an idea i had at the time:
instead of incrementing n by 1, use powers of 2. seems like an improvement.
(revised it to use relative difference with my current knowledge)

2.7182818284590446
accurate to 14 decimal places

"woo" might be a better english name than "urr". their japanese name is うーくん (ū-kun or uu-kun) with the honorific "-kun".
https://mastodon.social/@yuukikonno/113472756025081677