An asymptotically optimum Bernoulli factory for certain functions that can be expressed as power series

Given a sequence of independent Bernoulli variables with unknown parameter $p$, and a function $f$ that can be expressed as a power series with non-negative coefficients, an algorithm is presented that produces a Bernoulli random variable with parameter $f(p)$. In particular, the algorithm can simulate $f(p) = p^r$ for $0 < r < 1$. The algorithm requires, for general $f$, an average number of inputs that is asymptotically optimal in a precisely defined sense. In addition, the distribution of the number of inputs has an exponentially decaying tail. Some extensions of the algorithm are discussed.