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.
In the Imagick extension source, the function call is:
status = MagickSigmoidalContrastImageChannel(intern->magick_wand, channel, sharpen, alpha, beta);
Referring back to the binary API call gives:
foreign import ccall "MagickSigmoidalContrastImageChannel" magickSigmoidalContrastImageChannel
:: Ptr MagickWand
-> ChannelType -- ^ identify which channel to level: `redChannel`, `greenChannel`
-> MagickBooleanType -- ^ increase or decrease image contrast
-> CDouble -- ^ strength of the contrast, the larger the number the more 'threshold-like' it becomes
-> CDouble -- ^ midpoint of the function as a color value 0 to `quantumRange`
-> IO MagickBooleanType
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)