Re: ExactDotProduct
Ulrich Kulisch wrote:
During recent mails the question came up whether
a) an exact dot product (Motion 9) or
b) a tightly rounded dot product (Vienna proposal)
is to be required by IEEE P1788.
A few pros and cons are listed here which might help to vote for Motion 9.
* A pro for a): It is the key operation for variable precision
interval arithmetic (see the IFIP Working Group letters). Roughly:
With long interval arithmetic real arithmetic expressions usually
can be evaluated with sufficiently accurate bounds in a very
simple manner just by operator overloading (what sometimes is
called naïve interval arithmetic). This may make interval
arithmetic much more attractive.
This is not specific to a).
Long ago, I had programmed a variable (arbitrary) precision interval
arithmetic that provides least significant bit accuracy of real
arithmetic expressions (in my case for +,-,*,/ and sqrt) simulating
the accurately enclosed dot product (at most 2 ulp wide) using Brent's
multiprecision package. Actually, I learnt from Brent the multiprecision
triplex representation recommended in the Vienna proposal, which -
intelligently used - turned out to be a faster than evaluating the
expressions with the Rump and Boehm technique (based on ACRITH which
had the exact dot product).
The idea is to first work with a fixed precision, then fiogure out
from the result how much digits were lost through outward rounding,
then increase the precision by that amount plus a number of safety
digits and rerun the computation. This is usually much faster than
the iterative approach of Rump and Boehm, which essentially redo
much work in each iteration.
Therefore, the effort put into hardware for the exact dot product
is in my opinion better spent in hardware versions
of TwoSum and TwoProduct that figure in the algorithms for an
accurately enclosed dot product. These would also be valuable for
implementing fast variable precision arithmetic with all IEEE
rounding modes.
Thus it would help acceptance even outside the interval community.
Arnold Neumaier