BIT Numerical Mathematics

, Volume 49, Issue 2, pp 419–431 | Cite as

Computing predecessor and successor in rounding to nearest

  • Siegfried M. Rump
  • Paul Zimmermann
  • Sylvie Boldo
  • Guillaume Melquiond
Article

Abstract

We give simple and efficient methods to compute and/or estimate the predecessor and successor of a floating-point number using only floating-point operations in rounding to nearest. This may be used to simulate interval operations, in which case the quality in terms of the diameter of the result is significantly improved compared to existing approaches.

Keywords

Floating-point arithmetic Rounding to nearest Predecessor Successor Directed rounding 

Mathematics Subject Classification (2000)

68-04 68N30 

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References

  1. 1.
    American National Standards Institute, Institute of Electrical, and Electronic Engineers. IEEE Standard for Radix-Independent Floating-Point Arithmetic. ANSI/IEEE Standard, Std. 854–1987 (1987) Google Scholar
  2. 2.
    Boldo, S.: Preuves formelles en arithmétiques à virgule flottante. Ph.D. Dissertation, École Normale Supérieure de Lyon (2004) Google Scholar
  3. 3.
    Boldo, S., Muller, J.-M.: Some functions computable with a fused-mac. In: Montuschi, P., Schwarz, E. (eds.) Proceedings of the 17th Symposium on Computer Arithmetic, pp. 52–58, Cape Cod, USA (2005) Google Scholar
  4. 4.
    Cody, W.J. Jr., Coonen, J.T.: Algorithm 722: Functions to support the IEEE standard for binary floating-point arithmetic. ACM Trans. Math. Softw. 19(4), 443–451 (1993) MATHCrossRefGoogle Scholar
  5. 5.
    Daumas, M., Rideau, L., Théry, L.: A generic library of floating-point numbers and its application to exact computing. In: 14th International Conference on Theorem Proving in Higher Order Logics, Edinburgh, Scotland, pp. 169–184 (2001) Google Scholar
  6. 6.
    Institute of Electrical, and Electronic Engineers: IEEE Standard for Floating-Point Arithmetic. IEEE Standard 754-2008. Revision of ANSI-IEEE Standard 754-1985. Approved June 12, 2008: IEEE Standards Board Google Scholar
  7. 7.
    Kearfott, R.B., Dawande, M., Du, K., Hu, C.: Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Softw. 20(4), 447–459 (1994) MATHCrossRefGoogle Scholar
  8. 8.
    Rump, S.M., Ogita, T., Oishi, S.: Accurate floating-point summation part I: faithful rounding. SIAM J. Sci. Comput. (SISC) 31(1), 189–224 (2008) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  • Siegfried M. Rump
    • 1
    • 2
  • Paul Zimmermann
    • 3
  • Sylvie Boldo
    • 4
  • Guillaume Melquiond
    • 5
  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany
  2. 2.Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  3. 3.Centre de Recherche INRIA Nancy–Grand EstÉquipe-projet CACAO, Bâtiment AVillers-lès-NancyFrance
  4. 4.INRIA Saclay–Île-de-FranceParc Orsay Université–ZAC des VignesOrsay CedexFrance
  5. 5.Centre de recherche commun INRIA–Microsoft ResearchOrsay CedexFrance

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