On the computation of correctly-rounded sums

Peter Kornerup 1 Vincent Lefèvre 2, 3 Nicolas Louvet 3, 2 Jean-Michel Muller 3, 2
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
3 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : This paper presents a study of some basic blocks needed in the design of floating-point summa- tion algorithms. In particular, in radix-2 floating-point arithmetic, we show that among the set of the algo- rithms with no comparisons performing only floating- point additions/subtractions, the 2Sum algorithm in- troduced by Knuth is minimal, both in terms of number of operations and depth of the dependency graph. We investigate the possible use of another algorithm, Dekker's Fast2Sum algorithm, in radix-10 arithmetic. We give methods for computing, in radix 10, the floating-point number nearest the average value of two floating-point numbers. We also prove that un- der reasonable conditions, an algorithm performing only round-to-nearest additions/subtractions cannot compute the round-to-nearest sum of at least three floating-point numbers. Starting from an algorithm due to Boldo and Melquiond, we also present new results about the computation of the correctly-rounded sum of three floating-point numbers. For a few of our algorithms, we assume new operations defined by the recent IEEE 754-2008 Standard are available.
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Peter Kornerup, Vincent Lefèvre, Nicolas Louvet, Jean-Michel Muller. On the computation of correctly-rounded sums. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2012, 61 (3), p. 289-298. ⟨10.1109/TC.2011.27⟩. ⟨ensl-00331519v2⟩

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