311 lines
8.4 KiB
C
311 lines
8.4 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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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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#ifndef lint
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static char sccsid[] = "@(#)lgamma.c 8.2 (Berkeley) 11/30/93";
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#endif /* not lint */
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/*
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* Coded by Peter McIlroy, Nov 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 <errno.h>
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#include "mathimpl.h"
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/* Log gamma function.
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* Error: x > 0 error < 1.3ulp.
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* x > 4, error < 1ulp.
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* x > 9, error < .6ulp.
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* x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
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* Method:
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* x > 6:
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* Use the asymptotic expansion (Stirling's Formula)
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* 0 < x < 6:
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* Use gamma(x+1) = x*gamma(x) for argument reduction.
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* Use rational approximation in
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* the range 1.2, 2.5
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* Two approximations are used, one centered at the
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* minimum to ensure monotonicity; one centered at 2
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* to maintain small relative error.
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* x < 0:
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* Use the reflection formula,
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* G(1-x)G(x) = PI/sin(PI*x)
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* Special values:
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* non-positive integer returns +Inf.
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* NaN returns NaN
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*/
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static int endian;
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#if defined(vax) || defined(tahoe)
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#define _IEEE 0
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/* double and float have same size exponent field */
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#define TRUNC(x) x = (double) (float) (x)
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#else
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#define _IEEE 1
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#define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
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#define infnan(x) 0.0
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#endif
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static double small_lgam(double);
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static double large_lgam(double);
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static double neg_lgam(double);
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static double zero = 0.0, one = 1.0;
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int signgam;
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#define UNDERFL (1e-1020 * 1e-1020)
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#define LEFT (1.0 - (x0 + .25))
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#define RIGHT (x0 - .218)
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/*
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* Constants for approximation in [1.244,1.712]
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*/
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#define x0 0.461632144968362356785
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#define x0_lo -.000000000000000015522348162858676890521
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#define a0_hi -0.12148629128932952880859
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#define a0_lo .0000000007534799204229502
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#define r0 -2.771227512955130520e-002
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#define r1 -2.980729795228150847e-001
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#define r2 -3.257411333183093394e-001
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#define r3 -1.126814387531706041e-001
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#define r4 -1.129130057170225562e-002
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#define r5 -2.259650588213369095e-005
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#define s0 1.714457160001714442e+000
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#define s1 2.786469504618194648e+000
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#define s2 1.564546365519179805e+000
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#define s3 3.485846389981109850e-001
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#define s4 2.467759345363656348e-002
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/*
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* Constants for approximation in [1.71, 2.5]
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*/
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#define a1_hi 4.227843350984671344505727574870e-01
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#define a1_lo 4.670126436531227189e-18
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#define p0 3.224670334241133695662995251041e-01
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#define p1 3.569659696950364669021382724168e-01
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#define p2 1.342918716072560025853732668111e-01
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#define p3 1.950702176409779831089963408886e-02
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#define p4 8.546740251667538090796227834289e-04
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#define q0 1.000000000000000444089209850062e+00
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#define q1 1.315850076960161985084596381057e+00
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#define q2 6.274644311862156431658377186977e-01
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#define q3 1.304706631926259297049597307705e-01
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#define q4 1.102815279606722369265536798366e-02
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#define q5 2.512690594856678929537585620579e-04
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#define q6 -1.003597548112371003358107325598e-06
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/*
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* Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
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*/
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#define lns2pi .418938533204672741780329736405
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#define pb0 8.33333333333333148296162562474e-02
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#define pb1 -2.77777777774548123579378966497e-03
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#define pb2 7.93650778754435631476282786423e-04
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#define pb3 -5.95235082566672847950717262222e-04
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#define pb4 8.41428560346653702135821806252e-04
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#define pb5 -1.89773526463879200348872089421e-03
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#define pb6 5.69394463439411649408050664078e-03
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#define pb7 -1.44705562421428915453880392761e-02
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__pure double
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lgamma(double x)
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{
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double r;
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signgam = 1;
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endian = ((*(int *) &one)) ? 1 : 0;
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if (!finite(x))
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if (_IEEE)
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return (x+x);
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else return (infnan(EDOM));
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if (x > 6 + RIGHT) {
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r = large_lgam(x);
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return (r);
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} else if (x > 1e-16)
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return (small_lgam(x));
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else if (x > -1e-16) {
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if (x < 0)
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signgam = -1, x = -x;
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return (-log(x));
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} else
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return (neg_lgam(x));
