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

[Carter Schonwald]

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 <jjwchoy at gmail.com> 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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