you don’t have to use the actual, exact sigmoid function in a neural network algorithm but can replace it with an approximated version that has similar properties but is faster the compute.
For example, you can use the “fast sigmoid” function
f(x) = x / (1 + abs(x))
Using first terms of the series expansion for exp(x) won’t help too much if the arguments to f(x) are not near zero, and you have the same problem with a series expansion of the sigmoid function if the arguments are “large”.
An alternative is to use table lookup. That is, you precalculate the values of the sigmoid function for a given number of data points, and then do fast (linear) interpolation between them if you want.