I think the other replies are overcomplicating this.
+ is a binary operation, and a+b+c can’t be interpreted without knowing whether one treats + as left-associative or right-associative. Let’s assume the former: a+b+c really means (a+b)+c.
If + is commutative, you can turn (a+b)+c into (b+a)+c or c+(a+b) or (commuting twice) c+(b+a).
But that last expression is not the same thing as (c+b)+a. Getting there requires associativity, and floating point addition is not associative.
"a+b+c" doesn't describe a unique evaluation order. You need some parentheses to disambiguate which changes are due to associativity vs commutativity. a+(b+c)=(c+b)+a should be true of floating point numbers, due to commutativity. a+(b+c)=(a+b)+c may fail due to the lack of associativity.
For those to be equal you need both associativity and commutativity.
Commutativity says that a*b = b*a, but that's not enough to allow arbitrary reordering. When you write a*b*c depending on whether * is left or right associative that either means a*(b*c) or (a*b)*c. If those are equal we say the operation is associative. You need both to allow arbitrary reordering. If an operation is only commutative you can turn a*(b*c) into a*(c*b) or (b*c)*a but there is no way to put a in the middle.
We’re in very nitpicky terminology weeds here (and I’m not the person you’re replying to), but my understanding is “commutative” is specifically about reordering operands of one binary op (4+3 == 3+4), while “associative” is about reordering a longer chain of the same operation (1+2+3 == 1+3+2).
Edit: Wikipedia actually says associativity is definitionally about changing parens[0]. Mostly amounts to the same thing for standard arithmetic operators, but it’s an interesting distinction.
IEEE 754 floating-point addition and multiplication are commutative in practice, even if there are exceptions with NaNs etc..
But remember that commutative is on the operations (+,x) which are binary operations, a+b=b+a and ab=ba, you can get accumulated rounding errors on iterated forms of those binary operations.
vlovich123|3 months ago
a+b+c != c+b+a
That’s why you need techniques like Kahan summation.
amluto|3 months ago
+ is a binary operation, and a+b+c can’t be interpreted without knowing whether one treats + as left-associative or right-associative. Let’s assume the former: a+b+c really means (a+b)+c.
If + is commutative, you can turn (a+b)+c into (b+a)+c or c+(a+b) or (commuting twice) c+(b+a).
But that last expression is not the same thing as (c+b)+a. Getting there requires associativity, and floating point addition is not associative.
wtallis|3 months ago
dataangel|3 months ago
Commutativity says that a*b = b*a, but that's not enough to allow arbitrary reordering. When you write a*b*c depending on whether * is left or right associative that either means a*(b*c) or (a*b)*c. If those are equal we say the operation is associative. You need both to allow arbitrary reordering. If an operation is only commutative you can turn a*(b*c) into a*(c*b) or (b*c)*a but there is no way to put a in the middle.
wfleming|3 months ago
Edit: Wikipedia actually says associativity is definitionally about changing parens[0]. Mostly amounts to the same thing for standard arithmetic operators, but it’s an interesting distinction.
[0]: https://en.wikipedia.org/wiki/Associative_property
nyrikki|3 months ago
But remember that commutative is on the operations (+,x) which are binary operations, a+b=b+a and ab=ba, you can get accumulated rounding errors on iterated forms of those binary operations.