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Issue No.02 - February (2011 vol.60)
pp: 282-291
Jean-Michel Muller , CNRS-LIP, Arénaire, Lyon
Peter Kornerup , University of Southern Denmark, Odense
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.
INDEX TERMS
Signed-digit, round-to-nearest, constant-time rounding and sign-inversion, floating-point representation, double-rounding.
CITATION
Jean-Michel Muller, Peter Kornerup, "Performing Arithmetic Operations on Round-to-Nearest Representations", IEEE Transactions on Computers, vol.60, no. 2, pp. 282-291, February 2011, doi:10.1109/TC.2010.134
REFERENCES
[1] P. Kornerup and J.-M. Muller, "RN-Coding of Numbers: Definition and Some Properties," Proc. Int'l Meeting on Automated Compliance Systems (IMACS '05), July 2005.
[2] A. Booth, "A Signed Binary Multiplication Technique," Quarterly J. Mechanics and Applied Math., vol. 4, pp. 236-240, 1951.
[3] J.-L. Beuchat and J.-M. Muller, "Multiplication Algorithms for Radix-2 RN-Codings and Two's Complement Numbers," technical report, INRIA, http://www.inria.fr/rrrtrr-5511.html, Feb. 2005.
[4] IEEE, IEEE Std. 754-2008 Standard for Floating-Point Arithmetic, Aug. 2008.
[5] C. Lee, "Multistep Gradual Rounding," IEEE Trans. Computers, vol. 38, no. 4, pp. 595-600, Apr. 1989.
[6] A.M. Nielsen, D. Matula, C. Lyu, and G. Even, "An IEEE Compliant Floating-Point Adder that Conforms with the Pipelined Packet-Forwarding Paradigm," IEEE Trans. Computers, vol. 49, no. 1, pp. 33-47, Jan. 2000.
[7] R.E. Moore, Interval Analysis. Prentice Hall, 1963.
[8] P. Farmwald, "On the Design of High Performance Digital Arithmetic Units," PhD dissertation, Stanford Univ., Aug. 1981.
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