Error function: erf

### 5.8.11 Error function: erf

The error function erf is defined by:

erf(x)= |
2
√ |
π
| ∫ |
x
0
| e−t2dt

where the constant 2/√π is chosen so that

erf(+∞)=1, erf(−∞)=−1

since

∫ |
+∞
0
| e−t2dt= |
√ |
π
2

The erf command computes the error function.

- erf takes one argument:
  a, a number.
- erf(a) returns the value of erf(a).

#### Examples.

- Input:
  erf(1)
  Output:

  erf | ⎛ ⎝ | 1 | ⎞ ⎠

- Input:
  erf(1.0)
  Output:

  0.84270079295

- Input:
  erf(1/(sqrt(2)))*1/2+0.5
  Output:

  0.841344746069

Remark.
The relation between erf and normal_cdf (see Section 8.4.7) 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:
Input:

normal_cdf(1.0)

Output:

0.841344746069