  An algorithm is presented that, taking a sequence of independent Bernoulli
random variables with parameter $1/2$ as inputs and using only rational
arithmetic, simulates a Bernoulli random variable with possibly irrational
parameter $\tau$. It requires a series representation of $\tau$ with positive,
rational terms, and a rational bound on its truncation error that converges to
$0$. The number of required inputs has an exponentially bounded tail, and its
mean is at most $3$. The number of arithmetic operations has a tail that can be
bounded in terms of the sequence of truncation error bounds. The algorithm is
applied to two specific values of $\tau$, including Euler's constant, for which
obtaining a simple simulation algorithm was an open problem.
