TITLE: Can the golden ratio accurately be expressed in terms of $e$ and $\pi$
QUESTION [93 upvotes]: I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$
 And so, I took it upon myself to try to find a way to express the golden ratio in terms of the infamous values, $\large\pi$ and $\large e$
The closest that I've come so far is:
$$
\varphi \approx \sqrt e - \frac{\pi}{(e+\pi)^e - \sqrt e}
$$
My question is, Is there a better (more precise and accurate) way of expressing $\phi$ in terms of $e$ and $\pi$ ?

REPLY [3 votes]: I have found this very close to $\phi$:
$\phi \sim \dfrac{e^{e^2}}{\big(\pi^2-\dfrac{10e^2-16e-4}{6e^2-116e+69}\big)^3}$ and the error is close to $3.13\times 10^{-10}$