The "standard" method should be along the lines of "forming" an approximation of a and b from their digit representations, then computing exp(b* log(a)) using standard routines for approximations. You should look at routines used in scientific calculators (the relevant search terms escape my memory).
Something that might be of interest: decide on a low precision (say bits=16), and compute $a^b$ where each has so many bits to twice the precision, and a and b are numbers in (0,1). Then run a logic minimizer on the table to see if you can build a small Boolean circuit that computes this. It may be interesting to see how much you can speed up numeric computations with a (minimized for depth) logic circuit.
Gerhard "Make Computations Wider And Faster" Paseman, 2015.11.05
