) 109 #define Pa3 (Pa3u.extu_ld) 110 #define Pa4 (Pa4u.extu_ld) 111 #define Pa5 (Pa5u.extu_ld) 112 #define Pa6 (Pa6u.extu_ld) 113 #define Pa7 (Pa7u.extu_ld) 114 #define Pa8 (Pa8u.extu_ld) 115 #define Pa9 (Pa9u.extu_ld) 116 117 static struct LDouble 118 large_gam(long double x) 119 { 120 long double p, z, thi, tlo, xhi, xlo; 121 struct LDouble 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 + z * (Pa8 + z * Pa9)))))))); 126 p = p / x; 127 128 u = __log__LD(x); 129 u.a -= 1; 130 131 /* Split (x - 0.5) in high and low parts. */ 132 x -= 0.5L; 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 union ieee_ext_u 157 a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L), 158 a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L); 159 #define a0_hi (a0_hiu.extu_ld) 160 #define a0_lo (a0_lou.extu_ld) 161 162 static const union ieee_ext_u 163 P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L), 164 P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L), 165 P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L), 166 P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L), 167 P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L), 168 P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L), 169 P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L), 170 P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L), 171 P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L), 172 Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L), 173 Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L), 174 Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L), 175 Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L), 176 Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L), 177 Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L), 178 Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L), 179 Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L); 180 #define P0 (P0u.extu_ld) 181 #define P1 (P1u.extu_ld) 182 #define P2 (P2u.extu_ld) 183 #define P3 (P3u.extu_ld) 184 #define P4 (P4u.extu_ld) 185 #define P5 (P5u.extu_ld) 186 #define P6 (P6u.extu_ld) 187 #define P7 (P7u.extu_ld) 188 #define P8 (P8u.extu_ld) 189 #define Q1 (Q1u.extu_ld) 190 #define Q2 (Q2u.extu_ld) 191 #define Q3 (Q3u.extu_ld) 192 #define Q4 (Q4u.extu_ld) 193 #define Q5 (Q5u.extu_ld) 194 #define Q6 (Q6u.extu_ld) 195 #define Q7 (Q7u.extu_ld) 196 #define Q8 (Q8u.extu_ld) 197 198 static struct LDouble 199 ratfun_gam(long double z, long double c) 200 { 201 long double p, q, thi, tlo; 202 struct LDouble r; 203 204 q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + 205 z * (Q6 + z * (Q7 + z * Q8))))))); 206 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + 207 z * (P6 + z * (P7 + z * P8))))))); 208 p = p / q; 209 210 /* Split z into high and low parts. */ 211 thi = (float)z; 212 tlo = (z - thi) + c; 213 tlo *= (thi + z); 214 215 /* Split (z+c)^2 into high and low parts. */ 216 thi *= thi; 217 q = thi; 218 thi = (float)thi; 219 tlo += (q - thi); 220 221 /* Split p/q into high and low parts. */ 222 r.a = (float)p; 223 r.b = p - r.a; 224 225 tlo = tlo * p + thi * r.b + a0_lo; 226 thi *= r.a; /* t = (z+c)^2*(P/Q) */ 227 r.a = (float)(thi + a0_hi); 228 r.b = ((a0_hi - r.a) + thi) + tlo; 229 return (r); /* r = a0 + t */ 230 } 231 /* 232 * x < 6 233 * 234 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, 235 * 2.066124]. Use a rational approximation centered at the minimum 236 * (x0+1) to ensure monotonicity. 237 * 238 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) 239 * It also has correct monotonicity. 