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Given a $p$-coin that lands heads with unknown probability $p$, we wish toproduce an $f(p)$-coin for a given function $f: (0,1) \rightarrow (0,1)$. Thisproblem is commonly known as the Bernoulli Factory and results on itssolvability and complexity have already been obtained. Nevertheless, genericways to design a practical Bernoulli Factory for a given function $f$ existonly in a few special cases. We present a constructive way to build anefficient Bernoulli Factory when $f(p)$ is a rational function withcoefficients in $\mathbb{R}$. Moreover, we extend the Bernoulli Factory problemto a more general setting where we have access to an $m$-sided die and we wishto roll a $v$-sided one; i.e., we consider rational functions between openprobability simplices. Our construction consists of rephrasing the originalproblem as simulating from the stationary distribution of a certain class ofMarkov chains - a task that we show can be achieved using perfect simulationtechniques with the original $m$-sided die as the only source of randomness. Inthe Bernoulli Factory case, the number of tosses needed by the algorithm hasexponential tails and its expected value can be bounded uniformly in $p$. Enroute to optimizing the algorithm we show a fact of independent interest: everyfinite, integer valued, random variable will eventually become log-concaveafter convolving with enough Bernoulli trials.

From the Bernoulli Factory to a Dice Enterprise via Perfect Sampling of Markov Chains