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https://git.planet-casio.com/Lephenixnoir/OpenLibm.git
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71f60ec632
The finite() function has been superseded by isfinite(). There is also no need to use scalb(), as the exponent is also an integer value. We can simply use scalbn(). There is also no need to use __isnanf(). The values passed are guaranteed to be of type float, meaning we can safely use the standard isnan().
313 lines
8.5 KiB
C
313 lines
8.5 KiB
C
/*-
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* Copyright (c) 1992, 1993
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* The Regents of the University of California. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* 3. Neither the name of the University nor the names of its contributors
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* may be used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*/
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/* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */
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#include "cdefs-compat.h"
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//__FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_tgamma.c,v 1.10 2008/02/22 02:26:51 das Exp $");
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/*
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* This code by P. McIlroy, Oct 1992;
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*
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* The financial support of UUNET Communications Services is greatfully
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* acknowledged.
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*/
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#include <openlibm.h>
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#include "mathimpl.h"
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/* METHOD:
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* x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
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* At negative integers, return NaN and raise invalid.
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*
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* x < 6.5:
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* Use argument reduction G(x+1) = xG(x) to reach the
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* range [1.066124,2.066124]. Use a rational
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* approximation centered at the minimum (x0+1) to
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* ensure monotonicity.
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*
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* x >= 6.5: Use the asymptotic approximation (Stirling's formula)
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* adjusted for equal-ripples:
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*
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* log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
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*
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* Keep extra precision in multiplying (x-.5)(log(x)-1), to
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* avoid premature round-off.
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*
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* Special values:
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* -Inf: return NaN and raise invalid;
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* negative integer: return NaN and raise invalid;
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* other x ~< 177.79: return +-0 and raise underflow;
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* +-0: return +-Inf and raise divide-by-zero;
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* finite x ~> 171.63: return +Inf and raise overflow;
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* +Inf: return +Inf;
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* NaN: return NaN.
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*
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* Accuracy: tgamma(x) is accurate to within
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* x > 0: error provably < 0.9ulp.
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* Maximum observed in 1,000,000 trials was .87ulp.
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* x < 0:
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* Maximum observed error < 4ulp in 1,000,000 trials.
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*/
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static double neg_gam(double);
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static double small_gam(double);
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static double smaller_gam(double);
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static struct Double large_gam(double);
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static struct Double ratfun_gam(double, double);
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/*
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* Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
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* [1.066.., 2.066..] accurate to 4.25e-19.
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*/
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#define LEFT -.3955078125 /* left boundary for rat. approx */
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#define x0 .461632144968362356785 /* xmin - 1 */
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#define a0_hi 0.88560319441088874992
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#define a0_lo -.00000000000000004996427036469019695
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#define P0 6.21389571821820863029017800727e-01
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#define P1 2.65757198651533466104979197553e-01
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#define P2 5.53859446429917461063308081748e-03
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#define P3 1.38456698304096573887145282811e-03
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#define P4 2.40659950032711365819348969808e-03
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#define Q0 1.45019531250000000000000000000e+00
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#define Q1 1.06258521948016171343454061571e+00
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#define Q2 -2.07474561943859936441469926649e-01
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#define Q3 -1.46734131782005422506287573015e-01
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#define Q4 3.07878176156175520361557573779e-02
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#define Q5 5.12449347980666221336054633184e-03
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#define Q6 -1.76012741431666995019222898833e-03
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#define Q7 9.35021023573788935372153030556e-05
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#define Q8 6.13275507472443958924745652239e-06
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/*
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* Constants for large x approximation (x in [6, Inf])
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* (Accurate to 2.8*10^-19 absolute)
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*/
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#define lns2pi_hi 0.418945312500000
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#define lns2pi_lo -.000006779295327258219670263595
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#define Pa0 8.33333333333333148296162562474e-02
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#define Pa1 -2.77777777774548123579378966497e-03
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#define Pa2 7.93650778754435631476282786423e-04
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#define Pa3 -5.95235082566672847950717262222e-04
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#define Pa4 8.41428560346653702135821806252e-04
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#define Pa5 -1.89773526463879200348872089421e-03
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#define Pa6 5.69394463439411649408050664078e-03
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#define Pa7 -1.44705562421428915453880392761e-02
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static const double zero = 0., one = 1.0, tiny = 1e-300;
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DLLEXPORT double
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tgamma(x)
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double x;
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{
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struct Double u;
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if (x >= 6) {
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if(x > 171.63)
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return (x / zero);
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u = large_gam(x);
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return(__exp__D(u.a, u.b));
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} else if (x >= 1.0 + LEFT + x0)
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return (small_gam(x));
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else if (x > 1.e-17)
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return (smaller_gam(x));
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else if (x > -1.e-17) {
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if (x != 0.0)
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u.a = one - tiny; /* raise inexact */
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return (one/x);
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} else if (!isfinite(x))
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return (x - x); /* x is NaN or -Inf */
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else
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return (neg_gam(x));
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}
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/*
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* Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
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*/
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static struct Double
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large_gam(x)
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double x;
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{
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double z, p;
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struct Double t, u, v;
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z = one/(x*x);
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p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
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p = p/x;
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u = __log__D(x);
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u.a -= one;
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v.a = (x -= .5);
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TRUNC(v.a);
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v.b = x - v.a;
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t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
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t.b = v.b*u.a + x*u.b;
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/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
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t.b += lns2pi_lo; t.b += p;
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u.a = lns2pi_hi + t.b; u.a += t.a;
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u.b = t.a - u.a;
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u.b += lns2pi_hi; u.b += t.b;
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return (u);
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}
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/*
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* Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
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* It also has correct monotonicity.
