/*- * 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. All advertising materials mentioning features or use of this software * must display the following acknowledgement: * This product includes software developed by the University of * California, Berkeley and its contributors. * 4. 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. */ #ifndef lint static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93"; #endif /* not lint */ #include /* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_tgamma.c,v 1.7 2005/09/19 11:28:19 bde Exp $"); */ /* * This code by P. McIlroy, Oct 1992; * * The financial support of UUNET Communications Services is greatfully * acknowledged. */ //#include #include "../include/math.h" #include "mathimpl.h" #include /* METHOD: * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)) * At negative integers, return +Inf, and set errno. * * x < 6.5: * 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. * * x >= 6.5: Use the asymptotic approximation (Stirling's formula) * adjusted for equal-ripples: * * log(G(x)) ~= (x-.5)*(log(x)-1) + .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. * * Special values: * non-positive integer: Set overflow trap; return +Inf; * x > 171.63: Set overflow trap; return +Inf; * NaN: Set invalid trap; return NaN * * Accuracy: Gamma(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. */ static double neg_gam(double); static double small_gam(double); static double smaller_gam(double); static struct Double large_gam(double); static struct Double ratfun_gam(double, double); /* * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval * [1.066.., 2.066..] accurate to 4.25e-19. */ #define LEFT -.3955078125 /* left boundary for rat. approx */ #define x0 .461632144968362356785 /* xmin - 1 */ #define a0_hi 0.88560319441088874992 #define a0_lo -.00000000000000004996427036469019695 #define P0 6.21389571821820863029017800727e-01 #define P1 2.65757198651533466104979197553e-01 #define P2 5.53859446429917461063308081748e-03 #define P3 1.38456698304096573887145282811e-03 #define P4 2.40659950032711365819348969808e-03 #define Q0 1.45019531250000000000000000000e+00 #define Q1 1.06258521948016171343454061571e+00 #define Q2 -2.07474561943859936441469926649e-01 #define Q3 -1.46734131782005422506287573015e-01 #define Q4 3.07878176156175520361557573779e-02 #define Q5 5.12449347980666221336054633184e-03 #define Q6 -1.76012741431666995019222898833e-03 #define Q7 9.35021023573788935372153030556e-05 #define Q8 6.13275507472443958924745652239e-06 /* * Constants for large x approximation (x in [6, Inf]) * (Accurate to 2.8*10^-19 absolute) */ #define lns2pi_hi 0.418945312500000 #define lns2pi_lo -.000006779295327258219670263595 #define Pa0 8.33333333333333148296162562474e-02 #define Pa1 -2.77777777774548123579378966497e-03 #define Pa2 7.93650778754435631476282786423e-04 #define Pa3 -5.95235082566672847950717262222e-04 #define Pa4 8.41428560346653702135821806252e-04 #define Pa5 -1.89773526463879200348872089421e-03 #define Pa6 5.69394463439411649408050664078e-03 #define Pa7 -1.44705562421428915453880392761e-02 static const double zero = 0., one = 1.0, tiny = 1e-300; double tgamma(x) double x; { struct Double u; if (x >= 6) { if(x > 171.63) return(one/zero); u = large_gam(x); return(__exp__D(u.a, u.b)); } else if (x >= 1.0 + LEFT + x0) return (small_gam(x)); else if (x > 1.e-17) return (smaller_gam(x)); else if (x > -1.e-17) { if (x == 0.0) return (one/x); one+1e-20; /* Raise inexact flag. */ return (one/x); } else if (!finite(x)) return (x*x); /* x = NaN, -Inf */ else return (neg_gam(x)); } /* * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. */ static struct Double large_gam(x) double x; { double z, p; struct Double t, u, v; z = one/(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 -= one; v.a = (x -= .5); TRUNC(v.a); v.b = x - v.a; t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ t.b = v.b*u.a + x*u.b; /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */ t.b += lns2pi_lo; t.b += p; u.a = lns2pi_hi + t.b; u.a += t.a; u.b = t.a - u.a; u.b += lns2pi_hi; u.b += t.b; return (u); } /* * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) * It also has correct monotonicity. */ static double small_gam(x) double x; { double y, ym1, t; struct Double yy, r; y = x - one; ym1 = y - one; if (y <= 1.0 + (LEFT + x0)) { yy = ratfun_gam(y - x0, 0); return (yy.a + yy.b); } r.a = y; TRUNC(r.a); yy.a = r.a - one; y = ym1; yy.b = r.b = y - yy.a; /* Argument reduction: G(x+1) = x*G(x) */ for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) { t = r.a*yy.a; r.b = r.a*yy.b + y*r.b; r.a = t; TRUNC(r.a); 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 1ulp. */ static double smaller_gam(x) double x; { double t, d; struct Double r, xx; if (x < x0 + LEFT) { t = x, TRUNC(t); d = (t+x)*(x-t); t *= t; xx.a = (t + x), TRUNC(xx.a); xx.b = x - xx.a; xx.b += t; xx.b += d; t = (one-x0); t += x; d = (one-x0); d -= t; d += x; x = xx.a + xx.b; } else { xx.a = x, TRUNC(xx.a); xx.b = x - xx.a; t = x - x0; d = (-x0 -t); d += x; } r = ratfun_gam(t, d); d = r.a/x, TRUNC(d); r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b; return (d + r.a/x); } /* * returns (z+c)^2 * P(z)/Q(z) + a0 */ static struct Double ratfun_gam(z, c) double z, c; { double p, q; struct Double r, t; 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))); /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */ p = p/q; t.a = z, TRUNC(t.a); /* t ~= z + c */ t.b = (z - t.a) + c; t.b *= (t.a + z); q = (t.a *= t.a); /* t = (z+c)^2 */ TRUNC(t.a); t.b += (q - t.a); r.a = p, TRUNC(r.a); /* r = P/Q */ r.b = p - r.a; t.b = t.b*p + t.a*r.b + a0_lo; t.a *= r.a; /* t = (z+c)^2*(P/Q) */ r.a = t.a + a0_hi, TRUNC(r.a); r.b = ((a0_hi-r.a) + t.a) + t.b; return (r); /* r = a0 + t */ } static double neg_gam(x) double x; { int sgn = 1; struct Double lg, lsine; double y, z; y = floor(x + .5); if (y == x) /* Negative integer. */ return (one/zero); z = fabs(x - y); y = .5*ceil(x); if (y == ceil(y)) sgn = -1; if (z < .25) z = sin(M_PI*z); else z = cos(M_PI*(0.5-z)); /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ if (x < -170) { if (x < -190) return ((double)sgn*tiny*tiny); y = one - 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 = one-x; if (one-y == x) y = tgamma(y); else /* 1-x is inexact */ y = -x*tgamma(-x); if (sgn < 0) y = -y; return (M_PI / (y*z)); }