for equational laws to be sensible requires a sensible notion of equality,
the Eq for Floating point numbers is
meant for handling corner cases (eg: am i about to divide by zero), not
"semantic/denotational equivalence"
Exact equality is fundamentally incorrect for finite precision mathematical
computation.
You typically want to have something like
nearlyEq tolerance a b = if distance a b <= tolerance then True else False
Floating point is geometry, not exact things
https://hackage.haskell.org/package/ieee754-0.7.3/docs/Data-AEq.html
is a package that provides an approx equality notion.
Basically, floating points work the way they do because its a compromise
that works decently for those who really need it.
If you dont need to use floating point, dont! :)
On Fri, Sep 26, 2014 at 9:28 AM, Jason Choy
subject to certain caveats. It's not unfair to say that
floating point multiplication is (nearly) associative "within a few ulp".
I'm not disputing this.
However, you can't deny that this monoid law is broken for the floating point operations:
mappend x (mappend y z) = mappend (mappend x y) z
Perhaps I'm being pedantic, but this law should hold for all x, y, z, and it clearly doesn't.