I seek a simple algorithm or function that will produce an approximate value for $\sqrt{\pi}$, but none that begin with the actual value of $\pi$, then derive the root via approximation.
This is very specific. What (known?) natural function will produce this number without the usual methods of deriving $\pi$?
There may be an answer to this close at hand, but historical perspective is extremely appreciated.
Here is a formula of Ramanujan for $\sqrt{\frac{\pi e^x}{2x}} $: https://oeis.org/wiki/A_remarkable_formula_of_Ramanujan
Put $x=1$ or $x=2$ to get $\sqrt{\pi}$ as the sum of a continued fraction and a series, both of which can be evaluated iteratively.
Of course $e$ or $e^{1/2}$ has to be evaluated also, but these too can be done iteratively.