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Diffstat (limited to 'lib/msun/bsdsrc/b_tgamma.c')
| -rw-r--r-- | lib/msun/bsdsrc/b_tgamma.c | 396 |
1 files changed, 396 insertions, 0 deletions
diff --git a/lib/msun/bsdsrc/b_tgamma.c b/lib/msun/bsdsrc/b_tgamma.c new file mode 100644 index 000000000000..a7e97bc777c3 --- /dev/null +++ b/lib/msun/bsdsrc/b_tgamma.c @@ -0,0 +1,396 @@ +/*- + * SPDX-License-Identifier: BSD-3-Clause + * + * 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. 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. + */ + +/* + * The original code, FreeBSD's old svn r93211, contained the following + * attribution: + * + * This code by P. McIlroy, Oct 1992; + * + * The financial support of UUNET Communications Services is greatfully + * acknowledged. + * + * The algorithm remains, but the code has been re-arranged to facilitate + * porting to other precisions. + */ + +#include <float.h> + +#include "math.h" +#include "math_private.h" + +/* Used in b_log.c and below. */ +struct Double { + double a; + double b; +}; + +#include "b_log.c" +#include "b_exp.c" + +/* + * The range is broken into several subranges. Each is handled by its + * helper functions. + * + * x >= 6.0: large_gam(x) + * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0. + * xleft > x > iota: smaller_gam(x) where iota = 1e-17. + * iota > x > -itoa: Handle x near 0. + * -iota > x : neg_gam + * + * Special values: + * -Inf: return NaN and raise invalid; + * negative integer: return NaN and raise invalid; + * other x ~< 177.79: return +-0 and raise underflow; + * +-0: return +-Inf and raise divide-by-zero; + * finite x ~> 171.63: return +Inf and raise overflow; + * +Inf: return +Inf; + * NaN: return NaN. + * + * Accuracy: tgamma(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. + */ + +/* + * Constants for large x approximation (x in [6, Inf]) + * (Accurate to 2.8*10^-19 absolute) + */ + +static const double zero = 0.; +static const volatile double tiny = 1e-300; +/* + * x >= 6 + * + * Use the asymptotic approximation (Stirling's formula) adjusted fof + * equal-ripples: + * + * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(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. + * + * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error. + */ +static const double + ln2pi_hi = 0.41894531250000000, + ln2pi_lo = -6.7792953272582197e-6, + Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */ + Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */ + Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */ + Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */ + Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */ + Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */ + Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */ + Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */ + +static struct Double +large_gam(double x) +{ + double p, z, thi, tlo, xhi, xlo; + struct Double u; + + z = 1 / (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 -= 1; + + /* Split (x - 0.5) in high and low parts. */ + x -= 0.5; + xhi = (float)x; + xlo = x - xhi; + + /* Compute t = (x-.5)*(log(x)-1) in extra precision. */ + thi = xhi * u.a; + tlo = xlo * u.a + x * u.b; + + /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */ + tlo += ln2pi_lo; + tlo += p; + u.a = ln2pi_hi + tlo; + u.a += thi; + u.b = thi - u.a; + u.b += ln2pi_hi; + u.b += tlo; + return (u); +} +/* + * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval + * [1.066.., 2.066..] accurate to 4.25e-19. + * + * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated. + */ +static const double +#if 0 + a0_hi = 8.8560319441088875e-1, + a0_lo = -4.9964270364690197e-17, +#else + a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */ + a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */ +#endif + P0 = 6.2138957182182086e-1, + P1 = 2.6575719865153347e-1, + P2 = 5.5385944642991746e-3, + P3 = 1.3845669830409657e-3, + P4 = 2.4065995003271137e-3, + Q0 = 