Multiplication by rational constants: LIP research report 2011-3

Florent De Dinechin 1, 2
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Multiplications by simple rational constants often appear in fixed-point or floating-point application code, for instance in the form of division by an integer constant. The hardware implementation of such operations is of practical interest to FPGA-accelerated computing. It is well known that the binary representation of rational constants is eventually periodic. This article shows how this feature can be exploited to implement multiplication by a rational constant in a number of additions that is logarithmic in the precision. An open-source implementation of these techniques is provided, and is shown to be practically relevant for constants with small numerators and denominators, where it provides improvements of 20 to 40\% in area with respect to the state of the art. It is also shown that for such constants, the additional cost for a correctly rounded result is very small, and that correct rounding very often comes for free in practice.
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IEEE Transactions on Circuits and Systems Part 2 Analog and Digital Signal Processing, Institute of Electrical and Electronics Engineers (IEEE), 2012, pp.00. 〈10.1109/TCSII.2011.2177706〉
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Florent De Dinechin. Multiplication by rational constants: LIP research report 2011-3. IEEE Transactions on Circuits and Systems Part 2 Analog and Digital Signal Processing, Institute of Electrical and Electronics Engineers (IEEE), 2012, pp.00. 〈10.1109/TCSII.2011.2177706〉. 〈ensl-00610328v3〉

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