Error Function - Approximation With Elementary Functions

Approximation With Elementary Functions

Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are:

(maximum error: 5·10−4)

where a1 = 0.278393, a2 = 0.230389, a3 = 0.000972, a4 = 0.078108

(maximum error: 2.5·10−5)

where p = 0.47047, a1 = 0.3480242, a2 = −0.0958798, a3 = 0.7478556

(maximum error: 3·10−7)

where a1 = 0.0705230784, a2 = 0.0422820123, a3 = 0.0092705272, a4 = 0.0001520143, a5 = 0.0002765672, a6 = 0.0000430638

(maximum error: 1.5·10−7)

where p = 0.3275911, a1 = 0.254829592, a2 = −0.284496736, a3 = 1.421413741, a4 = −1.453152027, a5 = 1.061405429

All of these approximations are valid for x ≥ 0. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x) = −erf(−x).

Another approximation is given by

where

This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the error is less than 0.00035 for all x. Using the alternate value a ≈ 0.147 reduces the maximum error to about 0.00012.

This approximation can also be inverted to calculate the inverse error function:

\operatorname{erf}^{-1}(x)\approx \sgn(x) \sqrt{\sqrt{\left(\frac{2}{\pi a}+\frac{\ln(1-x^2)}{2}\right)^2 - \frac{\ln(1-x^2)}{a}} -\left(\frac{2}{\pi a}+\frac{\ln(1-x^2)}{2}\right)}

Read more about this topic:  Error Function

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