another Bernoulli factory

A question that came out on X validated is asking for help in figuring out the UMVUE (uniformly minimal variance unbiased estimator) of (1-θ)^½ when observing iid Bernoulli B(θ). As it happens, there is no unbiased estimator of this quantity and hence not UMVUE. But there exists a Bernoulli factory producing a coin with probability (1-θ)^½ from draws of a coin with probability θ, hence a mean to produce unbiased estimators of this quantity. Although of course UMVUE does not make sense in this sequential framework. While Nacu & Peres (2005) were uncertain there was a Bernoulli factory for θ^½, witness their Question #1, Mendo (2018) and Thomas and Blanchet (2018) showed that there does exist a Bernoulli factory solution for θ^a, 0≤a≤1, with constructive arguments that only require the series expansion of θ^½. In my answer to that question, using a straightforward R code, I tested the proposed algorithm, which indeed produces an unbiased estimate of θ^½… (Most surprisingly, the question got closed as a “self-study” question, which sounds absurd since it could not occur as an exercise or an exam question, unless the instructor is particularly clueless.)