Proposal: Add "fma" to the RealFloat class

[Mike Meyer]

On Sun, May 3, 2015 at 6:50 PM, Carter Schonwald <carter.schonwald at gmail.com
> wrote:

> .... how would you have an implementation of finite precision floating
> point that has the "expected" exact algebraic laws for * and +?
>

That's model #1 that we can't have. So you don't.


> I would argue that Float and Double do satisfy a form of the standard
> algebric laws where equality is approximate.
>
> eg  (a+(b+c)) - ((a+b)+c) <= \epsilon, where epsilon  is some constant
> multiple of max(ulp(a),ulp(b),ulp(c)).
> (a similar idea applies to pretty much any other algebraic law you can
> state, such as distributivity etc)
>

So how do you fix the fact that any comparison with a NaN and a non-NaN is
false? Among other IEEE oddities.


> I do think that it'd be useful if the RealFloat class provided an ulp
> function (unit of least precision), which is available as part of any IEEE
> compliant c float library.
>
> there are MANY computable number represntations where the *exact*
> algebraic laws dont hold, but this *approximate* form which provides some
> notion of bounded forwards/backwards relative/absolute error bound
> guarantee in a particularly strong way.
>

True. That's the root of the problem the proposal is trying to solve.


> i think we should figure out articulating laws that play nice for both the
> *exact* and *approximate* universes.
>

We also need laws that play nice for the IEEE universe, because people
doing serious numerical work want that one. I believe you will wind up with
two different sets of laws, which is why I proposed taking the parts that
don't agree out of the Prelude, and letting users import the ones they want
to use.
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