Can the golden ratio accurately be expressed in terms of $e$ and $\pi$ 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 $\pi$ ?
The following is exact. :-)

$$\phi=\frac{\frac{\pi}{\pi}+\sqrt{\frac{e+e+e+e+e}{e}}}{\frac{e}{e}+\frac{\pi}{\pi}}$$
An approximation with an accuracy similar to that of $\pi\approx3$ (error&lt;5%) is given by the sixth root of Gelfond's constant,

$$e^{\frac{\pi}{6}}\approx \phi$$

with rational term series

$$e^{\frac{\pi}{6}}=\sum_{k=0}^{\infty}\frac{\left(e^{\frac{\pi}{2}} - (-1)^k e^{-\frac{\pi}{2}}\right)\Gamma\left(\frac{k+i}{2}\right)\Gamma\left(\frac{k-i}{2}\right)}{4\pi k!}$$

Rational series representation of $e^\pi$