Performing Arithmetic Operations on Round-to-Nearest Representations - ENS de Lyon - École normale supérieure de Lyon Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2010

Performing Arithmetic Operations on Round-to-Nearest Representations

Résumé

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 un-biased 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 2'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 2'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.
Fichier principal
Vignette du fichier
05487497.pdf (366.3 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

ensl-00548988 , version 1 (21-12-2010)
ensl-00548988 , version 2 (04-01-2011)

Identifiants

  • HAL Id : ensl-00548988 , version 1

Citer

Peter Kornerup, Jean-Michel Muller, Adrien Panhaleux. Performing Arithmetic Operations on Round-to-Nearest Representations. 2010. ⟨ensl-00548988v1⟩
130 Consultations
336 Téléchargements

Partager

Gmail Facebook X LinkedIn More