[kvakil]

> Maksymilian Piskorowski found that if you happen to have a spare eight 9s, you can compute 𝑒 = (9/9 + 9^(-9^9))^(9^(9^9)), which is accurate to a little over 369 million decimal places.

Sure, because 9/9 = 1 and if you take x = 9^9^9, you get back (1 + x^(-1))^x, i.e. the first formula. It's cute, but I don't know if you could call it a "discovery".

[shakna]

You can see the formula Piskorowski used here [0], under "Best Approximations to e with n Copies of the Digit k". (All formulas marked with "MP").

He did in fact build up (1 + x^(-1))^x to slowly increase the accuracy of the formula.

[0] https://www2.stetson.edu/~efriedma/mathmagic/0804.html (The page may say 2004, but has been actively updated, and has references to 2019.)