Re: Edge case conversions, exceptions to IEEE FPA

[Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>]

Michel Hack schrieb:
> On (Fri, 14 Nov 2008 09:52:48 +0100) Arnold Neumaier wrote:
> > The midpoint is usually used as is for subsequent point evaluation
> > and may be approximate only but must lie in the interval.
>
> Does this mean mid() would return signed realmax for a singly-bounded
> interval, and 0 for Entire?  Or would it return signed Inf resp. NaN?
> I suppose the former, so it can be used without restrictions.  Oh, I
> forgot mid(Empty) -- surely that must be NaN, right?
>
> Radius is clear:  +Inf for unbounded (incl. Entire), 0 for Empty, and
> finite non-zero poitive otherwise.
>
> Is there a requirement on intervalabs(mid(xx), rad(xx)) ?

I'd favor the following:


   There are an operation mid(xx) that returns for any interval xx
   an IA numeral m called the midpoint of xx, an operation rad(xx)
   that returns for any interval xx an IA numeral r called the radius
   of xx, and an operation midRad(xx) that returns both m and r.

   If xx=[l,u] is compact, m and r shall have the values defined by
       set round up
       r = 0.5*(u-l);m=l+r.
   If xx has an infinite bound, m is the absolutely smallest number
   in xx, and r = inf.
   If xx is Empty, m = NaN, r = NaN.

   This guarantees that
       mid(x) in xx       if xx is nonempty,
       |x-mid(x)|<=rad(x) for all x in xx.
   Because of this,
   intervalabs(mid(xx), rad(xx)) will automatically contain xx.


Arnold Neumaier