You might enjoy a few of my favorite sigmoids: https://raphlinus.github.io/audio/2018/09/05/sigmoid.html https://raphlinus.github.io/audio/2018/09/05/sigmoid.ht

You might enjoy a few of my favorite sigmoids: https://raphlinus.github.io/audio/2018/09/05/sigmoid.html https://raphlinus.github.io/audio/2018/09/05/sigmoid.html That's not a deep exploration of the numerical analysis (especially compared with OP), but it does make one point: traditional "numerical analysis" techniques tend to assume the traditional four mathematical operators as canonical, but if your goal is to get something computed on actual computer hardware quickly and reasonably accurately, you have a wider palette to play with. In particular, inverse square root is a primitive about as efficient as reciprocal on modern hardware (or if using Carmack's wtf trick).