The documentation for this function really isn't useful or helpful at all. "Sharpen", "Contrast" and "Midpoint" aren't the names of the params, which are referred to as "alpha" and "beta".
The Imagemagick manual entry:
For those interested, the corrected formula for the 'sigmoidal non-linearity contrast control' is...
( 1/(1+exp(β*(α-u))) - 1/(1+exp(β)) ) / ( 1/(1+exp(β*(α-1))) - 1/(1+exp(β*α)) )
Where α is the threshold level, and β the contrast factor to be applied.
The formula is actually very simple exponential curve, with the bulk of the above formula is designed to ensure that 0 remains zero and 1 remains one. That is, the graph always goes though the points 0,0 and 1,1. And the highest gradient of change is at the given threshold.
Src: http://hackage.haskell.org/package/imagemagick-0.0.2/docs/src/Graphics-ImageMagick-MagickWand-FFI-WandImage.html
So the parameters should apparently be interpreted:
Sharpen: 0/1 (increase/decrease contrast)
Alpha: Strength of the contrast (typically 3-20)
Beta: Midpoint of the contrast (typically 50)