Simulating a coin with irrational bias using rational arithmetic

An algorithm is presented that, taking independent Bernoulli random variables with parameter 1/2 as inputs and using only rational arithmetic, simulates a Bernoulli random variable with possibly irrational parameter 𝜏. It requires a series representation of 𝜏 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 𝜏, including Euler's constant, for which obtaining a simple simulation algorithm was an open problem.