something about softmax

function softmax(z)
  #z = z - maximum(z)
  o = exp(z)
  return o / sum(o)
end
function gradient_together(z, y)
  o = softmax(z)
  o[y] -= 1.0
  return o
end
function gradient_separated(z, y)
  o = softmax(z)
  ∂o_∂z = diagm(o) - o*o'
  ∂f_∂o = zeros(size(o))
  ∂f_∂o[y] = -1.0 / o[y]
  return ∂o_∂z * ∂f_∂o
end