Having event with probability $p$, how can I get an event with probability $\sqrt{p}$? I'd like a design a circuit that, given a random bit with probability $p$ to be zero, it outputs a random bit with probability $\sqrt{p}$ to be zero. Actually, I am rather looking for $\sqrt{p(1-p)}$, in case it makes things easier.
Let's assume that we have an unlimited but finite stream of independent random bits and all the common logic gates (AND, OR, NOT, XOR...).
If an exact solution wouldn't be possible in a finite setting. I would also be happy with an approximation, preferably one that can be arbitrarily expanded to reduce the error.
I have already looked into a Taylor expansion around $p=1/2$, and it is a good option. The second degree expansion of $\sqrt{p(1-p)}$ is $\frac{1}{2} - (p - \frac{1}{2})^2 + O((p - \frac{1}{2}))^4$ which goes to $\frac{3}{4} + p(1-p)$ which is nice and easy to implement.
I am here to see if there is a better solution.
This question is highly related to the following question, but the answer is not general enough to satisfy my needs.
Understanding what $\sqrt{p}$ means for an event of probability $p$
 A: From (the abstract of) this paper [1] by Elchanan Mossel and Yuval Peres: 

however, pushdown automata can simulate $f(p)$-coins for certain non-rational functions such as the square root of $p$. These results complement the work of Keane and O'Brien (1994), who determined the functions $f$ for which an $f(p)$-coin can be simulated when there are no computational restrictions on the simulation scheme.

You may want to look at Section 3.2 of the arXiv version linked above, which describes a pushdown automaton for simulating an $\sqrt{p}$-coin. Moreover, as they discuss in Section 1.1, their Theorem 2.2 implies that no finite automaton can simulate an $\sqrt{p}$-coin.
Also very relevant is this other paper, by Wästlund: as discussed in this blog post, the discussion after Theorem 2.1 (in Section 2) describes a procedure to simulate an $\sqrt{p}$-coin.
[1] New Coins From Old: Computing With Unknown Bias. Elchanan Mossel, Yuval Peres,  Combinatorica 25(6): 707-724 (2005)
[2] Functions arising by coin flipping. Johan Wästlund, 1999.
