Recall for a minute that floating-point arithmetic is inherently inexact, and it is thus pretty rare that you query for equality using an == operator rather than a relative or absolute error check (both of which involve < or > against a non-NaN value and thus would fail anyways for NaN). No, the main reason to ever actually use == on floating-point is if you're doing a bunch of tests to make sure expressions get the correct exact result (so you're expecting the inexactness to resolve in a particular way) or if you're doing non-computational stuff with floats (such as using them as keys in a map), at which point it becomes a problem that x != x is ever true.
> If you would want such a facility in a program, then it is not enough to propagate just a floating-point value. You would have to define a custom data type and accompany the FP value by another value that encodes its provenance.
IEEE 754 was way ahead of you. The entire rationale for the existing of NaN payloads was to be able to encode the diagnostic details about operation failure. And this suggests that you really want is the rule that NaNs are equal only to other NaNs with the same payload but are unordered with respect to everything else.
From the research I've done, it seems that the main reason that x != x for NaNs isn't some deep theoretical aspect about the properties of NaNs, but rather because IEEE 754 wanted a cheap way to test for NaN without having to call a function.
> If you would want such a facility in a program, then it is not enough to propagate just a floating-point value. You would have to define a custom data type and accompany the FP value by another value that encodes its provenance.
IEEE 754 was way ahead of you. The entire rationale for the existing of NaN payloads was to be able to encode the diagnostic details about operation failure. And this suggests that you really want is the rule that NaNs are equal only to other NaNs with the same payload but are unordered with respect to everything else.
From the research I've done, it seems that the main reason that x != x for NaNs isn't some deep theoretical aspect about the properties of NaNs, but rather because IEEE 754 wanted a cheap way to test for NaN without having to call a function.