240 */ 241 static const union ieee_ext_u 242 xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L); 243 #define x0 (xm1u.extu_ld) 244 245 static const double 246 left = -0.3955078125; /* left boundary for rat. approx */ 247 248 static long double 249 small_gam(long double x) 250 { 251 long double t, y, ym1; 252 struct LDouble yy, r; 253 254 y = x - 1; 255 256 if (y <= 1 + (left + x0)) { 257 yy = ratfun_gam(y - x0, 0); 258 return (yy.a + yy.b); 259 } 260 261 r.a = (float)y; 262 yy.a = r.a - 1; 263 y = y - 1 ; 264 r.b = yy.b = y - yy.a; 265 266 /* Argument reduction: G(x+1) = x*G(x) */ 267 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { 268 t = r.a * yy.a; 269 r.b = r.a * yy.b + y * r.b; 270 r.a = (float)t; 271 r.b += (t - r.a); 272 } 273 274 /* Return r*tgamma(y). */ 275 yy = ratfun_gam(y - x0, 0); 276 y = r.b * (yy.a + yy.b) + r.a * yy.b; 277 y += yy.a * r.a; 278 return (y); 279 } 280 /* 281 * Good on (0, 1+x0+left]. Accurate to 1 ulp. 282 */ 283 static long double 284 smaller_gam(long double x) 285 { 286 long double d, t, xhi, xlo; 287 struct LDouble r; 288 289 if (x < x0 + left) { 290 t = (float)x; 291 d = (t + x) * (x - t); 292 t *= t; 293 xhi = (float)(t + x); 294 xlo = x - xhi; 295 xlo += t; 296 xlo += d; 297 t = 1 - x0; 298 t += x; 299 d = 1 - x0; 300 d -= t; 301 d += x; 302 x = xhi + xlo; 303 } else { 304 xhi = (float)x; 305 xlo = x - xhi; 306 t = x - x0; 307 d = - x0 - t; 308 d += x; 309 } 310 311 r = ratfun_gam(t, d); 312 d = (float)(r.a / x); 313 r.a -= d * xhi; 314 r.a -= d * xlo; 315 r.a += r.b; 316 317 return (d + r.a / x); 318 } 319 /* 320 * x < 0 321 * 322 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). 323 * At negative integers, return NaN and raise invalid. 324 */ 325 static const union ieee_ext_u 326 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L); 327 #define pi (piu.extu_ld) 328 329 static long double 330 neg_gam(long double x) 331 { 332 int sgn = 1; 333 long double y, z; 334 335 y = ceill(x); 336 if (y == x) /* Negative integer. */ 337 return ((x - x) / zero); 338 339 z = y - x; 340 if (z > 0.5) 341 z = 1 - z; 342 343 y = y / 2; 344 if (y == ceill(y)) 345 sgn = -1; 346 347 if (z < 0.25) 348 z = sinpil(z); 349 else 350 z = cospil(0.5 - z); 351 352 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ 353 if (x < -1753) { 354 355 if (x < -1760) 356 return (sgn * tiny * tiny); 357 y = expl(lgammal(x) / 2); 358 y *= y; 359 return (sgn < 0 ? -y : y); 360 } 361 362 363 y = 1 - x; 364 if (1 - y == x) 365 y = tgammal(y); 366 else /* 1-x is inexact */ 367 y = - x * tgammal(-x); 368 369 if (sgn < 0) y = -y; 370 return (pi / (y * z)); 371 } 372 /* 373 * xmax comes from lgamma(xmax) - emax * log(2) = 0. 374 * static const float xmax = 35.040095f 375 * static const double xmax = 171.624376956302725; 376 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), 377 * ld128: 1.75554834290446291700388921607020320e+03L, 378 * 379 * iota is a sloppy threshold to isolate x = 0. 380 */ 381 static const double xmax = 1755.54834290446291689; 382 static const double iota = 0x1p-116; 383 384 long double 385 tgammal(long double x) 386 { 387 struct LDouble u; 388 389 ENTERI(); 390 391 if (x >= 6) { 392 if (x > xmax) 393 RETURNI(x / zero); 394 u = large_gam(x); 395 RETURNI(__exp__LD(u.a, u.b)); 396 } 397 398 if (x >= 1 + left + x0) 399 RETURNI(small_gam(x)); 400 401 if (x > iota) 402 RETURNI(smaller_gam(x)); 403 404 if (x > -iota) { 405 if (x != 0) 406 u.a = 1 - tiny; /* raise inexact */ 407 RETURNI(1 / x); 408 } 409 410 if (!isfinite(x)) 411 RETURNI(x - x); /* x is NaN or -Inf */ 412 413 RETURNI(neg_gam(x)); 414 } 415
Indexes created Sat Sep 20 22:09:52 GMT 2025