Re: How to merge required operations under a common name

[Walter Mascarenhas <walter.mascarenhas@xxxxxxxxx>]

Dear Ulrich,

   Some statements in your message, and the previous ones,

are simply not right.

  You keep saying that people should read your work, and I

believe than that it is only fair to ask you to please read

the standard before making such statements.

  Your comments regarding +oo/-oo reflect your misunderstanding
of the purpose of the simplified part of the standard. The standard

as a whole has nothing whatsoever to do with +oo and -oo, and

the simplified part considers the **use** of IEEE arithmetic in
order to  implement an interval arithmetic which  is much like
what you propose,  ie., without +oo/-oo.

In order words, +oo/-oo are only used as convenient tools to
represent unbounded intervals. The abstractions +oo/-oo
are not part of such intervals, and this is written explicitly
in the standard.

  Therefore, the standard DOES NOT SUGGEST that one should

take +oo and -oo as numbers, and there is absolutely no need to

define operations involving them. The standard only defines

operations on unbounded intervals, and people who choose to
implement such intervals using the facilities available

in 99.99% of the computer used today may need to think a bit

on how to adapt what is available to what is mandated by

the standard.

 You do have a point regarding +/- 0 though. In this case, the

simplified part of the standard is (and must be) specific. In other

words, the simplified standard does mandate what should be

the sign of the zero returned by some functions.

  However, even in the case of signed zeros, the standard
is quite reasonable and I believe that having the standard
is much better than having nothing. I much rather have a
reasonable standard to guide implementers now than keep
waiting forever for a mythical ideal of what could be
implemented in the perfect hardware.

On Wed, Jan 27, 2016 at 7:42 AM, Ulrich Kulisch <ulrich.kulisch@xxxxxxx> wrote:

Dear colleagues,

just before the discussion period for IEEE 1788 closes let me make a final remark.

Interval arithmetic defined over the real and floating-point numbers leads to an exception-free calculus.

I have said this repeatedly. But I feel it is not yet really understood. Developing interval arithmetic over the IEEE 754 numbers pulls all the IEEE 754 exceptions  like -oo, +oo, NaN, +0, -0 into interval arithmetic where they do not occur and definitely are not needed.

In IEEE 754 -oo and +oo are numbers. In interval arithmetic -oo and +oo are just used to describe unbounded sets of real numbers. But they are themselves not elements of these intervals. This is a subtle difference! Interval arithmetic is an arithmetic for connected sets of real numbers!

If you take -oo and +oo as numbers, you have to study and provide operations like oo - oo, 0 times oo, oo/oo which in IEEE 754 are set to NaN. Then you have to define operations for NaNs and so on. All these operations do not occur in interval arithmetic if it is defined over the real and floating-point numbers. All this, of course, has to be proved. The proof can be found in my book Computer Arithmetic and Validity, in its second edition in sections 4.9 to 4.12, for instance.  Let me still mention that the book was published before IEEE 1788 was founded.

Best regards
Ulrich

-- 
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
KIT Distinguished Senior Fellow

Telefon: +49 721 608-42680 
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/

KIT - Universität des Landes Baden-Württemberg 
und nationales Großforschungszentrum in der 
Helmholtz-Gesellschaft