## On the precision attainable with various floating-point
number systems

17. R. P. Brent,
On the precision attainable with various floating-point number systems,
* IEEE Transactions on Computers* C-22 (1973), 601-607.
CR 14#25960.
Also appeared as Report TR RC 3751, IBM Research (February 1972), 28 pages.
Retyped 2000.
arXiv:1004.3374v1
Abstract:
dvi (2K),
pdf (76K).

Original paper:
pdf (1447K).

Retyped paper:
dvi (24K),
pdf (201K),
ps (80K).

## Abstract

For scientific computations on a digital computer the set
of real number is usually approximated by a finite set *F* of
"floating-point" numbers. We compare the numerical accuracy possible with
difference choices of *F* having approximately the same range and
requiring the same word length. In particular, we compare different
choices of base (or radix) in the usual floating-point systems.
The emphasis is on the choice of *F*, not on the details of the number
representation or the arithmetic, but both rounded and truncated arithmetic
are considered. Theoretical results are given, and some simulations of
typical floating-point computations (forming sums, solving systems of
linear equations, finding eigenvalues) are described. If the leading
fraction bit of a normalized base 2 number is not stored explicitly
(saving a bit), and the criterion is to minimize the mean square roundoff
error, then base 2 is best. If unnormalized numbers are allowed,
so the first bit must be stored explicitly, then base 4
(or sometimes base 8) is the best of the usual systems.
## Comments

This paper was written in the days when popular IBM machines used base 16,
well before the IEEE floating point standard.
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