Optimal linear Bernoulli factories for small mean problems

Suppose a coin with unknown probability $p$ of heads can be flipped as often as desired. A Bernoulli factory for a function $f$ is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability $f(p)$ of heads. Applications include near perfect sampling from the stationary distribution of regenerative processes. When $f$ is analytic, the problem can be reduced to a Bernoulli factory of the form $f(p) = Cp$ for constant $C$. Presented here is a new algorithm where for small values of $Cp$, requires roughly only $C$ coin flips to generate a $Cp$ coin. From information theory considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For $Cp$ large, the new algorithm can also be used to build a new Bernoulli factory that uses only 80\% of the expected coin flips of the older method, and applies to the more general problem of a multivariate Bernoulli factory, where there are $k$ coins, the $k$th coin has unknown probability $p_k$ of heads, and the goal is to simulate a coin flip with probability $C_1 p_1 + \cdots + C_k p_k$ of heads.