No: commutativity of multiplication and addition still holds, but associativity indeed does not. That associativity does not hold is a much more unpleasant than if associativity would hold but commutativity wouldn't (the mathematics of common associative, but non-commutative structures is understood quite well, say (non-abelian) groups and monoids).
#include <stdio.h>
int main() {
unsigned long long bits = 0x7FFC000000000000;
double x = *(double *)&bits;
bits += 1;
double y = *(double *)&bits;
double z = x + y, w = y + x;
printf("%llx %llx\n", *(unsigned long long *)&z, *(unsigned long long *)&w);
}
The one that surprised me was that, in some cases:
Large + small + small + small =! small + small + small + = Large
Really underscores how complex things are.
a ( b + c ) != a * b + a * c