Originally Posted by Rafal Lukawiecki
Here is another function that does include a shoulder:
f(x) = a1*log(1+exp(a2*x+a3))/(a4+log(1+exp(a2*x+a3)))
This is based on my original function for the toe and linear regions, but then modifies the result of that function by a rectangular hyperbolic function to form the shoulder at the larger values. I suggested this in the mega-thread on curve fitting. Here is a link to an image showing a fit of this function to a set of data for enlarging paper that Ralph Lambrecht provided.