[labawi]

I think definitions vary slightly, but the quality you mentioned (a == b => f(a) == f(b)) would be called a deterministic function [1] - very useful, often applicable or even type check enforced in functional programming languages.

Having no side effects is a different very useful quality - function doing exactly as specified and no more (I/O, setting variables ..). I'm not sure if it means not accessing global state, though it is usually better if both inputs and outputs are explicit.

All in, it is usually easier to reason about functions that are explicit, deterministic and side-effect free, yet I find it profoundly more valuable if it can actually be relied on (a known subset of) functions having those qualities.

[1] https://maksimivanov.com/posts/pure-functions-and-side-effec...