  Let S\subset (0,1). Given a known function f:S\to (0,1), we consider the
problem of using independent tosses of a coin with probability of heads p
(where p\in S is unknown) to simulate a coin with probability of heads f(p). We
prove that if S is a closed interval and f is real analytic on S, then f has a
fast simulation on S (the number of p-coin tosses needed has exponential
tails). Conversely, if a function f has a fast simulation on an open set, then
it is real analytic on that set.
