/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* Long double expansions are
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
and are incorporated herein by permission of the author. The author
reserves the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
/* __ieee754_lgammal_r(x, signgamp)
* Reentrant version of the logarithm of the Gamma function
* with user provide pointer for the sign of Gamma(x).
*
* Method:
* 1. Argument Reduction for 0 < x <= 8
* Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
* reduce x to a number in [1.5,2.5] by
* lgamma(1+s) = log(s) + lgamma(s)
* for example,
* lgamma(7.3) = log(6.3) + lgamma(6.3)
* = log(6.3*5.3) + lgamma(5.3)
* = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
* 2. Polynomial approximation of lgamma around its
* minimun ymin=1.461632144968362245 to maintain monotonicity.
* On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
* Let z = x-ymin;
* lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
* 2. Rational approximation in the primary interval [2,3]
* We use the following approximation:
* s = x-2.0;
* lgamma(x) = 0.5*s + s*P(s)/Q(s)
* Our algorithms are based on the following observation
*
* zeta(2)-1 2 zeta(3)-1 3
* lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
* 2 3
*
* where Euler = 0.5771... is the Euler constant, which is very
* close to 0.5.
*
* 3. For x>=8, we have
* lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
* (better formula:
* lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
* Let z = 1/x, then we approximation
* f(z) = lgamma(x) - (x-0.5)(log(x)-1)
* by
* 3 5 11
* w = w0 + w1*z + w2*z + w3*z + ... + w6*z
*
* 4. For negative x, since (G is gamma function)
* -x*G(-x)*G(x) = pi/sin(pi*x),
* we have
* G(x) = pi/(sin(pi*x)*(-x)*G(-x))
* since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
* Hence, for x<0, signgam = sign(sin(pi*x)) and
* lgamma(x) = log(|Gamma(x)|)
* = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
* Note: one should avoid compute pi*(-x) directly in the
* computation of sin(pi*(-x)).
*
* 5. Special Cases
* lgamma(2+s) ~ s*(1-Euler) for tiny s
* lgamma(1)=lgamma(2)=0
* lgamma(x) ~ -log(x) for tiny x
* lgamma(0) = lgamma(inf) = inf
* lgamma(-integer) = +-inf
*
*/
#include <math.h>
#include <math_private.h>
#include <libc-diag.h>
static const long double
half = 0.5L,
one = 1.0L,
pi = 3.14159265358979323846264L,
two63 = 9.223372036854775808e18L,
/* lgam(1+x) = 0.5 x + x a(x)/b(x)
-0.268402099609375 <= x <= 0
peak relative error 6.6e-22 */
a0 = -6.343246574721079391729402781192128239938E2L,
a1 = 1.856560238672465796768677717168371401378E3L,
a2 = 2.404733102163746263689288466865843408429E3L,
a3 = 8.804188795790383497379532868917517596322E2L,
a4 = 1.135361354097447729740103745999661157426E2L,
a5 = 3.766956539107615557608581581190400021285E0L,
b0 = 8.214973713960928795704317259806842490498E3L,
b1 = 1.026343508841367384879065363925870888012E4L,
b2 = 4.553337477045763320522762343132210919277E3L,
b3 = 8.506975785032585797446253359230031874803E2L,
b4 = 6.042447899703295436820744186992189445813E1L,
/* b5 = 1.000000000000000000000000000000000000000E0 */
tc = 1.4616321449683623412626595423257213284682E0L,
tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
/* tt = (tail of tf), i.e. tf + tt has extended precision. */
tt = 3.3649914684731379602768989080467587736363E-18L,
/* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
/* lgam (x + tc) = tf + tt + x g(x)/h(x)
- 0.230003726999612341262659542325721328468 <= x
