### 5.9.5 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
| e−t2dt

The normalization is chosen so that:

erf(+∞)=1, erf(−∞)=−1

since :

∫ |
+∞
0
| e−t2dt= |
√ |
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
√ |
2π
| ∫ |
x
0
| e−t2/2dt

gives :

normal_cdf(x)= |
1
2
| + |
1
√ |
π
| ∫ |
x
√ |
2
0
| e−u2du= |
1
2
| + |
1
2
| erf( |
x
√ |
2
| )

Check :
normal_cdf(1)=0.841344746069