''' gamma function and its natural logarithm'''
##/* @(#)er_lgamma.c 5.1 93/09/24 */
##/*
## * ====================================================
## * 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.
## * ====================================================
## *
## */
##original code from: http://www.sourceware.org/cgi-bin/cvsweb.cgi/~checkout~/src/newlib/libm/mathfp/er_lgamma.c?rev=1.6&content-type=text/plain&cvsroot=src
##/* 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)
## * where
## * poly(z) is a 14 degree polynomial.
## * 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)
## * with accuracy
## * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
## * 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
## * where
## * |w - f(z)| < 2**-58.74
## *
## * 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
## *
## */
##
from math import *
two52= 4.50359962737049600000e+15
half= 5.00000000000000000000e-01
one = 1.00000000000000000000e+00
pi = 3.14159265358979311600e+00
a0 = 7.72156649015328655494e-02
a1 = 3.22467033424113591611e-01
a2 = 6.73523010531292681824e-02
a3 = 2.05808084325167332806e-02
a4 = 7.38555086081402883957e-03
a5 = 2.89051383673415629091e-03
a6 = 1.19270763183362067845e-03
a7 = 5.10069792153511336608e-04
a8 = 2.20862790713908385557e-04
a9 = 1.08011567247583939954e-04
a10 = 2.52144565451257326939e-05
a11 = 4.48640949618915160150e-05
tc = 1.46163214496836224576e+00
tf = -1.21486290535849611461e-01
##/* tt = -(tail of tf) */
tt = -3.63867699703950536541e-18
t0 = 4.83836122723810047042e-01
t1 = -1.47587722994593911752e-01
t2 = 6.46249402391333854778e-02
t3 = -3.27885410759859649565e-02
t4 = 1.79706750811820387126e-02
t5 = -1.03142241298341437450e-02
t6 = 6.10053870246291332635e-03
t7 = -3.68452016781138256760e-03
t8 = 2.25964780900612472250e-03
t9 = -1.40346469989232843813e-03
t10 = 8.81081882437654011382e-04
t11 = -5.38595305356740546715e-04
t12 = 3.15632070903625950361e-04
t13 = -3.12754168375120860518e-04
t14 = 3.35529192635519073543e-04
u0 = -7.72156649015328655494e-02
u1 = 6.32827064025093366517e-01
u2 = 1.45492250137234768737e+00
u3 = 9.77717527963372745603e-01
u4 = 2.28963728064692451092e-01
u5 = 1.33810918536787660377e-02
v1 = 2.45597793713041134822e+00
v2 = 2.12848976379893395361e+00
v3 = 7.69285150456672783825e-01
v4 = 1.04222645593369134254e-01
v5 = 3.21709242282423911810e-03
s0 = -7.72156649015328655494e-02
s1 = 2.14982415960608852501e-01
s2 = 3.25778796408930981787e-01
s3 = 1.46350472652464452805e-01
s4 = 2.66422703033638609560e-02
s5 = 1.84028451407337715652e-03
s6 = 3.19475326584100867617e-05
r1 = 1.39200533467621045958e+00
r2 = 7.21935547567138069525e-01
r3 = 1.71933865632803078993e-01
r4 = 1.86459191715652901344e-02
r5 = 7.77942496381893596434e-04
r6 = 7.32668430744625636189e-06
w0 = 4.18938533204672725052e-01
w1 = 8.33333333333329678849e-02
w2 = -2.77777777728775536470e-03
w3 = 7.93650558643019558500e-04
w4 = -5.95187557450339963135e-04
w5 = 8.36339918996282139126e-04
w6 = -1.63092934096575273989e-03
zero= 0.00000000000000000000e+00
##inf = float('inf')
##nan = float('nan')
inf = float(9e999)
def sin_pi(x):
x = float(x)
e,ix = frexp(x)
if(abs(x)<0.25):
return -sin(pi*x)
y = -x ##/* x is assume negative */
## * argument reduction, make sure inexact flag not raised if input
## * is an integer
z = floor(y)
if(z!=y):
y *= 0.5
y = 2.0*(y - floor(y)) ##/* y = |x| mod 2.0 */
n = int(y*4.0)
else:
if(abs(ix)>=53):
y = zero
n = 0 ##/* y must be even */
else:
if(abs(ix)<52):
z = y+two52 ##/* exact */
e,n=frexp(z)
n &= 1
y = n
n<<= 2
if n == 0:
y = sin(pi*y)
elif (n == 1 or n == 2):
y = cos(pi*(0.5-y))
elif (n == 3 or n == 4):
y = sin(pi*(one-y))
elif (n == 5 or n == 6):
y = -cos(pi*(y-1.5))
else:
y = sin(pi*(y-2.0))
z = cos(pi*(z+1.0));
return -y*z
def Lgamma(x):
'''return natural logarithm of gamma function of x
raise ValueError if x is negative integer'''
x = float(x)
##/* purge off +-inf, NaN, +-0, and negative arguments */
if ((x == inf) or (x == -inf)):
return inf
## if (x is nan):
## return nan
e,ix = frexp(x)
nadj = 0
signgamp = 1
if ((e==0.0) and (ix==0)):
return inf
if (ix>1020):
return inf
if ((e != 0.0) and (ix<-71)):
if (x<0):
return -log(-x)
else:
return -log(x)
if (e<0):
if (ix>52):
return inf ##one/zero
t = sin_pi(x)
if (t==zero):
##return inf
raise ValueError('gamma not defined for negative integer')
nadj = log(pi/fabs(t*x))
if (t<zero):
signgamp = -1
x = -x
##/* purge off 1 and 2 */
if ((x==2.0) or (x==1.0)):
r = 0.0
##/* for x < 2.0 */
elif (ix<2):
if(x<=0.9): ##/* lgamma(x) = lgamma(x+1)-log(x) */
r = -log(x)
if (x>=0.7316):
y = one-x
z = y*y
p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))))
p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))))
p = y*p1+p2
r += (p-0.5*y)
elif (x>=0.23164):
y= x-(tc-one)
z = y*y
w = z*y
p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))) ## /* parallel comp */
p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)))
p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)))
p = z*p1-(tt-w*(p2+y*p3))
r += (tf + p)
else:
y = x
p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))))
p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))))
r += (-0.5*y + p1/p2)
else:
r = zero
if(x>=1.7316):
y=2.0-x ##/* [1.7316,2] */
z = y*y
p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))))
p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))))
p = y*p1+p2
r += (p-0.5*y)
elif(x>=1.23164):
y=x-tc ##/* [1.23,1.73] */
z = y*y
w = z*y
p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))) ## /* parallel comp */
p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)))
p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)))
p = z*p1-(tt-w*(p2+y*p3))
r += (tf + p)
else:
y=x-one
p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))))
p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))))
r += (-0.5*y + p1/p2)
##/* x < 8.0 */
elif(ix<4):
i = int(x)
t = zero
y = x-i
p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))))
q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))))
r = half*y+p/q
z = one ##/* lgamma(1+s) = log(s) + lgamma(s) */
while (i>2):
i -=1
z *= (y+i)
r += log(z)
##/* 8.0 <= x < 2**58 */
elif (ix<58):
t = log(x)
z = one/x
y = z*z
w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))))
r = (x-half)*(t-one)+w
##/* 2**58 <= x <= inf */
else:
r = x*(log(x)-one)
if (e<0):
r = nadj - r
return signgamp*r
def Gamma(x):
'''return gamma function of x
raise ValueError if x is negative integer'''
x = float(x)
if x == 0.0:
return inf
s = 1.0
if (x<0):
s = cos(pi*floor(x))
return s*exp(Lgamma(x))
