Estimate $\pi$ using a 3D Random Walk.

Question

A 3D Random Walk according to WolframAlpha says:

$P(\text{{probability of eventual return to origin}}) = 1 - \frac{{16 \sqrt{{\frac{2}{3}}} \pi^3}}{{\Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\frac{7}{24}) \Gamma(\frac{11}{24})}} \approx 0.340537$

Rearranging gives

\begin{equation*} \pi = \left[ \frac{{0.659463 \cdot \Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\frac{7}{24}) \Gamma(\frac{11}{24})}}{{16 \sqrt{\frac{2}{3}}}} \right]^{1/3} \end{equation*}

Plugging that equation into WolframAlpha computes $\pi$ as

$$3.1415931769021159447130538115032990978757778167518560061468225968...$$

However, WolframAlpha for $\pi$ is

$$3.1415926535897932384626433832795028841971693993751058209749445923...$$

Question

Is there a way to tweak that equation to improve the accuracy of $\pi$?

Answer 1

Note the approximation sign in Wolfram alpha's expression $$ 1 - \frac{{16 \sqrt{{\frac{2}{3}}} \pi^3}}{{\Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\frac{7}{24}) \Gamma(\frac{11}{24})}} \color{red}{\approx} 0.340537 $$ for the probability, which means you can only expect the expession \begin{equation*} \left[ \frac{{0.659463 \cdot \Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\frac{7}{24}) \Gamma(\frac{11}{24})}}{{16 \sqrt{\frac{2}{3}}}} \right]^{1/3} \end{equation*} to approximate $\ \pi\ $ to about $\ 6\ $ decimal places at the most. Plugging the expresssion $$ a=1 - \frac{{16 \sqrt{{\frac{2}{3}}} \pi^3}}{{\Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\frac{7}{24}) \Gamma(\frac{11}{24})}} $$ into Wolfram alpha, however, gives a much more accurate value, $$ 0.3405373295509991428262731844329028967106082171243020977632361053 $$ for $\ a\ $. If you now submit the expression $$ \left[ \frac{{(1-a) \cdot \Gamma(\frac{1}{24}) \Gamma(\frac{5}{24}) \Gamma(\frac{7}{24}) \Gamma(\frac{11}{24})}}{{16 \sqrt{\frac{2}{3}}}} \right]^{1/3} $$ to Wolfram alpha, it returns the much more accurate approximation $$ 3.1415926535897932384626433832795028841971693993751058209749445924 $$ to $\ \pi\ $, differing from the value it gives for that constant only in the last of its $\ 64\ $ decimal places.