Simple approximation to error function

I recently ran across the fact that

is a remarkably good approximation for −1 ≤ x ≤ 1.

Since the integral above defines the error function erf(x), modulo a constant, this says we have a good approximation for the error function

again provided −1 ≤ x ≤ 1.

The error function is closely related to the Gaussian integral, i.e. the normal probability distribution CDF Φ. The relation between erf and Φ is simple but error-prone. I wrote up some a page notes for myself a few years ago so I wouldn’t make a mistake again moving between these functions and their inverses.

You can derive the approximation by writing out the power series for exp(t), substituting −t² for t, and integrating term-by-term from 0 to x. You’ll see that the result is the same as the power series for sine until you get to the x⁵ term, so the error is on the order of x⁵. Here’s a plot of the error.

The error is extremely small near 0, which is what you’d expect since the error is on the order of x⁵.