/*- * SPDX-License-Identifier: BSD-3-Clause * * Copyright (c) 1992, 1993 * The Regents of the University of California. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * 3. Neither the name of the University nor the names of its contributors * may be used to endorse or promote products derived from this software * without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. */ /* * The original code, FreeBSD's old svn r93211, contained the following * attribution: * * This code by P. McIlroy, Oct 1992; * * The financial support of UUNET Communications Services is greatfully * acknowledged. * * The algorithm remains, but the code has been re-arranged to facilitate * porting to other precisions. */ #include <float.h> #include "math.h" #include "math_private.h" /* Used in b_log.c and below. */ struct Double { double a; double b; }; #include "b_log.c" #include "b_exp.c" /* * The range is broken into several subranges. Each is handled by its * helper functions. * * x >= 6.0: large_gam(x) * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0. * xleft > x > iota: smaller_gam(x) where iota = 1e-17. * iota > x > -itoa: Handle x near 0. * -iota > x : neg_gam * * Special values: * -Inf: return NaN and raise invalid; * negative integer: return NaN and raise invalid; * other x ~< 177.79: return +-0 and raise underflow; * +-0: return +-Inf and raise divide-by-zero; * finite x ~> 171.63: return +Inf and raise overflow; * +Inf: return +Inf; * NaN: return NaN. * * Accuracy: tgamma(x) is accurate to within * x > 0: error provably < 0.9ulp. * Maximum observed in 1,000,000 trials was .87ulp. * x < 0: * Maximum observed error < 4ulp in 1,000,000 trials. */ /* * Constants for large x approximation (x in [6, Inf]) * (Accurate to 2.8*10^-19 absolute) */ static const double zero = 0.; static const volatile double tiny = 1e-300; /* * x >= 6 * * Use the asymptotic approximation (Stirling's formula) adjusted fof * equal-ripples: * * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x)) * * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid * premature round-off. * * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. */ static const double ln2pi_hi = 0.41894531250000000, ln2pi_lo = -6.7792953272582197e-6, Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */ Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */ Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */ Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */ Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */ Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */ Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */ static struct Double large_gam(double x) { double p, z, thi, tlo, xhi, xlo; struct Double u; z = 1 / (x * x); p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 + z * (Pa6 + z * Pa7)))))); p = p / x; u = __log__D(x); u.a -= 1; /* Split (x - 0.5) in high and low parts. */ x -= 0.5; xhi = (float)x; xlo = x - xhi; /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ thi = xhi * u.a; tlo = xlo * u.a + x * u.b; /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ tlo += ln2pi_lo; tlo += p; u.a = ln2pi_hi + tlo; u.a += thi; u.b = thi - u.a; u.b += ln2pi_hi; u.b += tlo; return (u); } /* * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval * [1.066.., 2.066..] accurate to 4.25e-19. * * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. */ static const double #if 0 a0_hi = 8.8560319441088875e-1, a0_lo = -4.9964270364690197e-17, #else a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */ a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */ #endif P0 = 6.2138957182182086e-1, P1 = 2.6575719865153347e-1, P2 = 5.5385944642991746e-3, P3 = 1.3845669830409657e-3, P4 = 2.4065995003271137e-3, Q0 = 1.4501953125000000e+0, Q1 = 1.0625852194801617e+0, Q2 = -2.0747456194385994e-1, Q3 = -1.4673413178200542e-1, Q4 = 3.0787817615617552e-2, Q5 = 5.1244934798066622e-3, Q6 = -1.7601274143166700e-3, Q7 = 