\(\text{sig}(x,k) = \dfrac{x-xk}{k-|x|2k+1}\)
Desmos
\(\begin{align} &k = \text{remap}(c,-1,1,1,-1)\\ &\text{sig_contrast}(x,c) = \dfrac{\text{sig}(2x-1,k)+1}{2} \end{align}\)