'''error function and complementary error function
'''
##/*
## * ====================================================
## * 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.
## * ====================================================
## */
##
##/* double erf(double x)
## * double erfc(double x)
## * original code from: http://sourceware.org/cgi-bin/cvsweb.cgi/~checkout~/src/newlib/libm/math/s_erf.c?rev=1.1.1.1&cvsroot=src
## * x
## * 2 |\
## * erf(x) = --------- | exp(-t*t)dt
## * sqrt(pi) \|
## * 0
## *
## * erfc(x) = 1-erf(x)
## * Note that
## * erf(-x) = -erf(x)
## * erfc(-x) = 2 - erfc(x)
## *
## * Method:
## * 1. For |x| in [0, 0.84375]
## * erf(x) = x + x*R(x^2)
## * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
## * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
## * where R = P/Q where P is an odd poly of degree 8 and
## * Q is an odd poly of degree 10.
## * -57.90
## * | R - (erf(x)-x)/x | <= 2
## *
## *
## * Remark. The formula is derived by noting
## * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
## * and that
## * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
## * is close to one. The interval is chosen because the fix
## * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
## * near 0.6174), and by some experiment, 0.84375 is chosen to
## * guarantee the error is less than one ulp for erf.
## *
## * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
## * c = 0.84506291151 rounded to single (24 bits)
## * erf(x) = sign(x) * (c + P1(s)/Q1(s))
## * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
## * 1+(c+P1(s)/Q1(s)) if x < 0
## * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
## * Remark: here we use the taylor series expansion at x=1.
## * erf(1+s) = erf(1) + s*Poly(s)
## * = 0.845.. + P1(s)/Q1(s)
## * That is, we use rational approximation to approximate
## * erf(1+s) - (c = (single)0.84506291151)
## * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
## * where
## * P1(s) = degree 6 poly in s
## * Q1(s) = degree 6 poly in s
## *
## * 3. For x in [1.25,1/0.35(~2.857143)],
## * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
## * erf(x) = 1 - erfc(x)
## * where
## * R1(z) = degree 7 poly in z, (z=1/x^2)
## * S1(z) = degree 8 poly in z
## *
## * 4. For x in [1/0.35,28]
## * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
## * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
## * = 2.0 - tiny (if x <= -6)
## * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
## * erf(x) = sign(x)*(1.0 - tiny)
## * where
## * R2(z) = degree 6 poly in z, (z=1/x^2)
## * S2(z) = degree 7 poly in z
## *
## * Note1:
## * To compute exp(-x*x-0.5625+R/S), let s be a single
## * precision number and s := x; then
## * -x*x = -s*s + (s-x)*(s+x)
## * exp(-x*x-0.5626+R/S) =
## * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
## * Note2:
## * Here 4 and 5 make use of the asymptotic series
## * exp(-x*x)
## * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
## * x*sqrt(pi)
## * We use rational approximation to approximate
## * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
## * Here is the error bound for R1/S1 and R2/S2
## * |R1/S1 - f(x)| < 2**(-62.57)
## * |R2/S2 - f(x)| < 2**(-61.52)
## *
## * 5. For inf > x >= 28
## * erf(x) = sign(x) *(1 - tiny) (raise inexact)
## * erfc(x) = tiny*tiny (raise underflow) if x > 0
## * = 2 - tiny if x<0
## *
## * 7. Special case:
## * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
## * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
## * erfc/erf(NaN) is NaN
## */
from math import *
tiny= 1e-300
half= 5.00000000000000000000e-01
one = 1.00000000000000000000e+00
two = 2.00000000000000000000e+00
erx = 8.45062911510467529297e-01
## Coefficients for approximation to erf on [0,0.84375]
efx = 1.28379167095512586316e-01
efx8= 1.02703333676410069053e+00
pp0 = 1.28379167095512558561e-01
pp1 = -3.25042107247001499370e-01
pp2 = -2.84817495755985104766e-02
pp3 = -5.77027029648944159157e-03
pp4 = -2.37630166566501626084e-05
qq1 = 3.97917223959155352819e-01
qq2 = 6.50222499887672944485e-02
qq3 = 5.08130628187576562776e-03
qq4 = 1.32494738004321644526e-04
