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) = $${\frac{{2}}{{\sqrt{\pi}}}}$$$$\int_{0}^{{x}}$$e^-t2dt

The normalization is choosen so that:

erf(+ $$\infty$$) = 1,    erf(- $$\infty$$) = - 1

since :

$$\int_{0}^{{+\infty}}$$e^-t2dt = $${\frac{{\sqrt{\pi}}}{{2}}}$$

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 :

$$\mbox{\tt normal\_cdf}(x)=\frac{1}{2}+\frac{1}{2}\*\mbox{\tt erf}(\frac{x}{\sqrt{2}})$$

Indeed, making the change of variable t = u*$\sqrt{{2}}$ in

$$\mbox{normal\_cdf}(x)=\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\int_0^{x}e^{-t^2/2}dt$$

gives :

$$\mbox{normal\_cdf}(x)=\frac{1}{2}+\frac{1}{\sqrt{\pi}}\int_0^... ...-u^2}du=\frac{1}{2}+\frac{1}{2}\*\mbox{erf}(\frac{x}{\sqrt{2}})$$

Check :
normal_cdf(1)=0.841344746069