0b42281ee9
to doubles as bits. fdlibm-1.1 had similar aliasing bugs, but these were fixed by NetBSD or Cygnus before a modified version of fdlibm was imported in 1994. TRUNC() is only used by tgamma() and some implementation-detail functions. The aliasing bugs were detected by compiling with gcc -O2 but don't seem to have broken tgamma() on i386's or amd64's. They broke my modified version of tgamma(). Moved the definition of TRUNC() to mathimpl.h so that it can be fixed in one place, although the general version is even slower than necessary because it has to operate on pointers to volatiles to handle its arg sometimes being volatile. Inefficiency of the fdlibm macros slows down libm generally, and tgamma() is a relatively unimportant part of libm. The macros act as if on 32-bit words in memory, so they are hard to optimize to direct actions on 64-bit double registers for (non-i386) machines where this is possible. The optimization is too hard for gcc on amd64's, and declaring variables as volatile makes it impossible.
316 lines
8.6 KiB
C
316 lines
8.6 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. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* This product includes software developed by the University of
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* California, Berkeley and its contributors.
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* 4. 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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#ifndef lint
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static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93";
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#endif /* not lint */
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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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 <math.h>
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#include "mathimpl.h"
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#include <errno.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 +Inf, and set errno.
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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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* non-positive integer: Set overflow trap; return +Inf;
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* x > 171.63: Set overflow trap; return +Inf;
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* NaN: Set invalid trap; return NaN
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*
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* Accuracy: Gamma(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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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(one/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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return (one/x);
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one+1e-20; /* Raise inexact flag. */
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return (one/x);
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} else if (!finite(x))
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return (x*x); /* x = NaN, -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 = floor(x + .5);
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if (y == x) /* Negative integer. */
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return (one/zero);
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z = fabs(x - y);
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y = .5*ceil(x);
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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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