Jean-Michel Muller
Abstract
In their book, Scientific Computing on the Itanium, Cornea et al. [2002] introduce an accurate algorithm for evaluating expressions of the form ab+cd in binary floating-point arithmetic, assuming an FMA instruction is available. They show that if p is the precision of the floating-point format and if u = 2-p, the relative error of the result is of order u. We improve their proof to show that the relative error is bounded by 2u+7u2+6u3. Furthermore, by building an example for which the relative error is asymptotically (as p → ∞ or, equivalently, as u → 0) equivalent to 2u, we show that our error bound is asymptotically optimal.
- M. Cornea, J. Harrison, and P. T. P. Tang. 2002. Scientific Computing on Itanium®-Based Systems. Intel Press, Hillsboro, OR. Google Scholar
Digital Library
- N. J. Higham. 1996. Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, PA. Google ScholarDigital Library
- IEEE Computer Society. 2008. IEEE standard for floating-point arithmetic. IEEE Standard 754-2008. http://ieeexplore.ieee.org/servlet/opac?punumber=4610933.Google Scholar
- C.-P. Jeannerod, N. Louvet, and J.-M. Muller. 2013. Further analysis of Kahan's algorithm for the accurate computation of 2×2 determinants. Math. Comp. 82 (2013).Google Scholar
- W. Kahan. 1996. Lecture notes on the status of IEEE-754. (1996). http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF.Google Scholar
- J.-M. Muller, N. Brisebarre, F. de Dinechin, C.-P. Jeannerod, V. Lefèvre, G. Melquiond, N. Revol, D. Stehlé, and S. Torres. 2010. Handbook of Floating-Point Arithmetic. Birkhäuser Boston. 572 pages. ISBN 978-0-8176-4704-9. Google ScholarDigital Library
- P. H. Sterbenz. 1974. Floating-Point Computation. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Index Terms
- On the Error of Computing ab+cd using Cornea, Harrison and Tang's Method
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