You keep saying that == on floating point is a partial relation, but that's only true because of the way NaN currently compares. Floating point exists as a mode

You keep saying that == on floating point is a partial relation, but that's only true because of the way NaN currently compares. Floating point exists as a model of the real numbers, on which equality is total, so a system modeling them should absolutely keep that if possible, which it is. It's fine to have a non-reflexive operation that behaves like == does for floats, but making that the primary equivalence relation is absolutely bonkers, since 2 NaN values with the same bit-pattern are completely indistinguishable from each other, yet compare non-equal. Furthermore, I can't think of any reason that Inf==Inf should be true, while NaN==Nan is false. In both cases, they can be modeling different numbers (although that is true of literally any floating point number). If there were a major advantage to making == non-reflexive, that would be one thing, but I've been programming for most of my life, (including a lot of floating point), and I've yet to come across a place where I was glad that NaN!=NaN.