1.4501953125000000e+0, + Q1 = 1.0625852194801617e+0, + Q2 = -2.0747456194385994e-1, + Q3 = -1.4673413178200542e-1, + Q4 = 3.0787817615617552e-2, + Q5 = 5.1244934798066622e-3, + Q6 = -1.7601274143166700e-3, + Q7 = 9.3502102357378894e-5, + Q8 = 6.1327550747244396e-6; + +static struct Double +ratfun_gam(double z, double c) +{ + double p, q, thi, tlo; + struct Double r; + + 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))); + p = p / q; + + /* Split z into high and low parts. */ + thi = (float)z; + tlo = (z - thi) + c; + tlo *= (thi + z); + + /* Split (z+c)^2 into high and low parts. */ + thi *= thi; + q = thi; + thi = (float)thi; + tlo += (q - thi); + + /* Split p/q into high and low parts. */ + r.a = (float)p; + r.b = p - r.a; + + tlo = tlo * p + thi * r.b + a0_lo; + thi *= r.a; /* t = (z+c)^2*(P/Q) */ + r.a = (float)(thi + a0_hi); + r.b = ((a0_hi - r.a) + thi) + tlo; + return (r); /* r = a0 + t */ +} +/* + * x < 6 + * + * 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. + * + * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.) + * It also has correct monotonicity. + */ +static const double + left = -0.3955078125, /* left boundary for rat. approx */ + x0 = 4.6163214496836236e-1; /* xmin - 1 */ + +static double +small_gam(double x) +{ + double t, y, ym1; + struct Double yy, r; + + y = x - 1; + if (y <= 1 + (left + x0)) { + yy = ratfun_gam(y - x0, 0); + return (yy.a + yy.b); + } + + r.a = (float)y; + yy.a = r.a - 1; + y = y - 1 ; + r.b = yy.b = y - yy.a; + + /* Argument reduction: G(x+1) = x*G(x) */ + for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) { + t = r.a * yy.a; + r.b = r.a * yy.b + y * r.b; + r.a = (float)t; + 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 1 ulp. + */ +static double +smaller_gam(double x) +{ + double d, t, xhi, xlo; + struct Double r; + + if (x < x0 + left) { + t = (float)x; + d = (t + x) * (x - t); + t *= t; + xhi = (float)(t + x); + xlo = x - xhi; + xlo += t; + xlo += d; + t = 1 - x0; + t += x; + d = 1 - x0; + d -= t; + d += x; + x = xhi + xlo; + } else { + xhi = (float)x; + xlo = x - xhi; + t = x - x0; + d = - x0 - t; + d += x; + } + + r = ratfun_gam(t, d); + d = (float)(r.a / x); + r.a -= d * xhi; + r.a -= d * xlo; + r.a += r.b; + + return (d + r.a / x); +} +/* + * x < 0 + * + * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)). + * At negative integers, return NaN and raise invalid. + */ +static double +neg_gam(double x) +{ + int sgn = 1; + struct Double lg, lsine; + double y, z; + + y = ceil(x); + if (y == x) /* Negative integer. */ + return ((x - x) / zero); + + z = y - x; + if (z > 0.5) + z = 1 - z; + + y = y / 2; + if (y == ceil(y)) + sgn = -1; + + if (z < 0.25) + z = sinpi(z); + else + z = cospi(0.5 - z); + + /* Special case: G(1-x) = Inf; G(x) may be nonzero. */ + if (x < -170) { + + if (x < -190) + return (sgn * tiny * tiny); + + y = 1 - 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 = 1 - x; + if (1 - y == x) + y = tgamma(y); + else /* 1-x is inexact */ + y = - x * tgamma(-x); + + if (sgn < 0) y = -y; + return (M_PI / (y * z)); +} +/* + * xmax comes from lgamma(xmax) - emax * log(2) = 0. + * static const float xmax = 35.040095f + * static const double xmax = 171.624376956302725; + * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L), + * ld128: 1.75554834290446291700388921607020320e+03L, + * + * iota is a sloppy threshold to isolate x = 0. + */ +static const double xmax = 171.624376956302725; +static const double iota = 0x1p-56; + +double +tgamma(double x) +{ + struct Double u; + + if (x >= 6) { + if (x > xmax) + return (x / zero); + u = large_gam(x); + return (__exp__D(u.a, u.b)); + } + + if (x >= 1 + left + x0) + return (small_gam(x)); + + if (x > iota) + return (smaller_gam(x)); + + if (x > -iota) { + if (x != 0.) + u.a = 1 - tiny; /* raise inexact */ + return (1 / x); + } + + if (!isfinite(x)) + return (x - x); /* x is NaN or -Inf */ + + return (neg_gam(x)); +} + +#if (LDBL_MANT_DIG == 53) +__weak_reference(tgamma, tgammal); +#endif |