Re: Motion P1788/M007.01_NaI: Discussion period begins
On 2009-08-24 14:34:11 +0100, John Pryce wrote:
> On 6 Aug 2009, at 00:35, Vincent Lefevre wrote:
> >Very partially. NaN can also mean [−∞,+∞] (any real number), as in
> >hypot(+∞,qNaN), or also the empty set (e.g. result of sqrt(-1)).
>
> Hmm. In 754, values are always point values, never sets of values.
> So sqrt(-1) = Empty is incorrect in the 754 context. (Remember that
> in the interval context, it is only correct inasmuch as sqrt(-1) is
> syntactic sugar for sqrt( {-1} ) .)
NaN isn't a value. It's a datum that can be used (in particular) to
signal an invalid construct (sqrt(-1) doesn't exist as a real number).
In interval arithmetic, the empty set can be used in the same way.
> Also, when you say NaN can also mean [−∞,+∞] (any real number), are
> you sure that's what "754 thinks"? Or is it (any _extended real
> number_)?
One can see extended real numbers just there for handling overflows
in a nice way. But most math/physical/... problems are expressed on
real numbers.
> If it's the former, then the same reasoning that leads to
> hypot(NaN,+∞) = +∞
> also leads to
> 0 * ∞ = 0,
No, you can't use the same reasoning because of the rule on the limits:
lim_{x -> 0, y -> +∞} x * y
doesn't exist (note that here I don't use the extended real numbers).
> which is not what 754 says. 754's arithmetic is based on the
> extended reals,
The extended reals are based on the real numbers (by using limits).
So, by transitivity, 754's arithmetic is based on the real numbers.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.org/>
100% accessible validated (X)HTML - Blog: <http://www.vinc17.org/blog/>
Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)