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2.9.5  Error function : erf

erf takes as argument a number a.
erf returns the floating point value of the error function at x=a, where the error function is defined by :

erf(x)=
2
π
x


0
et2dt 

The normalization is choosen so that:

erf(+∞)=1,     erf(−∞)=−1 

since :

+∞


0
et2dt=
π
2
 

Input :

erf(1)

Output :

0.84270079295

Input :

erf(1/(sqrt(2)))*1/2+0.5

Output :

0.841344746069

Remark
The relation between erf and normal_cdf is :

normal_cdf(x)=
1
2
+
1
2
erf(
x
2

Indeed, making the change of variable t=u*√2 in

normal_cdf(x)=
1
2
+
1
x


0
et2/2dt

gives :

normal_cdf(x)=
1
2
+
1
π
x
2


0
eu2du=
1
2
+
1
2
erf(
x
2
)

Check :
normal_cdf(1)=0.841344746069


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