Approximations to Error Functions

This paper addresses approximations to error functions and points out three representative approximations, each with its own merits. Cody's approximation is the most computationally intensive of the three, it is not overly so, and there is no arguing over its accuracy. The other two approximations are much simpler computationally, and they both yield accuracies that would be considered more than sufficient in most practical situations. Absolute relative error provides an effective measure of goodness, and, for approximations to the Q-function, it also places a loose bound on the absolute error in the approximation. Cody's approximation is an effective surrogate for the true error function; the values provided by that approximation match the actual values of the error function to within roughly the precision of double-precision floating point arithmetic.