qq5 = -3.96022827877536812320e-06
def erf1(x):
'''erf(x) for x in [0,0.84375]'''
e, i = frexp(x)
if abs(i)>28:
if abs(i)>57:
return 0.125*(8.0*x+efx8*x)
return x + efx*x
z = x*x
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)))
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
y = r/s
return x + x*y
def erfc1(x):
'''erfc(x)for x in [0,0.84375]'''
e,i = frexp(x)
if abs(i)>56:
return one-x
z = x*x
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)))
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
y = r/s
if (x<0.25):
return one-(x+x*y)
else:
r = x*y
r += (x-half)
return half - r
## Coefficients for approximation to erf in [0.84375,1.25]
pa0 = -2.36211856075265944077e-03
pa1 = 4.14856118683748331666e-01
pa2 = -3.72207876035701323847e-01
pa3 = 3.18346619901161753674e-01
pa4 = -1.10894694282396677476e-01
pa5 = 3.54783043256182359371e-02
pa6 = -2.16637559486879084300e-03
qa1 = 1.06420880400844228286e-01
qa2 = 5.40397917702171048937e-01
qa3 = 7.18286544141962662868e-02
qa4 = 1.26171219808761642112e-01
qa5 = 1.36370839120290507362e-02
qa6 = 1.19844998467991074170e-02
def erf2(x):
'''erf(x) for x in [0.84375,1.25]'''
s = fabs(x)-one
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
if x>=0:
return erx + P/Q
return -erx - P/Q
def erfc2(x):
'''erfc(x) for x in [0.84375, 1.25]'''
return one-erf2(x)
## Coefficients for approximation to erfc in [1.25,1/0.35]
ra0 = -9.86494403484714822705e-03
ra1 = -6.93858572707181764372e-01
ra2 = -1.05586262253232909814e+01
ra3 = -6.23753324503260060396e+01
ra4 = -1.62396669462573470355e+02
ra5 = -1.84605092906711035994e+02
ra6 = -8.12874355063065934246e+01
ra7 = -9.81432934416914548592e+00
sa1 = 1.96512716674392571292e+01
sa2 = 1.37657754143519042600e+02
sa3 = 4.34565877475229228821e+02
sa4 = 6.45387271733267880336e+02
sa5 = 4.29008140027567833386e+02
sa6 = 1.08635005541779435134e+02
sa7 = 6.57024977031928170135e+00
sa8 = -6.04244152148580987438e-02
def erf3(x):
'''erf(x) for x in [1.25,2.857142]'''
x0=x
x = fabs(x)
s = one/(x*x)
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
z = ldexp(x0,0)
r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)
if(x0>=0):
return one-r/x
else:
return r/x-one;
def erfc3(x):
'''erfc(x) for x in [1.25,1/0.35]'''
return one-erf3(x)
## Coefficients for approximation to erfc in [1/.35,28]
rb0 = -9.86494292470009928597e-03
rb1 = -7.99283237680523006574e-01
rb2 = -1.77579549177547519889e+01
rb3 = -1.60636384855821916062e+02
rb4 = -6.37566443368389627722e+02
rb5 = -1.02509513161107724954e+03
rb6 = -4.83519191608651397019e+02
sb1 = 3.03380607434824582924e+01
sb2 = 3.25792512996573918826e+02
sb3 = 1.53672958608443695994e+03
sb4 = 3.19985821950859553908e+03
sb5 = 2.55305040643316442583e+03
sb6 = 4.74528541206955367215e+02
sb7 = -2.24409524465858183362e+01
def erf4(x):
'''erf(x) for x in [1/.35,6]'''
x0=x
x = fabs(x)
s = one/(x*x)
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
z = ldexp(x0,0)
r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)
if(z>=0):
return one-r/x
else:
return r/x-one;
def erfc4(x):
'''erfc(x) for x in [2.857142,6]'''
return one-erf4(x)
def erf5(x):
'''erf(x) for |x| in [6,inf)'''
if x>0:
return one-tiny
return tiny-one
def erfc5(x):
'''erfc(x) for |x| in [6,inf)'''
if (x>0):
return tiny*tiny
return two-tiny
#############
##inf = float('inf')
##nan = float('nan')
###########
inf = float(9e999)
def Erf(x):
'''return the error function of x'''
f = float(x)
if (f == inf):
return 1.0
elif (f == -inf):
return -1.0
## elif (f is nan):
## return nan
else:
if (abs(x)<0.84375):
return erf1(x)
elif (0.84375<=abs(x)<1.25):
return erf2(x)
elif (1.25<=abs(x)<2.857142):
return erf3(x)
elif (2.857142<=abs(x)<6):
return erf4(x)
elif (abs(x)>=6):
return erf5(x)
def Erfc(x):
'''return the complementary of error function of x'''
f = float(x)
if (f == inf):
return 0.0
elif (f is -inf):
return 2.0
## elif (f == nan):
## return nan
else:
if (abs(x)<0.84375):
return erfc1(x)
elif (0.84375<=abs(x)<1.25):
return erfc2(x)
elif (1.25<=abs(x)<2.857142):
return erfc3(x)
elif (2.857142<=abs(x)<6):
return erfc4(x)
elif (abs(x)>=6):
return erfc5(x)
