Re: Fw: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
Ian,
Yes, I concede, based on both Dan's and your analysis, that multiple
internal representations from level 3 can correspond to the same exact object.
Talking about uniqueness seems more logical at level 2. However,
we still need to do some work here, since we not only have a finite
set of floating point numbers, but we have many ways of combining
elements of that set into objects that can be interpreted as intervals.
Perhaps one simple and non-problematic way would be to interpret
two floating point numbers in terms of inf and sup. You have illustrated
that there are various interpretations of two floating point numbers
as interval bounds, even at level 2. (However, I think they would all
generate the same set of mathematical intervals, if we the end points
come from 754 -- I'll need to double-check that.) Separately,
we could have mid-rad representation which might be similarly manageable,
if we don't get too arcane. Knowing that, for a particular underlying
floating point system, we are working with a uniquely defined set
of floating point intervals at level 2 seems very useful.
Best regards,
Baker
On 6/30/2010 15:19, Ian McIntosh wrote:
Ralph Baker Kearfott>> inf-sup representation and 754 arithmetic,
rounding the lower bound
Ralph Baker Kearfott>> down and rounding the upper bound up gives a
uniquely defined interval,
Dan Zuras> A uniquely defined INTERVAL, yes. It need not be a
Dan Zuras> uniquely defined REPRESENTATION of that interval, though.
. . .
Ralph Baker Kearfott>> mapping in the context of inf-sup and 754 is
unique (and hence would
Ralph Baker Kearfott>> lead to an arithmetic that is reproducible across
implementations).
Dan Zuras> Actually, except for having a representation for infinity
Dan Zuras> in our floating-point arithmetic, we need not mention 754
Dan Zuras> at all. And we would STILL be reproducible across any
Dan Zuras> implementations that share that floating-point type.
An example of multiple level 3 representations of the same interval
would be to store an inf-sup interval as one of the pairs
inf sup
sup inf
-inf -sup
-sup -inf
inf -sup
-sup inf
-inf sup
sup -inf
The last four representations, with one but not both of inf and sup
negated, let one use the same rounding mode for both when doing an add
or subtract, while the first four require opposite rounding modes. All
eight ways represent the same interval with the same lower and upper
bounds, and extracting those bounds will give the same values.
Another example is specific to IEEE 754-2008 decimal floating point,
where multiple representations (DPD = Densely Packed Decimal, BID =
Binary Integer Decimal, and others) are allowed. The exact same values
are stored in each representation, but the representations are very
different.
There are other examples but two is enough.
With just level 3 representation differences, basic arithmetic (but not
necessarily everything) should be reproducible across implementations.
- Ian McIntosh IBM Canada Lab Compiler Back End Support and Development
----- Forwarded by Ian McIntosh/Toronto/IBM on 06/30/2010 03:58 PM -----
From:
Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
To:
Ian McIntosh/Toronto/IBM@IBMCA
Date:
06/30/2010 02:26 PM
Subject:
Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus
applications
--
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R. Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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