One thing I didn't see mentioned in the article is the motivation for choosing the particular (exponential) mapping. If you imagine replacing your function with

One thing I didn't see mentioned in the article is the motivation for choosing the particular (exponential) mapping. If you imagine replacing your function with f(s)=1-1/g(s) for some g with g(0)=1 which goes to infinity as s does, then as long as g grows faster than polynomial you'll obtain the same properties. Something like x^x, or just 2^sqrt(x). 2^x certainly seems the simplest, but was there a further reason?