Consider approximations of the form, x>=0, p an integer:
C 1 exp(-x) ~ (-----)^p = ------------- (1) C + x (1 + 1/C x)^p
The complexity of the approximations is arguably independent of p, with only one optimizable constant. Execution time grows as O(log(p)) using exponentiation by squaring; then p being a power of 2 is particularly efficient.
Note that substituting variable x, e.g. with x^2/2 to create an approximation to the Gaussian, does not change the maximum |abs. error|.
With an optimal C-value, Copt, the maximum |abs. error| becomes roughly inversely proportional to p.
A simple asymptotic approximation to Copt is: C(p) = p-0.6627435.
The following approximation might be useful for precalculation of Copt(p):