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error functions are part of C99 standard library but not yet implemented on all platforms. this module makes them available.

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'''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)

errors functions and gammas functions are likely to be integrated into the Python 2.7/3.1 math module. posting pure Python codes for these functions was suggested, so here they are. the script need to be tested extensively and it would be nice if one can compare the results with other tools. comments are greatly appreciated so we can select the best approach to implement these functions.

the original code of this script can be found at: http://sourceware.org/cgi-bin/cvsweb.cgi/~checkout~/src/newlib/libm/math/s_erf.c?rev=1.1.1.1&cvsroot=src

if we know other algorithms (with appropriate licence) that can be used with Python, please let me know!