On the precision attainable with various floating-point
17. R. P. Brent,
On the precision attainable with various floating-point number systems,
IEEE Transactions on Computers C-22 (1973), 601-607.
Also appeared as Report TR RC 3751, IBM Research (February 1972), 28 pages.
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.
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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