Re: [Haskell-cafe] Sum and Product do not respect the Monoid laws

for equational laws to be sensible requires a sensible notion of equality, the Eq for Floating point numbers is

could it not be then that for floating points to be a monoid you must specify a satisfying notion of equality? (well, i guess nothing is stopping anyone from doing that themselves; and your point is that simply not having floats as a monoid is somehow "bad"?) On Sat, Sep 27, 2014 at 5:41 AM, Carter Schonwald < carter.schonwald@gmail.com> wrote:

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 wrote:

...

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.

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