NinjaKoala

Numerical solutions are usually much more accurate, yes. However, especially when computing with fragment shaders for example, closed form formulas tend to be faster. But the main reason for looking into it was because i had fun doing it.

Yes, the procedure you mention works, but it's a numerical method. I'm not saying such numerical method's are impractical or something, but depending on your general setup, closed form formulas might be faster.

Yes, you're right. I didn't find a proof for the fact that elliptic integrals have no closed form, though.

There was actually a 4k intro released this year which uses changing seeds: https://www.pouet.net/prod.php?which=94081