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 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.
Type de document :
Pré-publication, Document de travail
Pre-print, 24 pages, submitted to IEEE Transactions on Computers. 2009
Liste complète des métadonnées

Littérature citée [4 références]  Voir  Masquer  Télécharger

https://hal-ens-lyon.archives-ouvertes.fr/ensl-00441933
Contributeur : Adrien Panhaleux <>
Soumis le : jeudi 17 décembre 2009 - 16:22:09
Dernière modification le : samedi 21 avril 2018 - 01:27:15
Document(s) archivé(s) le : jeudi 18 octobre 2012 - 11:05:36

Fichier

KoMuPa2009.pdf
Fichiers éditeurs autorisés sur une archive ouverte

Identifiants

  • HAL Id : ensl-00441933, version 1

Collections

Citation

Adrien Panhaleux, Peter Kornerup, Jean-Michel Muller. Performing Arithmetic Operations on Round-to-Nearest Representations. Pre-print, 24 pages, submitted to IEEE Transactions on Computers. 2009. 〈ensl-00441933〉

Partager

Métriques

Consultations de la notice

276

Téléchargements de fichiers

117