Performing Arithmetic Operations on Round-to-Nearest Representations

Abstract

During any composite computation, there is a constant need for rounding intermediate results before they can participate in further processing. Recently, a class of number representations denoted RN-Codings were introduced, allowing an unbiased rounding-to-nearest to take place by a simple truncation, with the property that problems with double-roundings are avoided. In this paper, we first investigate a particular encoding of the binary representation. This encoding is generalized to any radix and digit set; however, radix complement representations for even values of the radix turn out to be particularly feasible. The encoding is essentially an ordinary radix complement representation with an appended round-bit, but still allowing rounding-to-nearest by truncation, and thus avoiding problems with double-roundings. Conversions from radix complement to these round-to-nearest representations can be performed in constant time, whereas conversion the other way, in general, takes at least logarithmic time. Not only is rounding-to-nearest a constant time operation, but so is also sign inversion, both of which are at best log-time operations on ordinary two's complement representations. Addition and multiplication on such fixed-point representations are first analyzed and defined in such a way that rounding information can be carried along in a meaningful way, at minimal cost. The analysis is carried through for a compact (canonical) encoding using two's complement representation, supplied with a round-bit. Based on the fixed-point encoding, it is shown possible to define floating-point representations, and a sketch of the implementation of an FPU is presented.