The formula for sigmoidal contrast given by Thyssen is missing a term. It should be:
( 1/(1+exp(β*(α-u))) - 1/(1+exp(β*α)) ) / ( 1/(1+exp(β*(α-1))) - 1/(1+exp(β*α)) )
where there was an α missing from the second term.
Note that a sigmoidal contrast with α = 6 and β = 0.46 is approximately equivalent to combining a "screen" overlay of the image on itself, followed by a "multiply" overlay.