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Issue No.02  February (2011 vol.60)
pp: 282291
JeanMichel Muller , CNRSLIP, Arénaire, Lyon
Peter Kornerup , University of Southern Denmark, Odense
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2010.134
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 RNCodings were introduced, allowing an unbiased roundingtonearest to take place by a simple truncation, with the property that problems with doubleroundings 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 roundbit, but still allowing roundingtonearest by truncation, and thus avoiding problems with doubleroundings. Conversions from radix complement to these roundtonearest representations can be performed in constant time, whereas conversion the other way, in general, takes at least logarithmic time. Not only is roundingtonearest a constant time operation, but so is also sign inversion, both of which are at best logtime operations on ordinary two's complement representations. Addition and multiplication on such fixedpoint 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 roundbit. Based on the fixedpoint encoding, it is shown possible to define floatingpoint representations, and a sketch of the implementation of an FPU is presented.
INDEX TERMS
Signeddigit, roundtonearest, constanttime rounding and signinversion, floatingpoint representation, doublerounding.
CITATION
JeanMichel Muller, Peter Kornerup, "Performing Arithmetic Operations on RoundtoNearest Representations", IEEE Transactions on Computers, vol.60, no. 2, pp. 282291, February 2011, doi:10.1109/TC.2010.134
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