Approximations of Euler's constant using $\pi$

Question

Disclaimer: This is for recreational purposes.

Hello MSE! So while my research paper about Euler's constant $\gamma$, I created this amazing approximation for it: $$\frac{\pi^2}{12000}-\ln((10^{-3})!)*1000$$Which is correct to almost six digits (where $!$ is extended to the real numbers). However, I don't like the look of the logarithm of a factorial. So out of curiosity, are there any other interesting approximations of $\gamma$ using $\pi$? I'm fine with logarithms but factorials are just too ugly.

Edit: I only asked for an approximation of $\gamma$ using $\pi$'s. There is no other question here.

Answer 1

What do you think about this one $\gamma\approx\dfrac{\pi}{2e}\times\dfrac{8907511}{8917511}$ with $10$ correct decimals.

I proceeded from the rough $\frac{\pi}{2e}$ approximation, then searched the decimals in the Plouffe inverter and choose something simple which was close.

I guess you can derive many fancy approximations using this method, provided you get something nice around the desired decimals in the inverter. If not try another starting approximation.

Answer 2

A very simple one $$\gamma \sim\frac{26685+102790 \pi }{20080+186403 \pi }$$ the obsolute error is $3.65\times 10^{-21}$

Answer 3

How about $$ \gamma \approx \frac{7 \pi^{14}-24 \pi^{13}-13 \pi^{12}-18 \pi^{11}-4 \pi^{10}+3 \pi^{9}+20 \pi^{8}-10 \pi^{7}+11 \pi^{6}-4 \pi^{5}-12 \pi^{4}+37 \pi^{3}+9 \pi^{2}-4}{-3 \pi^{14}-2 \pi^{13}-8 \pi^{12}+\pi^{11}-5 \pi^{10}+13 \pi^{9}+6 \pi^{8}}$$ with error about $2.76 \times 10^{-27}$ or $$ \gamma \approx \frac{15 \pi^{9}-47 \pi^{8}+31 \pi^{7}-39 \pi^{6}+56 \pi^{5}-21 \pi^{4}+13 \pi^{3}-12 \pi^{2}-12 \pi -44}{3 \pi^{9}+5 \pi^{8}-3 \pi^{7}+2 \pi^{6}-4 \pi^{5}-37 \pi^{4}+32 \pi^{3}-11 \pi^{2}-4 \pi +10}$$ with error about $1.87 \times 10^{-24}$.

Answer 4

Thanks to Wolfram Alpha, for an absolute error of $1.27\times 10^{-101}$

$$\gamma =-\frac{141670595684161}{89719457709126} \binom{\pi }{\pi !}+\frac{6168819408880 }{44859728854563}\binom{\pi !}{\pi }-$$ $$\frac{115396055527429 }{89719457709126}\binom{\pi !}{\log (\pi )}+\frac{144642478651913 }{89719457709126}\binom{\log (\pi )}{\pi !}+$$ $$\frac{57273545495261 }{29906485903042} \binom{\pi }{\log (\pi )}-\frac{2493037306729 }{44859728854563}\binom{\log (\pi )}{\pi }$$