Reliable Computing

, Volume 6, Issue 2, pp 193–205 | Cite as

Accelerated Shift-and-Add Algorithms

  • Nathalie Revol
  • Jean-Claude Yakoubsohn
Article

Abstract

The problem addressed in this paper is the computation of elementary functions (exponential, logarithm, trigonometric functions, hyperbolic functions and their reciprocals) in fixed precision, typically the computer single or double precision. The method proposed here combines Shift-and-Add algorithms and classical methods for the numerical integration of ODEs: it consists in performing the Shift-and-Add iteration until a point close enough to the argument is reached, thus only one step of Euler method or Runge-Kutta method is performed. This speeds up the computation while ensuring the desired accuracy is preserved. Time estimations on various processors are presented which illustrate the advantage of this hybrid method.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atkinson, K.: An Introduction to Numerical Analysis, 2nd edition, Wiley, 1989.Google Scholar
  2. 2.
    Avizienis, A.: Signed-Digit Representations for Fast Parallel Arithmetic, IRE Transactions on Electronic Computers 10, pp. 389-400. Reprinted in: Swartzlander, E. E., Computer Arithmetic, vol. 2, IEEE Computer Society Press, 1990.Google Scholar
  3. 3.
    Brent, R. P.: Fast Multiple Precision Evaluation of Elementary Functions, Journal of the ACM 23 (1976), pp. 242-251.Google Scholar
  4. 4.
    Cody, W. J.: Implementation and Testing of Function Software, in: Problems and Methodologies in Mathematical Software Production (Lecture Notes in Computer Science 142), Springer-Verlag, Berlin etc., 1982.Google Scholar
  5. 5.
    Cody, W. J. and Waite, W.: Software Manual for the Elementary Functions, Prentice Hall, 1980.Google Scholar
  6. 6.
    Daumas, M., Mazenc, C., Merrheim, X., and Muller, J.-M.: Modular Range Reduction: A New Algorithm for Fast and Accurate Computation of the Elementary Functions, Journal of Universal Computer Science 1(3) (1995), pp. 162-175.Google Scholar
  7. 7.
    Ercegovac, M. D. and Land, T.: Division and Square-Root: Digit-Recurrence Algorithms and Implementations, Kluwer Academic Publishers, 1994.Google Scholar
  8. 8.
    Muller, J.-M.: Arithmétique des ordinateurs, Masson, 1989 (in French).Google Scholar
  9. 9.
    Muller, J.-M.: Elementary Functions, Birkhaüser, 1997.Google Scholar
  10. 10.
    Parahmi, B.: Generalized Signed-Digit Number Systems: A Unifying Framework for Redundant Number Representations, IEEE Transactions on Computers 39(1) (1990), pp. 89-98.Google Scholar
  11. 11.
    Payne, M. and Hanek, R.: Radian Reduction for Trigonometric Functions, SIGNUM Newsletter 18 (1983), pp. 19-24.Google Scholar
  12. 12.
    Revol, N. and Yakoubsohn, J.-C.: Accelerated Shift-and-Add algorithms, Research Report, extended version, ftp://ano.univ-lillel.fr/pub/1999/ano395.ps.z.Google Scholar
  13. 13.
    Takagi, N., Asada, T., and Yajima, S.: Redundant CORDIC Methods with a Constant Scale Factor, IEEE Transactions on Computers 40(9) (1991), pp. 989-995.Google Scholar
  14. 14.
    Volder, J.: The CORDIC Computing Technique, IRE Transactions on Electronic Computers (1959), pp. 14-17. Reprinted in: Swartzlander, E. E., Computer Arithmetic, vol. 1, IEEE Computer Society Press, 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Nathalie Revol
    • 1
  • Jean-Claude Yakoubsohn
    • 2
  1. 1.Lab. ANOUniv. de Lille I, UFR IEEAVilleneuve d'Ascq CedexFrance
  2. 2.Lab. d'Analyse Numérique et OptimisationUniv. Paul SabatierToulouse Cedex 4France

Personalised recommendations