I'm looking for a (computationally cheap) way to compute a function that behaves similar to the skew normal distribution, i.e. it has a shape like the normal distribution when some parameter a = c, while allowing to set some skew by varying a.
The background is that I'm using a Gaussian model for a shape manipulation tool but there are some edge cases that would be better served by an asymmetric function. Ideally, the function should be able to fit a positively shifted Gaussian G(x,mu,sigma), mu > 0 for the range of positive x.
The logistic curve $\dfrac1{1+e^{-ax}} $ very closely matches the cumulative normal distribution especially if $a=\dfrac{4}{\sqrt{2\pi}}$, which matches the slope at the origin. The maximum difference is at most $0.017$.
Perhaps you can modify this to do what you want.