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).


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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