<= 0.2699962730003876587373404576742786715318
peak relative error 2.1e-21 */
g0 = 3.645529916721223331888305293534095553827E-18L,
g1 = 5.126654642791082497002594216163574795690E3L,
g2 = 8.828603575854624811911631336122070070327E3L,
g3 = 5.464186426932117031234820886525701595203E3L,
g4 = 1.455427403530884193180776558102868592293E3L,
g5 = 1.541735456969245924860307497029155838446E2L,
g6 = 4.335498275274822298341872707453445815118E0L,
h0 = 1.059584930106085509696730443974495979641E4L,
h1 = 2.147921653490043010629481226937850618860E4L,
h2 = 1.643014770044524804175197151958100656728E4L,
h3 = 5.869021995186925517228323497501767586078E3L,
h4 = 9.764244777714344488787381271643502742293E2L,
h5 = 6.442485441570592541741092969581997002349E1L,
/* h6 = 1.000000000000000000000000000000000000000E0 */
/* lgam (x+1) = -0.5 x + x u(x)/v(x)
-0.100006103515625 <= x <= 0.231639862060546875
peak relative error 1.3e-21 */
u0 = -8.886217500092090678492242071879342025627E1L,
u1 = 6.840109978129177639438792958320783599310E2L,
u2 = 2.042626104514127267855588786511809932433E3L,
u3 = 1.911723903442667422201651063009856064275E3L,
u4 = 7.447065275665887457628865263491667767695E2L,
u5 = 1.132256494121790736268471016493103952637E2L,
u6 = 4.484398885516614191003094714505960972894E0L,
v0 = 1.150830924194461522996462401210374632929E3L,
v1 = 3.399692260848747447377972081399737098610E3L,
v2 = 3.786631705644460255229513563657226008015E3L,
v3 = 1.966450123004478374557778781564114347876E3L,
v4 = 4.741359068914069299837355438370682773122E2L,
v5 = 4.508989649747184050907206782117647852364E1L,
/* v6 = 1.000000000000000000000000000000000000000E0 */
/* lgam (x+2) = .5 x + x s(x)/r(x)
0 <= x <= 1
peak relative error 7.2e-22 */
s0 = 1.454726263410661942989109455292824853344E6L,
s1 = -3.901428390086348447890408306153378922752E6L,
s2 = -6.573568698209374121847873064292963089438E6L,
s3 = -3.319055881485044417245964508099095984643E6L,
s4 = -7.094891568758439227560184618114707107977E5L,
s5 = -6.263426646464505837422314539808112478303E4L,
s6 = -1.684926520999477529949915657519454051529E3L,
r0 = -1.883978160734303518163008696712983134698E7L,
r1 = -2.815206082812062064902202753264922306830E7L,
r2 = -1.600245495251915899081846093343626358398E7L,
r3 = -4.310526301881305003489257052083370058799E6L,
r4 = -5.563807682263923279438235987186184968542E5L,
r5 = -3.027734654434169996032905158145259713083E4L,
r6 = -4.501995652861105629217250715790764371267E2L,
/* r6 = 1.000000000000000000000000000000000000000E0 */
/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
x >= 8
Peak relative error 1.51e-21
w0 = LS2PI - 0.5 */
w0 = 4.189385332046727417803e-1L,
w1 = 8.333333333333331447505E-2L,
w2 = -2.777777777750349603440E-3L,
w3 = 7.936507795855070755671E-4L,
w4 = -5.952345851765688514613E-4L,
w5 = 8.412723297322498080632E-4L,
w6 = -1.880801938119376907179E-3L,
w7 = 4.885026142432270781165E-3L;
static const long double zero = 0.0L;
static long double
sin_pi (long double x)
{
long double y, z;
int n, ix;
uint32_t se, i0, i1;
GET_LDOUBLE_WORDS (se, i0, i1, x);
ix = se & 0x7fff;
ix = (ix << 16) | (i0 >> 16);
if (ix < 0x3ffd8000) /* 0.25 */
return __sinl (pi * x);
y = -x; /* x is assume negative */
/*
* argument reduction, make sure inexact flag not raised if input
* is an integer
*/
z = __floorl (y);
if (z != y)
{ /* inexact anyway */
y *= 0.5;
y = 2.0*(y - __floorl(y)); /* y = |x| mod 2.0 */
n = (int) (y*4.0);
}
else
{
if (ix >= 0x403f8000) /* 2^64 */
{
y = zero; n = 0; /* y must be even */
