2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
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15 * This product includes software developed by the University of
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30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * @(#)gamma.c 8.1 (Berkeley) 6/4/93
34 * FreeBSD SVN: 176449 (2008-02-22)
38 * This code by P. McIlroy, Oct 1992;
40 * The financial support of UUNET Communications Services is greatfully
48 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
49 * At negative integers, return NaN and raise invalid.
52 * Use argument reduction G(x+1) = xG(x) to reach the
53 * range [1.066124,2.066124]. Use a rational
54 * approximation centered at the minimum (x0+1) to
55 * ensure monotonicity.
57 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
58 * adjusted for equal-ripples:
60 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
62 * Keep extra precision in multiplying (x-.5)(log(x)-1), to
63 * avoid premature round-off.
66 * -Inf: return NaN and raise invalid;
67 * negative integer: return NaN and raise invalid;
68 * other x ~< 177.79: return +-0 and raise underflow;
69 * +-0: return +-Inf and raise divide-by-zero;
70 * finite x ~> 171.63: return +Inf and raise overflow;
74 * Accuracy: tgamma(x) is accurate to within
75 * x > 0: error provably < 0.9ulp.
76 * Maximum observed in 1,000,000 trials was .87ulp.
78 * Maximum observed error < 4ulp in 1,000,000 trials.
81 static double neg_gam(double);
82 static double small_gam(double);
83 static double smaller_gam(double);
84 static struct Double large_gam(double);
85 static struct Double ratfun_gam(double, double);
88 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
89 * [1.066.., 2.066..] accurate to 4.25e-19.
91 #define LEFT -.3955078125 /* left boundary for rat. approx */
92 #define x0 .461632144968362356785 /* xmin - 1 */
94 #define a0_hi 0.88560319441088874992
95 #define a0_lo -.00000000000000004996427036469019695
96 #define P0 6.21389571821820863029017800727e-01
97 #define P1 2.65757198651533466104979197553e-01
98 #define P2 5.53859446429917461063308081748e-03
99 #define P3 1.38456698304096573887145282811e-03
100 #define P4 2.40659950032711365819348969808e-03
101 #define Q0 1.45019531250000000000000000000e+00
102 #define Q1 1.06258521948016171343454061571e+00
103 #define Q2 -2.07474561943859936441469926649e-01
104 #define Q3 -1.46734131782005422506287573015e-01
105 #define Q4 3.07878176156175520361557573779e-02
106 #define Q5 5.12449347980666221336054633184e-03
107 #define Q6 -1.76012741431666995019222898833e-03
108 #define Q7 9.35021023573788935372153030556e-05
109 #define Q8 6.13275507472443958924745652239e-06
111 * Constants for large x approximation (x in [6, Inf])
112 * (Accurate to 2.8*10^-19 absolute)
114 #define lns2pi_hi 0.418945312500000
115 #define lns2pi_lo -.000006779295327258219670263595
116 #define Pa0 8.33333333333333148296162562474e-02
117 #define Pa1 -2.77777777774548123579378966497e-03
118 #define Pa2 7.93650778754435631476282786423e-04
119 #define Pa3 -5.95235082566672847950717262222e-04
120 #define Pa4 8.41428560346653702135821806252e-04
121 #define Pa5 -1.89773526463879200348872089421e-03
122 #define Pa6 5.69394463439411649408050664078e-03
123 #define Pa7 -1.44705562421428915453880392761e-02
125 static const double zero = 0., one = 1.0, tiny = 1e-300;
137 return(__exp__D(u.a, u.b));
138 } else if (x >= 1.0 + LEFT + x0)
139 return (small_gam(x));
141 return (smaller_gam(x));
142 else if (x > -1.e-17) {
144 u.a = one - tiny; /* raise inexact */
146 } else if (!finite(x))
147 return (x - x); /* x is NaN or -Inf */
152 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
159 struct Double t, u, v;
162 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
170 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
171 t.b = v.b*u.a + x*u.b;
172 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
173 t.b += lns2pi_lo; t.b += p;
174 u.a = lns2pi_hi + t.b; u.a += t.a;
176 u.b += lns2pi_hi; u.b += t.b;
180 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
181 * It also has correct monotonicity.
191 if (y <= 1.0 + (LEFT + x0)) {
192 yy = ratfun_gam(y - x0, 0);
193 return (yy.a + yy.b);
199 yy.b = r.b = y - yy.a;
200 /* Argument reduction: G(x+1) = x*G(x) */
201 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
203 r.b = r.a*yy.b + y*r.b;
208 /* Return r*tgamma(y). */
209 yy = ratfun_gam(y - x0, 0);
210 y = r.b*(yy.a + yy.b) + r.a*yy.b;
215 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
227 xx.a = (t + x), TRUNC(xx.a);
228 xx.b = x - xx.a; xx.b += t; xx.b += d;
229 t = (one-x0); t += x;
230 d = (one-x0); d -= t; d += x;
233 xx.a = x, TRUNC(xx.a);
236 d = (-x0 -t); d += x;
238 r = ratfun_gam(t, d);
240 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
244 * returns (z+c)^2 * P(z)/Q(z) + a0
253 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
254 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
256 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
258 t.a = z, TRUNC(t.a); /* t ~= z + c */
261 q = (t.a *= t.a); /* t = (z+c)^2 */
264 r.a = p, TRUNC(r.a); /* r = P/Q */
266 t.b = t.b*p + t.a*r.b + a0_lo;
267 t.a *= r.a; /* t = (z+c)^2*(P/Q) */
268 r.a = t.a + a0_hi, TRUNC(r.a);
269 r.b = ((a0_hi-r.a) + t.a) + t.b;
270 return (r); /* r = a0 + t */
278 struct Double lg, lsine;
282 if (y == x) /* Negative integer. */
283 return ((x - x) / zero);
293 z = cos(M_PI*(0.5-z));
294 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
297 return ((double)sgn*tiny*tiny);
298 y = one - x; /* exact: 128 < |x| < 255 */
300 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
301 lg.a -= lsine.a; /* exact (opposite signs) */
304 z = (y + lg.a) + lg.b;
312 else /* 1-x is inexact */
315 return (M_PI / (y*z));