Date: Sat, 18 May 2013 15:00:22 +0200
From: Ulrich Kulisch <ulrich.kulisch@xxxxxxx>
To: stds-1788 <STDS-1788@xxxxxxxxxxxxxxxxx>
Subject: exact dot product
Dear colleagues,
In a mail of May 15, 2013 Vincent Lefevre wrote among others:
And about 11.11.11, "Complete arithmetic, dot product function"
(for 754-conforming types): I completely disagree on it being a
requirement. IMHO, it should not even be normative. P1788 is a
standard on interval arithmetic. I don't see the point of having a
single paragraph on a different arithmetic (which, BTW, would rather
belong to IEEE 754). If the only goal is to have an interval version
of the dot product, then the dot product could be specified like the
other operations, without enforcing an algorithm.
In general I appreciate Vincent's mails, but not this one. Nevertheless
I am grateful to have it. It gives me a chance to answer. There may be
others who share his view.
. . .
I agree that IEEE P1788 is called a standard for interval arithmetic.
But I think we would make a big mistake if we would unnecessarily
restrict it to the four basic arithmetic operations for real intervals.
Techniques for computing /close/ bounds for solutions should also be
considered in IEEE P1788.
I am sure that many numerical analysts share this view. See the letter
of the IFIP WG 2.5 on Numerical Software to the standard committee IEEE
754 (attachment 1). It requires an exact dot product for the data format
double precision. IEEE 754 did not reject this requirement. Their
primary goal was a revision of the floating-point standards 754 and 854.
In an answer they said: /we feel we do not have enough information at
this time to make a useful definition of these extensions. We have been
considering proposals along these lines for the past 6 years and nothing
has so far risen to the level of consensus of any kind./
. . .
Folks,
I fear I must disagree with Vincent & agree with Ulrich. Not only
did the 754 group "not reject (the) requirement" for correctly
rounded dot product, we supported it. Even in 754 we included
definitions for dot products, sums, sum of squares, & sum of
absolute values. (Clause 9.4, reduction operations.)
Alas, Vincent may point out that we did NOT specify them as being
exactly computed. Indeed, we were not able to at the time, due
to there being so many 754s out there that did not support it.
So these functions are defined but without specified result
precision. (Although we DID specify exceptions in a standard way.)
However, correctly rounded versions WERE our intention. Indeed,
it is silly to define them just as they would come out of any
C-compiler.
Then, shortly after the standard was published a number of papers
appeared that show you how to correctly round your result in all
cases using 754 arithmetic, just a bit of extra precision, &
some sorting. It is quite clever & not at all difficult to do.
And, indeed, had we known they were possible (& fast) at the time
we WOULD have required them as many (perhaps most) existing 754
implementations have done now.
So it is my hope that we do it under 1788 as well.
After all, we should NOT be less accurate than 754, now should we. :-)
Yours,
Dan