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}
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static double
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large_lgam(double x)
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{
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double z, p, x1;
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int i;
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struct Double t, u, v;
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u = __log__D(x);
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u.a -= 1.0;
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if (x > 1e15) {
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v.a = x - 0.5;
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TRUNC(v.a);
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v.b = (x - v.a) - 0.5;
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t.a = u.a*v.a;
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t.b = x*u.b + v.b*u.a;
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if (_IEEE == 0 && !finite(t.a))
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return(infnan(ERANGE));
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return(t.a + t.b);
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}
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x1 = 1./x;
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z = x1*x1;
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p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
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/* error in approximation = 2.8e-19 */
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p = p*x1; /* error < 2.3e-18 absolute */
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/* 0 < p < 1/64 (at x = 5.5) */
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v.a = x = x - 0.5;
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TRUNC(v.a); /* truncate v.a to 26 bits. */
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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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t.b += p; t.b += lns2pi; /* return t + lns2pi + p */
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return (t.a + t.b);
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}
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static double
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small_lgam(double x)
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{
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int x_int;
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double y, z, t, r = 0, p, q, hi, lo;
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struct Double rr;
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x_int = (x + .5);
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y = x - x_int;
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if (x_int <= 2 && y > RIGHT) {
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t = y - x0;
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y--; x_int++;
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goto CONTINUE;
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} else if (y < -LEFT) {
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t = y +(1.0-x0);
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CONTINUE:
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z = t - x0_lo;
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p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
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q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
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r = t*(z*(p/q) - x0_lo);
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t = .5*t*t;
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z = 1.0;
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switch (x_int) {
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case 6: z = (y + 5);
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case 5: z *= (y + 4);
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case 4: z *= (y + 3);
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case 3: z *= (y + 2);
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rr = __log__D(z);
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rr.b += a0_lo; rr.a += a0_hi;
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return(((r+rr.b)+t+rr.a));
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case 2: return(((r+a0_lo)+t)+a0_hi);
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case 0: r -= log1p(x);
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default: rr = __log__D(x);
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rr.a -= a0_hi; rr.b -= a0_lo;
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return(((r - rr.b) + t) - rr.a);
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}
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} else {
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p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
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q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
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p = p*(y/q);
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t = (double)(float) y;
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z = y-t;
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hi = (double)(float) (p+a1_hi);
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lo = a1_hi - hi; lo += p; lo += a1_lo;
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r = lo*y + z*hi; /* q + r = y*(a0+p/q) */
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q = hi*t;
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z = 1.0;
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switch (x_int) {
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case 6: z = (y + 5);
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case 5: z *= (y + 4);
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case 4: z *= (y + 3);
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case 3: z *= (y + 2);
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rr = __log__D(z);
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r += rr.b; r += q;
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return(rr.a + r);
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case 2: return (q+ r);
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case 0: rr = __log__D(x);
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r -= rr.b; r -= log1p(x);
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r += q; r-= rr.a;
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return(r);
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default: rr = __log__D(x);
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r -= rr.b;
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q -= rr.a;
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return (r+q);
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}
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}
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}
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static double
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neg_lgam(double x)
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{
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int xi;
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double y, z, one = 1.0, zero = 0.0;
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extern double gamma();
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/* avoid destructive cancellation as much as possible */
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if (x > -170) {
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xi = x;
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if (xi == x)
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if (_IEEE)
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return(one/zero);
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else
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return(infnan(ERANGE));
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y = gamma(x);
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if (y < 0)
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y = -y, signgam = -1;
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return (log(y));
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}
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z = floor(x + .5);
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if (z == x) { /* convention: G(-(integer)) -> +Inf */
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if (_IEEE)
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return (one/zero);
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else
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return (infnan(ERANGE));
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}
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y = .5*ceil(x);
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if (y == ceil(y))
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signgam = -1;
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x = -x;
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z = fabs(x + z); /* 0 < z <= .5 */
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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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z = log(M_PI/(z*x));
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y = large_lgam(x);
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return (z - y);
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}
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