> I'm still wondering if there could exist an alternative world where efficient addition over decimal numbers that we developers use on a day to day basis is associative. Is that even possible or is there perhaps some fundamental limit that forces us to trade associativity for performance?

You're always welcome to use a weaker notion of associativity than bitwise equality (e.g., -ffast-math pretends many operations are associative to reorder them for speed, and that only gives approximately correct results on well-conditioned problems).

In general though, yes, such a limit does exist. Imagine, for the sake of argument, an xxx.yyy fixed-point system. What's the result of 100 * 0.01 * 0.01? You either get 0.01 or 0, depending on where you place the parentheses.

The general problem is in throwing away information. Trashing bits doesn't necessarily mean your operations won't be associative (imagine as a counter-example the infix operator x+y==1 for all x,y). It doesn't take many extra conditions to violate associativity though, and trashed bits for addition and multiplication are going to fit that description.

How do you gain associativity then? At a minimum, you can't throw information away. Your fast machine operations use an unbounded amount of RAM and don't fit in registers. Being floating-point vs fixed-point only affects that conclusion in extremely specialized cases (like only doing addition without overflow -- which sometimes applies to the financial industry, but even then you need to think twice about the machine representation of what you're doing).