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Issue No.02 - February (2011 vol.60)

pp: 228-241

Claude-Pierre Jeannerod , INRIA Grenoble - Rhône-Alpes, Universit de Lyon, Lyon

Nicolas Louvet , Université Claude Bernard Lyon 1, Universit de Lyon, Lyon

Jean-Michel Muller , CNRS, Universit de Lyon, Lyon

Adrien Panhaleux , École Normale Supérieure de Lyon, Universit de Lyon, Lyon

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2010.144

ABSTRACT

When implementing a function f in floating-point arithmetic, if we wish correct rounding and good performance, it is important to know if there are input floating-point values x such that f(x) is either the middle of two consecutive floating-point numbers (assuming rounded-to-nearest arithmetic), or a floating-point number (assuming rounded toward \pm \infty or toward 0 arithmetic). In the first case, we say that f(x) is a midpoint, and in the second case, we say that f(x) is an exact point. For some usual algebraic functions and various floating-point formats, we prove whether or not there exist midpoints or exact points. When there exist midpoints or exact points, we characterize them or list all of them (if there are not too many). The results and the techniques presented in this paper can be used in particular to deal with both the binary and the decimal formats defined in the IEEE 754-2008 standard for floating-point arithmetic.

INDEX TERMS

Floating-point arithmetic, correct rounding, algebraic function.

CITATION

Claude-Pierre Jeannerod, Nicolas Louvet, Jean-Michel Muller, Adrien Panhaleux, "Midpoints and Exact Points of Some Algebraic Functions in Floating-Point Arithmetic",

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