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Correct rounding of algebraic functions

Published online by Cambridge University Press:  24 April 2007

Nicolas Brisebarre  and
Jean-Michel Muller
Nicolas Brisebarre
Affiliation:
Laboratoire LIP (CNRS/ENS Lyon/INRIA/Univ. Lyon 1), Projet Arénaire, 46 allée d'Italie, 69364 Lyon Cedex 07, France; Nicolas.Brisebarre@ens-lyon.fr; Jean-Michel.Muller@ens-lyon.fr Laboratoire LaMUSE, Université J. Monnet (Saint-Étienne), 23, rue du Dr P. Michelon, 42023 Saint-Étienne Cedex 02, France.
Jean-Michel Muller
Affiliation:
Laboratoire LIP (CNRS/ENS Lyon/INRIA/Univ. Lyon 1), Projet Arénaire, 46 allée d'Italie, 69364 Lyon Cedex 07, France; Nicolas.Brisebarre@ens-lyon.fr; Jean-Michel.Muller@ens-lyon.fr
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Abstract

We explicit the link between the computer arithmetic problem of providing correctly rounded algebraic functions and some diophantine approximation issues. This allows to get bounds on the accuracy with which intermediate calculations must be performed to correctly round these functions.

Keywords

Floating-point arithmeticcomputer arithmeticalgebraic functionscorrect roundingdiophantine approximation.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 1: Real Numbers , January 2007 , pp. 71 - 83
Copyright
© EDP Sciences, 2007

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References

American National Standards Institute and Institute of Electrical and Electronic Engineers. IEEE standard for binary floating-point arithmetic. ANSI/IEEE Standard, Std 754-1985, New York (1985).
N.P. Brousentsov, S.P. Maslov, J. Ramil Alvarez and E.A. Zhogolev, Development of ternary computers at Moscow State University. Technical report, Dept. VMK MGU (2000). Available at http://www.computer-museum.ru/english/setun.htm.
Cody, W.J., A proposed radix and word length independent standard for floating-point arithmetic. ACM SIGNUM Newsletter 20 (1985) 37–51. CrossRef
Cody, W.J., Coonen, J.T., Gay, D.M., Hanson, K., Hough, D., Kahan, W., Karpinski, R., Palmer, J., Ris, F.N. and Stevenson, D., A proposed radix-and-word-length-independent standard for floating-point arithmetic. IEEE MICRO 4 (1984) 86–100. CrossRef
Croot, E., Li, R.-C., and Zhu, H.J., The abc conjecture and correctly rounded reciprocal square roots. Theor. Comput. Sci. 315 (2004) 405–417. CrossRef
Dunham, C.B., Feasibility of “perfect” function evaluation. SIGNUM Newsletter 25 (1990) 25–26. CrossRef
Gal, S. and Bachelis, B., An accurate elementary mathematical library for the IEEE floating point standard. ACM Trans. Math. Software 17 (1991) 26–45. CrossRef
Hurwitz, A., Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Math. Ann. 46 (1891) 279–284. CrossRef
American National Standards Institute, Institute of Electrical, and Electronic Engineers. IEEE standard for radix independent floating-point arithmetic. ANSI/IEEE Standard, Std 854-1987, New York (1987).
C. Iordache and D.W. Matula, On infinitely precise rounding for division, square root, reciprocal and square root reciprocal, in Proceedings of the 14th IEEE Symposium on Computer Arithmetic (Adelaide, Australia), edited by Koren and Kornerup, IEEE Computer Society Press, Los Alamitos, CA (1999) 233–240.
A.Ya. Khintchine, Continued fractions. Translated by Peter Wynn. P. Noordhoff Ltd., Groningen (1963).
T. Lang and J.-M. Muller, Bound on runs of zeros and ones for algebraic functions, in Proc. of the 15th IEEE Symposium on Computer Arithmetic (Arith-15), edited by Burgess and Ciminiera, IEEE Computer Society Press (2001).
V. Lefèvre, Moyens arithmétiques pour un calcul fiable. Thèse, École normale supérieure de Lyon, Lyon, France (2000).
V. Lefèvre and J.-M. Muller, Worst cases for correct rounding of the elementary functions in double precision, in Proc. of the 15th IEEE Symposium on Computer Arithmetic (Arith-15), edited by Burgess and Ciminiera, IEEE Computer Society Press (2001).
A.-M. Legendre, Essai sur la théorie des nombres. Duprat, Paris, An VI (1798).
J. Liouville, Nouvelle démonstration d'un théorème sur les irrationnelles algébriques inséré dans le compte rendu de la dernière séance. C.R. Acad. Sci. Paris, Sér. A 18 (1844) 910–911.
J. Liouville, Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques. C.R. Acad. Sci. Paris, Sér. A 18 (1844) 883–885.
Liouville, J., Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques. J. Math. Pures Appl. 16 (1851) 133–142.
Roth, K.F., Rational approximations to algebraic numbers. Mathematika 2 (1955) 1–20; corrigendum 168 (1955). CrossRef
Stehlé, D., Lefèvre, V. and Zimmermann, P., Searching worst cases of a one-variable function using lattice reduction. IEEE Trans. Comput. 54 (2005) 340–346. CrossRef