/*
* ====================================================
* 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.
* ====================================================
*/
/*
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
*
* Permission to use, copy, modify, and distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
*/
/* 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 <openlibm_math.h>
#include "math_private.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;
u_int32_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
lgammal_r(long double x, int *signgamp)
{
long double t, y, z, nadj, p, p1, p2, q, r, w;
int i, ix;
u_int32_t se, i0, i1;
*signgamp = 1;
GET_LDOUBLE_WORDS (se, i0, i1, x);
ix = se & 0x7fff;
if ((ix | i0 | i1) == 0)
{
if (se & 0x8000)
*signgamp = -1;
return one / fabsl (x);
}
ix = (ix << 16) | (i0 >> 16);
/* purge off +-inf, NaN, +-0, and negative arguments */
if (ix >= 0x7fff0000)
return x * x;
if (ix < 0x3fc08000) /* 2^-63 */
{ /* |x|<2**-63, return -log(|x|) */
if (se & 0x8000)
{
*signgamp = -1;
return -logl (-x);
}
else
return -logl (x);
}
if (se & 0x8000)
{
t = sin_pi (x);
if (t == zero)
return one / fabsl (t); /* -integer */
nadj = 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 = -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 += logl (z);
break;
}
}
else if (ix < 0x40418000) /* 2^66 */
{
/* 8.0 <= x < 2**66 */
t = 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 * (logl (x) - one);
if (se & 0x8000)
r = nadj - r;
return r;
}