# [Haskell-cafe] Why not some subclass of Floating to model NaNs as some handleable bottom?

Richard O'Keefe raoknz at gmail.com
Thu Aug 5 07:26:27 UTC 2021

```You quoted
"> Note that due to the presence of @NaN@, not all elements of 'Float'

Let x y and z be finite floating-point numbers such that x + y ==> z.
Does there always exist neg(y) such that z + neg(y) ==> x?
NO.

And the presence or absence of NaN in the system makes no difference.
If, for example, you add 1.0e-18 to 1.0e0, the answer is 1.0e0 exactly.
That is, (x + y) - y == 0, but x is not 0.

In the presence of rounding, additive inverses do not in general exist.
Neither do multiplicative inverses.

Also addition and multiplication are not associative, but you knew that.
The only reason Float and Double are in Num is because Haskell doesn't
refactoring for years.

The main thing that NaN wrecks that wasn't already broken is Eq.  I would
argue that the right decision there would have been to rule that x == y
(when x and y are floating point numbers) precisely when x and y are
represented by the same bit pattern, with a separate operation for IEEE
"ordered and equal".

At some point, Haskell should make provision for decimal floating point,
as the current versions of IEEE 754 and C do, and that might be a good
reorganisation time.

On Thu, 5 Aug 2021 at 02:05, YueCompl via Haskell-Cafe
>
> Thanks Michał,
>
> I feel less confused as I realized the non-halting possibility per bottoms, from your hint.
>
> I too think the signaling NaN is dreadful enough, so fortunately it's rarely seen nowadays.
>
> Actually what's on my mind was roughly something like "Maybe on steroids", I'm aware that NaN semantics breaks `Num` (or descendants) laws, as seen at https://gitlab.haskell.org/ghc/ghc/-/blob/master/libraries/base/GHC/Float.hs
>
> > Note that due to the presence of @NaN@, not all elements of 'Float' have an additive inverse.
>
> > Also note that due to the presence of -0, Float's 'Num' instance doesn't have an additive identity
>
> > Note that due to the presence of @NaN@, not all elements of 'Float' have an multiplicative inverse.
>
> So it should have been another family of `Num` classes, within which, various NaN related semantics can be legal, amongst which I'd think:
>
> * Silent propagation of NaN in arithmetics, like `Maybe` monad does, seems quite acceptable
> * Identity test, namely `NaN` /= `NaN` - this lacks theoretical ground or not?
> * Comparison, neither `NaN` > 1 nor `NaN` <= 1 - whether or not there's a theoretical framework for this to hold? Maybe `Boolean` type needs enhancement too to do it?
>
> No such family of `Num` classes exists to my aware by now, I just can't help wondering why.
>
> Cheers,
> Compl
>
> On 2021-08-04, at 02:38, Michał J Gajda <mjgajda at gmail.com> wrote:
>
> Dear Yue,
>
> Bottom has much weaker semantics than an exception: it means You may never get a result and thus will never handle it!
>
> Another reason is convenience: it is frequently the case that giving NaN in a row of numbers is much more informative than crashing a program with an exception and never printing the result anyway.
>
> Finally IEEE special values have clear propagation semantics: they are basically Maybe on steroids.
>
> The problem with this approach is indeed a silent handling.
>
> But in order to fix this, it is better to add preconditions to specific algorithms that do not allow IEEE special value on input (`isFinite` or `isNotNaN`) and then track the origin of the special value with the methods like those described here: https://skillsmatter.com/skillscasts/14905-agile-functional-data-pipeline-in-haskell-a-case-study-of-multicloud-api-binding
>
> Never throw an error without telling exactly why it happened and exactly where to fix it :-). Using bottom is last resort; exceptions likewise.
> --
>   Cheers
>     Michał
>
>
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