da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Copyright (c) 1992, 1993
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * The Regents of the University of California. All rights reserved.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Redistribution and use in source and binary forms, with or without
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * modification, are permitted provided that the following conditions
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * are met:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * 1. Redistributions of source code must retain the above copyright
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * notice, this list of conditions and the following disclaimer.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * 2. Redistributions in binary form must reproduce the above copyright
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * notice, this list of conditions and the following disclaimer in the
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * documentation and/or other materials provided with the distribution.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * 3. Neither the name of the University nor the names of its contributors
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * may be used to endorse or promote products derived from this software
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * without specific prior written permission.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * SUCH DAMAGE.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93";
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#endif /* not lint */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * This code by P. McIlroy, Oct 1992;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * The financial support of UUNET Communications Services is greatfully
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * acknowledged.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * At negative integers, return +Inf, and set errno.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x < 6.5:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Use argument reduction G(x+1) = xG(x) to reach the
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * range [1.066124,2.066124]. Use a rational
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * approximation centered at the minimum (x0+1) to
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * ensure monotonicity.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * adjusted for equal-ripples:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Keep extra precision in multiplying (x-.5)(log(x)-1), to
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * avoid premature round-off.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Special values:
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * non-positive integer: Set overflow trap; return +Inf;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x > 171.63: Set overflow trap; return +Inf;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * NaN: Set invalid trap; return NaN
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Accuracy: Gamma(x) is accurate to within
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * x > 0: error provably < 0.9ulp.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Maximum observed in 1,000,000 trials was .87ulp.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Maximum observed error < 4ulp in 1,000,000 trials.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * [1.066.., 2.066..] accurate to 4.25e-19.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin#define LEFT -.3955078125 /* left boundary for rat. approx */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Constants for large x approximation (x in [6, Inf])
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * (Accurate to 2.8*10^-19 absolute)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic const double zero = 0., one = 1.0, tiny = 1e-300;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * TRUNC sets trailing bits in a floating-point number to zero.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * is a temporary variable.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x >= 6) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if(x > 171.63)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return(__exp__D(u.a, u.b));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (small_gam(x));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (smaller_gam(x));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x == 0.0)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else return (one/x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (one/x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin } else if (!finite(x)) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (x*x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (neg_gam(x));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic struct Double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double z, p;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double t, u, v;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = one/(x*x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin v.a = (x -= .5);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin v.b = x - v.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b = v.b*u.a + x*u.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b += lns2pi_lo; t.b += p;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.a = lns2pi_hi + t.b; u.a += t.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.b = t.a - u.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin u.b += lns2pi_hi; u.b += t.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (u);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * It also has correct monotonicity.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* Argument reduction: G(x+1) = x*G(x) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t = r.a*yy.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b = r.a*yy.b + y*r.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b += (t - r.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* Return r*gamma(y). */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y += yy.a*r.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double t, d;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = (t+x)*(x-t);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = (-x0 -t); d += x;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin d = r.a/x, TRUNC(d);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (d + r.a/x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin * returns (z+c)^2 * P(z)/Q(z) + a0
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic struct Double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double z, c;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double p, q;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin struct Double r, t;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b = (z - t.a) + c;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b *= (t.a + z);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin q = (t.a *= t.a); /* t = (z+c)^2 */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b += (q - t.a);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b = p - r.a;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.b = t.b*p + t.a*r.b + a0_lo;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin t.a *= r.a; /* t = (z+c)^2*(P/Q) */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin r.b = ((a0_hi-r.a) + t.a) + t.b;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (r); /* r = a0 + t */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chinstatic double
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin double y, z;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (y == x) /* Negative integer. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin z = fabs(x - y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (y == ceil(y))
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (z < .25)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x < -170) {
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (x < -190)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (sgn < 0) y = -y;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (y);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (one-y == x)
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin else /* 1-x is inexact */
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin y = -x*gamma(-x);
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin if (sgn < 0) y = -y;
da2e3ebdc1edfbc5028edf1354e7dd2fa69a7968chin return (M_PI / (y*z));