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For community wiki, On the occasion of the World day of $\pi$, I want to look for simple formulas which involves $\pi$ using known constants like $e, \phi,\gamma,\cdots$ , For instance just a simple attempt I got the following formula with approximation of $10^{-5}$ such that :$\pi \sim \dfrac{2e-\frac{3\sqrt{2}\gamma^2}{4}}{\phi}$ , Are there others ?

Note: $\phi$ is the golden ratio

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    $\begingroup$ All these constants are irrational numbers, you can approximate $\pi$ pretty well, that's KAT. $\endgroup$ – rtybase Mar 6 at 0:08
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    $\begingroup$ Another example: $$ \pi \approx \dfrac{2\sqrt{10}}{\gamma^2 + e/\varphi} = \underline{3.14159265}\color{gray}{26267...} $$ $\endgroup$ – Oleg567 Mar 9 at 15:35
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    $\begingroup$ Yet another approximation: $$ \pi \approx \dfrac{50(75+\gamma)}{11(17^2-e)(2-\varphi)} = \underline{3.14159265358}\color{gray}{25265...} $$ $\endgroup$ – Oleg567 Mar 10 at 0:12
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    $\begingroup$ All of these formulas does not look very "efficient". For example it takes $44$ characters to type out your formula and evaluating it would also require knowing these constants and still it only gives $\pi$ to 5 digits. The approximation $\pi \approx 3.14159265$ on the other hand is superior, just $10$ characters and correct to $8\cdot 10^{-9}$. It's just not very pretty. $\endgroup$ – Winther Mar 10 at 8:14
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    $\begingroup$ I wrote a very simple python3 code to estimate the value of $\pi$ by linear combination of $e$, $\phi$, $\log 2$ and $\gamma$ with simple rational denominator. An example is $\pi\approx (10 e -7 \log 2 -14 \phi + 9\gamma)/5 = \underline{3.141592}7880168177\cdots$. Also, wolfram|alpha gave $(-3 e e! + 32 + 14 e + e^2)/(5 e) \approx \underline{3.141592}503375$. $\endgroup$ – dust05 Mar 10 at 9:53
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Few examples (using relatively small integers and constants $e$, $\varphi$, $\gamma$) with increasing precision (and, of course, increasing size of the expression):

$$ \pi \approx \dfrac{5}{7}\varphi e = \underline{3.141}\color{gray}{623...}\tag{1} $$

$$ \pi \approx \dfrac{8}{e-1}-\dfrac{\sqrt{2}}{\varphi \gamma} = \underline{3.141592}\color{gray}{711...}\tag{2} $$

$$ \pi \approx \dfrac{2\sqrt{10}}{\gamma^2 + e/\varphi} = \underline{3.14159265}\color{gray}{262...}\tag{3} $$

$$ \pi \approx \dfrac{50(75+\gamma)}{11(17^2-e)(2-\varphi)} = \underline{3.14159265358}\color{gray}{252...}\tag{4} $$

$$ \pi \approx \dfrac{28 + \left(e-\dfrac{7}{6}\right)\left(\dfrac{7}{1+\varphi}+\gamma + 3\right)}{12} = \underline{3.14159265358979}\color{gray}{773...}\tag{5} $$

The WolframAlpha link to check last approximation.


Happy $\pi$ day!

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