9.3502102357378894e-5, Q8 = 6.1327550747244396e-6; static struct Double ratfun_gam(double z, double c) { double p, q, thi, tlo; struct Double r; q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 + z * (Q6 + z * (Q7 + z * Q8))))))); p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4))); p = p / q; /* Split z into high and low parts. */ thi = (float)z; tlo = (z - thi) + c; tlo *= (thi + z); /* Split (z+c)^2 into high and low parts. */ thi *= thi; q = thi; thi = (float)thi; tlo += (q - thi); /* Split p/q into high and low parts. */ r.a = (float)p; r.b = p - r.a; tlo = tlo * p + thi * r.b + a0_lo; thi *= r.a; /* t = (z+c)^2*(P/Q) */ r.a = (float)(thi + a0_hi); r.b = ((a0_hi - r.a) + thi) + tlo; return (r); /* r = a0 + t */ } /* * x < 6 * * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124, * 2.066124]. Use a rational approximation centered at the minimum * (x0+1) to ensure monotonicity. * * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) * It also has correct monotonicity. */ static const double left = -0.3955078125, /* left boundary for rat. approx */ x0 = 4.6163214496836236e-1; /* xmin - 1 */ static double small_gam(double x) { double t, y, ym1; struct Double yy, r; y = x - 1; if (y <= 1 + (left + x0)) { yy = ratfun_gam(y - x0, 0); return (yy.a + yy.b); } r.a = (float)y; yy.a = r.a - 1; y = y - 1 ; r.b = yy.b = y - yy.a; /* Argument reduction: G(x+1) = x*G(x) */ for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { t = r.a * yy.a; r.b = r.a * yy.b + y * r.b; r.a = (float)t; r.b += (t - r.a); } /* Return r*tgamma(y). */ yy = ratfun_gam(y - x0, 0); y = r.b * (yy.a + yy.b) + r.a * yy.b; y += yy.a * r.a; return (y); } /* * Good on (0, 1+x0+left]. Accurate to 1 ulp. */ static double smaller_gam(double x) { double d, t, xhi, xlo; struct Double r; if (x < x0 + left) { t = (float)x; d = (t + x) * (x - t); t *= t; xhi = (float)(t + x); xlo = x - xhi; xlo += t; xlo += d; t = 1 - x0; t += x; d = 1 - x0; d -= t; d += x; x = xhi + xlo; } else { xhi = (float)x; xlo = x - xhi; t = x - x0; d = - x0 - t; d += x; } r = ratfun_gam(t, d); d = (float)(r.a / x); r.a -= d * xhi; r.a -= d * xlo; r.a += r.b; return (d + r.a / x); } /* * x < 0 * * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). * At negative integers, return NaN and raise invalid. */ static double neg_gam(double x) { int sgn = 1; struct Double lg, lsine; double y, z; y = ceil(x); if (y == x) /* Negative integer. */ return ((x - x) / zero); z = y - x; if (z > 0.5) z = 1 - z; y = y / 2; if (y == ceil(y)) sgn = -1; if (z < 0.25) z = sinpi(z); else z = cospi(0.5 - z); /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ if (x < -170) { if (x < -190) return (sgn * tiny * tiny); y = 1 - x; /* exact: 128 < |x| < 255 */ lg = large_gam(y); lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */ lg.a -= lsine.a; /* exact (opposite signs) */ lg.b -= lsine.b; y = -(lg.a + lg.b); z = (y + lg.a) + lg.b; y = __exp__D(y, z); if (sgn < 0) y = -y; return (y); } y = 1 - x; if (1 - y == x) y = tgamma(y); else /* 1-x is inexact */ y = - x * tgamma(-x); if (sgn < 0) y = -y; return (M_PI / (y * z)); } /* * xmax comes from lgamma(xmax) - emax * log(2) = 0. * static const float xmax = 35.040095f * static const double xmax = 171.624376956302725; * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), * ld128: 1.75554834290446291700388921607020320e+03L, * * iota is a sloppy threshold to isolate x = 0. */ static const double xmax = 171.624376956302725; static const double iota = 0x1p-56; double tgamma(double x) { struct Double u; if (x >= 6) { if (x > xmax) return (x / zero); u = large_gam(x); return (__exp__D(u.a, u.b)); } if (x >= 1 + left + x0) return (small_gam(x)); if (x > iota) return (smaller_gam(x)); if (x > -iota) { if (x != 0.) u.a = 1 - tiny; /* raise inexact */ return (1 / x); } if (!isfinite(x)) return (x - x); /* x is NaN or -Inf */ return (neg_gam(x)); } #if (LDBL_MANT_DIG == 53) __weak_reference(tgamma, tgammal); #endif
