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from math import * | ||
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tiny= 1e-300 | ||
half= 5.00000000000000000000e-01 | ||
one = 1.00000000000000000000e+00 | ||
two = 2.00000000000000000000e+00 | ||
erx = 8.45062911510467529297e-01 | ||
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## Coefficients for approximation to erf on [0,0.84375] | ||
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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 | ||
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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 | ||
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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 | ||
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## Coefficients for approximation to erf in [0.84375,1.25] | ||
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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 | ||
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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 | ||
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def erfc2(x): | ||
'''erfc(x) for x in [0.84375, 1.25]''' | ||
return one-erf2(x) | ||
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## Coefficients for approximation to erfc in [1.25,1/0.35] | ||
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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 | ||
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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; | ||
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def erfc3(x): | ||
'''erfc(x) for x in [1.25,1/0.35]''' | ||
return one-erf3(x) | ||
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## Coefficients for approximation to erfc in [1/.35,28] | ||
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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 | ||
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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; | ||
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def erfc4(x): | ||
'''erfc(x) for x in [2.857142,6]''' | ||
return one-erf4(x) | ||
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def erf5(x): | ||
'''erf(x) for |x| in [6,inf)''' | ||
if x>0: | ||
return one-tiny | ||
return tiny-one | ||
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def erfc5(x): | ||
'''erfc(x) for |x| in [6,inf)''' | ||
if (x>0): | ||
return tiny*tiny | ||
return two-tiny | ||
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############# | ||
##inf = float('inf') | ||
##nan = float('nan') | ||
########### | ||
inf = float(9e999) | ||
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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) | ||
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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) |