}
else
{
if (ix < 0x403e8000) /* 2^63 */
z = y + two63; /* exact */
GET_LDOUBLE_WORDS (se, i0, i1, z);
n = i1 & 1;
y = n;
n <<= 2;
}
}
switch (n)
{
case 0:
y = __sinl (pi * y);
break;
case 1:
case 2:
y = __cosl (pi * (half - y));
break;
case 3:
case 4:
y = __sinl (pi * (one - y));
break;
case 5:
case 6:
y = -__cosl (pi * (y - 1.5));
break;
default:
y = __sinl (pi * (y - 2.0));
break;
}
return -y;
}
long double
__ieee754_lgammal_r (long double x, int *signgamp)
{
long double t, y, z, nadj, p, p1, p2, q, r, w;
int i, ix;
uint32_t se, i0, i1;
*signgamp = 1;
GET_LDOUBLE_WORDS (se, i0, i1, x);
ix = se & 0x7fff;
if (__builtin_expect((ix | i0 | i1) == 0, 0))
{
if (se & 0x8000)
*signgamp = -1;
return one / fabsl (x);
}
ix = (ix << 16) | (i0 >> 16);
/* purge off +-inf, NaN, +-0, and negative arguments */
if (__builtin_expect(ix >= 0x7fff0000, 0))
return x * x;
if (__builtin_expect(ix < 0x3fc08000, 0)) /* 2^-63 */
{ /* |x|<2**-63, return -log(|x|) */
if (se & 0x8000)
{
*signgamp = -1;
return -__ieee754_logl (-x);
}
else
return -__ieee754_logl (x);
}
if (se & 0x8000)
{
if (x < -2.0L && x > -33.0L)
return __lgamma_negl (x, signgamp);
t = sin_pi (x);
if (t == zero)
return one / fabsl (t); /* -integer */
nadj = __ieee754_logl (pi / fabsl (t * x));
if (t < zero)
*signgamp = -1;
x = -x;
}
/* purge off 1 and 2 */
if ((((ix - 0x3fff8000) | i0 | i1) == 0)
|| (((ix - 0x40008000) | i0 | i1) == 0))
r = 0;
else if (ix < 0x40008000) /* 2.0 */
{
/* x < 2.0 */
if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
{
/* lgamma(x) = lgamma(x+1) - log(x) */
r = -__ieee754_logl (x);
if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
{
y = x - one;
i = 0;
}
else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
{
y = x - (tc - one);
i = 1;
}
else
{
/* x < 0.23 */
y = x;
i = 2;
}
}
else
{
r = zero;
if (ix >= 0x3fffdda6) /* 1.73162841796875 */
{
/* [1.7316,2] */
y = x - 2.0;
i = 0;
}
else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
{
/* [1.23,1.73] */
y = x - tc;
i = 1;
}
else
{
/* [0.9, 1.23] */
y = x - one;
i = 2;
}
}
switch (i)
{
case 0:
p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
r += half * y + y * p1/p2;
break;
case 1:
p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
p = tt + y * p1/p2;
r += (tf + p);
break;
case 2:
p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
r += (-half * y + p1 / p2);
}
}
else if (ix < 0x40028000) /* 8.0 */
{
/* x < 8.0 */
i = (int) x;
t = zero;
y = x - (double) i;
p = y *
(s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
r = half * y + p / q;
z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
switch (i)
{
case 7:
z *= (y + 6.0); /* FALLTHRU */
case 6:
z *= (y + 5.0); /* FALLTHRU */
case 5:
z *= (y + 4.0); /* FALLTHRU */
case 4:
z *= (y + 3.0); /* FALLTHRU */
case 3:
z *= (y + 2.0); /* FALLTHRU */
r += __ieee754_logl (z);
break;
}
}
else if (ix < 0x40418000) /* 2^66 */
{
/* 8.0 <= x < 2**66 */
t = __ieee754_logl (x);
z = one / x;
y = z * z;
w = w0 + z * (w1
+ y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
r = (x - half) * (t - one) + w;
}
else
/* 2**66 <= x <= inf */
r = x * (__ieee754_logl (x) - one);
/* NADJ is set for negative arguments but not otherwise, resulting
in warnings that it may be used uninitialized although in the
cases where it is used it has always been set. */
DIAG_PUSH_NEEDS_COMMENT;
DIAG_IGNORE_NEEDS_COMMENT (4.9, "-Wmaybe-uninitialized");
if (se & 0x8000)
r = nadj - r;
DIAG_POP_NEEDS_COMMENT;
return r;
}
strong_alias (__ieee754_lgammal_r, __lgammal_r_finite)