There's a C++26 paper about compile time math optimizations with a good overview and discussion about some of these issues [P1383]. The paper explicitly states:
1. It is acceptable for evaluation of mathematical functions to differ between translation time and runtime.
2. It is acceptable for constant evaluation of mathematical functions to differ between platforms.
So C++ has very much accepted the fact that floating point functions should not be presumed to give identical results in all circumstances.
Now, it is of course possible to ensure that floating point-related functions give identical results on all your target machines, but it's usually not worth the hassle.
[P1383]: https://www.open-std.org/jtc1/sc22/wg21/docs/papers/2023/p13...
Intel Compiler for e.g. uses less than IEEE764 precision for floating point ops by default, for example.
It is commutative (except for NaN). It isn't associative though.
Why? This is well specified by IEEE 754. Many runtimes (e.g. for Javascript) use NaN boxing. Treating floats as a semi-arbitrary selection of rational numbers plus a handful of special values is /more/ correct than treating them as real numbers, but treating them as actually specified does give more flexibility and power.
But I've never seen them depend on those NaNs surviving the FPU. Hell, they could use the same trick on bit patterns that overlap with valid float values if they really wanted to.
My understanding is the exact opposite - that it allows implementations to return any NaN value at all. It need not be any that were inputs.
It may be that JavaScript relies on it and that has become more binding than the actual spec, but I don't think the spec actually guarantees this.
Edit: actually it turns out nan-boxing does not involve arithmetic, which is why it works. I think my original point stands, if you are doing something that relies on how bit values of NaNs are propagated during arithmetic, you are on shaky ground.
> 6.2.3 NaN propagation
> An operation that propagates a NaN operand to its result and has a single NaN as an input should produce a NaN with the payload of the input NaN if representable in the destination format.
> If two or more inputs are NaN, then the payload of the resulting NaN should be identical to the payload of one of the input NaNs if representable in the destination format. This standard does not specify which of the input NaNs will provide the payload.
https://rust-lang.github.io/rfcs/3514-float-semantics.html
See also this section of wikipedia https://en.wikipedia.org/wiki/NaN#Canonical_NaN
"On RISC-V, most floating-point operations only ever generate the canonical NaN, even if a NaN is given as the operand (the payload is not propagated)."
And from the same article:
"IEEE 754-2008 recommends, but does not require, propagation of the NaN payload." (Emphasis mine)
I call bullshit on the statement "specifically binary operations combining two NaN inputs must result in one of the input NaNs." It is definitely not in the spec.
> For an operation with quiet NaN inputs, other than maximum and minimum operations, if a floating-point result is to be delivered the result shall be a quiet NaN which should be one of the input NaNs.
The same document say:
> shall -- indicates mandatory requirements strictly to be followed in order to conform to the standard and from which no deviation is permitted (“shall” means “is required to”)
> should -- indicates that among several possibilities, one is recommended as particularly suitable, without mentioning or excluding others; or that a certain course of action is preferred but not necessarily required; or that (in the negative form) a certain course of action is deprecated but not prohibited (“should” means “is recommended to”)
i.e. It required to be a quiet NaN, and recommended to use one of the input NaN.
I'm wondering if we couldn't re-think "bit" to the computer science usage instead of the thing that goes in the horse's mouth, and what it would mean for an AI agent to "champ at the bit"?
What new sayings will we want?
a * b = b * a for all "normal" floating point numbers.