On the precision attainable with various floating-point number systems
Provides theoretical and simulation-based guidance for choosing floating-point base to minimize roundoff error in scientific computing.
This paper compares the numerical accuracy of different floating-point number systems (bases 2, 4, 8, etc.) with similar range and word length. It finds that base 2 is best when the leading bit is implicit (saving a bit) and mean square roundoff error is minimized, while base 4 (or 8) is best when unnormalized numbers require explicit storage of the first bit.
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.