[kzrdude]

It looks like -fassociative-math is "safe" in the sense that it can not be used to get UB in working code? That's a good property to make it easier to use in the right context.

[mbauman]

See the one footnote: you can re-associate a list of 2046 numbers such that they sum to _any_ floating point number between 0 and 2^970.

https://discourse.julialang.org/t/array-ordering-and-naive-s...

[StefanKarpinski]

To be fair though, as noted further down in that thread, naive left-to-right summation is the worst case here since the tree of operations is as skewed as possible. I think that any other tree shape is better and compilers will tend to make the tree more balanced, not less, when they use associativity to enable SIMD optimizations. So while reassociation is theoretically arbitrarily bad, in practice it's probably mostly ok. Probably.

[stephencanon]

This is exactly right. Also, _all_ of the summation orderings satisfy the usual backwards-stability bounds, and so all of them are perfectly stable in normal computations.

[evilotto]

Floating point math is fun:

  def re_add(a,b,c,d,e):
      return (a+b+c+d+e) == (2 * (c+a+b+d+e))
  
  print(re_add(1e17, -1e17, 3, 2, 1))

[simonbyrne]

Not necessarily: if your cospi(x) function is always returning 1.0 (https://github.com/JuliaLang/julia/issues/30073#issuecomment...), but you wrote your code assuming the result was in a different interval, then you could quite easily invoke undefined behavior.

[adgjlsfhk1]

Yeah. It's safe in that you won't get UB, but it's bad in that you can get arbitrarily wrong answers.

[wiml]

To be fair, if you're using floating point at all you can get arbitrarily wrong answers. The nice thing about ieee754 conformance is that you can, with a lot of expertise, somewhat reason about the kinds of error you're getting. But for code that wasn't written by someone skilled in numerical techniques, and that's the vast majority of fp code, is fast-math worse than the status quo?

[simonbyrne]

From personal experience, yes: I've seen multiple cases of scientists finding the ultimate cause of their bugs was some fast-math-related optimization.

The problem isn't necessarily the code they wrote themselves: it is often that they've compiled someone else's code or an open source library with fast-math, which broke some internal piece.