Dear Baker:
As Ramon Moore said it in the
attached letter interval arithmetic aims that machine
computing can be done
correctly and rigorously. Necessary for reaching this goal
certainly is that it
is kept simple and provided at high speed.
In March 2008 I published my book: Computer
Arithmetic and Validity
-- Theory, Implementation and Applications.
It develops interval arithmetic over the sets R and F of
real and
floating-point numbers. It leads to a closed calculus that is
totally free of
exceptions, i.e., the result of any of the operations plus,
minus, multiply,
divide and the dot product for intervals of real numbers
including elementary
functions always delivers a real interval again. Let me just
call this interval
arithmetic CIRF.
In contrast to this IEEE 1788
develops interval arithmetic over the set of IEEE 754 numbers.
Since these
include R and F this leads to a superset SIRF of CIRF. Much of
the recent
discussion within IEEE 1788 deals with elements of the set
SIRF\CIRF. Since
CIRF is a closed calculus, elements of SIRF\CIRF do not occur
for computations
in CIRF. So developing IEEE 1788 over IEEE 754 unnecessarily
complicates
interval arithmetic.
Interval arithmetic of CIRF needs
not to be slower than conventional floating-point arithmetic.
The contrary is
the case. The exact dot product provides very fast matrix and
matrix-vector
operations as well as defect correction or iterative
refinement techniques
which frequently are absolutely necessary for obtaining high
accuracy and close
bounds in floating-point and interval arithmetic. It also
allows very fast long
real and long interval arithmetic. A hardware implementation
of the exact dot
product in 1993 computed it in 1/4 of the time the Intel
processor needed for
computing the dot product in conventional floating-point
arithmetic. A
corresponding hardware implementation of the exact dot product
at Berkeley
in 2015 even
reaches a speed increase by a factor of 6.
In the early days of the standard
development I prepared several motions which all have been
accepted. My last
motion required an exact dot product. It only got a small
majority of the
yes-votes. In an early draft of the standard the exact dot
product (EDP) was
listed in §4 as basic arithmetic operation where indeed it
needs to be listed.
However, a renowned colleague
repeatedly argued against the EDP, perhaps motivated by some
commercial
interest. So it was not listed under basic arithmetic
operations anymore and
promised it would much better fit into the standard later
towards its end. Finally
a motion was placed not requiring the EDP anymore. This motion
was accepted.
Stupid arguments like the EDP has nothing to do with
interval
arithmetic or it needs a standard of its own appeared
again and again.
Many members of IEEE 1788 -- unwilling to study the literature
-- still
consider computing the dot product exactly as more complicated
than computing
it in conventional floating-point arithmetic or computing it
correctly rounded,
while the contrary is the case. Fixed-point accumulation of
the dot product is
the simplest thing in the world. By pipelining it can be done
in the time the
processor needs to read the data, i.e., no method for
computing a dot product
can be faster than computing it exactly.
Shifting the requirement for an EDP to a revision of IEEE 754
is a terrible
mistake. The way how the operations with the directed
roundings are
handled in IEEE 754 shows that a majority of its members is
not really
interested in making interval arithmetic a success. They are
rather afraid of
damaging their baby.
The understanding of the meaning of the exact dot product for
success of
interval arithmetic within the members of IEEE 1788 may have
increased during
recent months. I appreciate Ned Nedialcov's shortened version
of the standard.
The tragic present situation practically could be repaired to
a certain extent
by simply adding the dot product into this version after fma.
I hope that a
corresponding motion today would be accepted.
Best
wishes
Ulrich
Kulisch
--
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