definition of the sigmoid    
  sig(x) = 1.0 / (1.0 + exp(-x))

value in 0
  sig(0) = 0.5


derivative of sig(x)
  sig'(x) = sig(x)*(1.0-sig(x))

slope in 0
  sig'(0) = sig(0)*(1.0-sig(0)) = 0.5*0.5 = 0.25


symmetrisation of the sigmoid, with an output range [-1, 1] 
  ssig(x) = 2.0*sig(x)-1.0 = (1.0 - exp(-x)) / (1.0 + exp(-x))

derivative of ssig(x)
  ssig'(x)= 2.0 * sig'(x)

slope in 0
  ssig'(0) = 0.5


normalization of the sigmoid, with a slope 1
  nsig(x) = ssig(2*x) = (1.0 - exp(-2*x)) / (1.0 + exp(-2*x))

derivative of nsig(x)
  nsig'(x)= 4.0 * sig'(x)

slope in 0
  nsig'(0) = 1.0


generic sigmoid, symmetric, with an output range [-1 1] and a slope s
  gsig(x, s) = nsig(s*x) = (1.0 - exp(-2*s*x)) / (1.0 + exp(-2*s*x))

slope in 0
  (d gsig(x,s) / dx)(0) = s