This is an awesome article! One of my favorite SO answers I remember researching dealt with the Padé approximation of tanh[1] (which I found was significantly better than the Taylor expansion), but the caveat was that it only worked within a very narrow neighborhood.
I will say that the article didn't really touch on techniques that minimize IEEE 754 subnormal[2] performance impact, which is a very interesting problem in itself. A lot of real-life implementations of something like e^x will have various clever ways to avoid subnormals.