I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. 

**My question**: What continued fraction for $e$ is most computationally efficient? A proof or a link to one for why a method is optimal would be of great interest to me.

Here, my metric for "computational efficiency" is achieving the most precise decimal places with a fixed number of terms generated in the continued fraction.