  Given a sequence of independent Bernoulli variables with unknown parameter
$p$, and a function $f$ expressed as a power series with non-negative
coefficients that sum to at most $1$, an algorithm is presented that produces a
Bernoulli variable with parameter $f(p)$. In particular, the algorithm can
simulate $f(p)=p^a$, $a\in(0,1)$. For functions with a derivative growing at
least as $f(p)/p$ for $p\rightarrow 0$, the average number of inputs required
by the algorithm is asymptotically optimal among all simulations that are fast
in the sense of Nacu and Peres. A non-randomized version of the algorithm is
also given. Some extensions are discussed.
