Cumulative distribution function


Returns the area under the Gaussian probability density function, integrated from minus infinity to x.

Parameters:x (float) – a real scalar.


Area under the curve:

\[\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x \exp(-t^2/2) dt\]

Equivalently, we have:

ndtr(x) = ( 1 + erf(z) ) / 2 =  erfc(z) / 2

where \(z = x/\sqrt{2}\). Computation is done via the functions erf() and erfc() with care to avoid error amplification in computing \(\exp{(-x^2)}\).


  x   relative error
arithmetic domain # trials peak rms
IEEE -13, 0 30000 1.3e-15 2.2e-16

Error messages

message condition value returned
erfc underflow x > 37.519379347 0.0