2 * SPDX-License-Identifier: BSD-3-Clause
4 * Copyright (c) 1992, 1993
5 * The Regents of the University of California. All rights reserved.
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
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.
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
33 * The original code, FreeBSD's old svn r93211, contain the following
36 * This code by P. McIlroy, Oct 1992;
38 * The financial support of UUNET Communications Services is greatfully
41 * bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
45 * See bsdsrc/t_tgamma.c for implementation details.
50 #if LDBL_MAX_EXP != 0x4000
51 #error "Unsupported long double format"
60 #include "math_private.h"
62 /* Used in b_log.c and below. */
71 static const double zero = 0.;
72 static const volatile double tiny = 1e-300;
76 * Use the asymptotic approximation (Stirling's formula) adjusted for
79 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
81 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
82 * premature round-off.
84 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
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.
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)
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);
121 large_gam(long double x)
123 long double p, z, thi, tlo, xhi, xlo;
128 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
129 z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
135 /* Split (x - 0.5) in high and low parts. */
140 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */
142 tlo = xlo * u.a + x * u.b;
144 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
147 u.a = ln2pi_hi + tlo;
155 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
156 * [1.066.., 2.066..] accurate to 4.25e-19.
158 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
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)
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);
203 ratfun_gam(long double z, long double c)
205 long double p, q, thi, tlo;
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)))))));
214 /* Split z into high and low parts. */
219 /* Split (z+c)^2 into high and low parts. */
225 /* Split p/q into high and low parts. */
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 */
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.
242 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
243 * It also has correct monotonicity.
245 static const union IEEEl2bits
246 xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
250 left = -0.3955078125; /* left boundary for rat. approx */
253 small_gam(long double x)
255 long double t, y, ym1;
260 if (y <= 1 + (left + x0)) {
261 yy = ratfun_gam(y - x0, 0);
262 return (yy.a + yy.b);
268 r.b = yy.b = y - yy.a;
270 /* Argument reduction: G(x+1) = x*G(x) */
271 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
273 r.b = r.a * yy.b + y * r.b;
278 /* Return r*tgamma(y). */
279 yy = ratfun_gam(y - x0, 0);
280 y = r.b * (yy.a + yy.b) + r.a * yy.b;
285 * Good on (0, 1+x0+left]. Accurate to 1 ulp.
288 smaller_gam(long double x)
290 long double d, rhi, rlo, t, xhi, xlo;
295 d = (t + x) * (x - t);
297 xhi = (float)(t + x);
315 r = ratfun_gam(t, d);
316 d = (float)(r.a / x);
321 return (d + r.a / x);
326 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
327 * At negative integers, return NaN and raise invalid.
329 static const union IEEEl2bits
330 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
334 neg_gam(long double x)
337 struct Double lg, lsine;
341 if (y == x) /* Negative integer. */
342 return ((x - x) / zero);
357 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
361 return (sgn * tiny * tiny);
362 y = expl(lgammal(x) / 2);
364 return (sgn < 0 ? -y : y);
371 else /* 1-x is inexact */
372 y = - x * tgammal(-x);
375 return (pi / (y * z));
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,
384 * iota is a sloppy threshold to isolate x = 0.
386 static const double xmax = 1755.54834290446291689;
387 static const double iota = 0x1p-116;
390 tgammal(long double x)
400 RETURNI(__exp__D(u.a, u.b));
403 if (x >= 1 + left + x0)
404 RETURNI(small_gam(x));
407 RETURNI(smaller_gam(x));
411 u.a = 1 - tiny; /* raise inexact */
416 RETURNI(x - x); /* x is NaN or -Inf */