Wonderful post! I just want to throw a minor wrench in: ajb@spamcop.net wrote:
"Eq" is not merely a function of type a -> a -> Bool. It's a concept with semantics. It must be an equivalence relation, and it also must mean semantic equality. Functions that respect semantic equality have the property that x == y implies f x == f y.
(No, none of these restrictions are in the language report. But I'd be annoyed if someone defined a version of Eq that didn't have these properties.)
From what you argue above, and reading the IEEE 754 standard correctly, instance Eq Float and instance Eq Double should be *removed* ! That is because +0. == -0. is true (according to IEEE 754) but it is legitimate to have functions f where f +0. <> f -0. [this mostly shows up for complex rather than real functions, but this is still in IEEE 754]. The easiest interpretation is that == as defined in IEEE 754 is not in fact a semantic equality predicate. Very messy. But having a good language (Haskell) that does not respect established standards (IEEE 754) when it claims to implement 'floating point' computations is a real drawback. Jacques