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Re: Up-to date Interval Arithmetic



On 2015-03-24 21:19:16 +0100, Ulrich Kulisch wrote:
> I attach an unpublished paper entitled:
> 
> /Up-to-date Interval Arithmetic - From closed intervals to connected sets of
> real numbers//
> /
> which on request recently was sent to the Reliable Computing Group. You may
> find it interesting.

I haven't had much time to look at it closely, but there's an
important remark I would like to make. You are saying in §3.3:

  There is absolutely no need for introducing non mathematical entities
  like -0, +0, NaN (not a number) or NaI (not an interval) in this new
  computing environment.

-0, +0 and NaN in IEEE 754 and NaI in P1788 *are* mathematical entities
in the sense there is a full mathematical and consistent specification
behind. Perhaps what you mean is that they don't fit in your way of
seeing things. Note also that NaI in the set-based flavor of P1788 is
not really useful: in short, it is just there to detect what could be
regarded as input errors (in the system itself, it is not needed). But
this would be true for any arithmetic or computing system in general:
if you want to detect input errors, you must have something like NaI;
that would be NaN in IEEE 754 (but here it is overloaded), or undef in
Perl.

In the same way, using the point of view of purely numerical systems
(where the elements are numbers), I could say that the unums for which
ubit = 1 are non mathematical entities because they do not correspond
to numbers.

Now, every arithmetic / computing system has its own advantages and
drawbacks. At least, floating point has some regularity, and in the
IEEE 754 specification, one can use it to build algorithms like
TwoSum, FastTwoSum, and others to control the error, said otherwise,
do some kind of multiple precision internally. I'm waiting to see
something like that with the unum format.

-- 
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <https://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)