Can the golden ratio accurately be expressed in terms of ee and π\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 $\pi$ ?

Answer

Attribution
Source : Link , Question Author : Nick , Answer Author : vadim123