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*/
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static double
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small_gam(x)
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double x;
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{
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double y, ym1, t;
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struct Double yy, r;
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y = x - one;
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ym1 = y - one;
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if (y <= 1.0 + (LEFT + x0)) {
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yy = ratfun_gam(y - x0, 0);
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return (yy.a + yy.b);
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}
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r.a = y;
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TRUNC(r.a);
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yy.a = r.a - one;
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y = ym1;
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yy.b = r.b = y - yy.a;
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/* Argument reduction: G(x+1) = x*G(x) */
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for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
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t = r.a*yy.a;
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r.b = r.a*yy.b + y*r.b;
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r.a = t;
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TRUNC(r.a);
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r.b += (t - r.a);
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}
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/* Return r*tgamma(y). */
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yy = ratfun_gam(y - x0, 0);
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y = r.b*(yy.a + yy.b) + r.a*yy.b;
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y += yy.a*r.a;
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return (y);
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}
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/*
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* Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
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*/
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static double
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smaller_gam(x)
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double x;
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{
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double t, d;
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struct Double r, xx;
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if (x < x0 + LEFT) {
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t = x, TRUNC(t);
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d = (t+x)*(x-t);
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t *= t;
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xx.a = (t + x), TRUNC(xx.a);
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xx.b = x - xx.a; xx.b += t; xx.b += d;
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t = (one-x0); t += x;
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d = (one-x0); d -= t; d += x;
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x = xx.a + xx.b;
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} else {
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xx.a = x, TRUNC(xx.a);
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xx.b = x - xx.a;
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t = x - x0;
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d = (-x0 -t); d += x;
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}
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r = ratfun_gam(t, d);
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d = r.a/x, TRUNC(d);
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r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
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return (d + r.a/x);
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}
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/*
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* returns (z+c)^2 * P(z)/Q(z) + a0
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*/
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static struct Double
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ratfun_gam(z, c)
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double z, c;
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{
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double p, q;
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struct Double r, t;
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q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
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p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
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/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
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p = p/q;
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t.a = z, TRUNC(t.a); /* t ~= z + c */
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t.b = (z - t.a) + c;
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t.b *= (t.a + z);
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q = (t.a *= t.a); /* t = (z+c)^2 */
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TRUNC(t.a);
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t.b += (q - t.a);
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r.a = p, TRUNC(r.a); /* r = P/Q */
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r.b = p - r.a;
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t.b = t.b*p + t.a*r.b + a0_lo;
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t.a *= r.a; /* t = (z+c)^2*(P/Q) */
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r.a = t.a + a0_hi, TRUNC(r.a);
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r.b = ((a0_hi-r.a) + t.a) + t.b;
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return (r); /* r = a0 + t */
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}
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static double
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neg_gam(x)
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double x;
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{
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int sgn = 1;
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struct Double lg, lsine;
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double y, z;
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y = ceil(x);
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if (y == x) /* Negative integer. */
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return ((x - x) / zero);
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z = y - x;
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if (z > 0.5)
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z = one - z;
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y = 0.5 * y;
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if (y == ceil(y))
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sgn = -1;
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if (z < .25)
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z = sin(M_PI*z);
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else
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z = cos(M_PI*(0.5-z));
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/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
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if (x < -170) {
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if (x < -190)
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return ((double)sgn*tiny*tiny);
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y = one - x; /* exact: 128 < |x| < 255 */
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lg = large_gam(y);
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lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
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lg.a -= lsine.a; /* exact (opposite signs) */
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lg.b -= lsine.b;
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y = -(lg.a + lg.b);
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z = (y + lg.a) + lg.b;
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y = __exp__D(y, z);
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if (sgn < 0) y = -y;
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return (y);
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}
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y = one-x;
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if (one-y == x)
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y = tgamma(y);
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else /* 1-x is inexact */
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y = -x*tgamma(-x);
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if (sgn < 0) y = -y;
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return (M_PI / (y*z));
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}
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