BIT Numerical Mathematics

, Volume 35, Issue 3, pp 352–360 | Cite as

Highly accurate tables for elementary functions

  • Wolfram Luther
Article
  • 25 Downloads

Abstract

In this article we describe a fast method to obtain highly accurate tables for all elementary functions by using Bresenham's algorithm. For nearly equally spaced table-points {x i } we construct pairs {f(x i ),g(x i )} such thatf(x i ) is a machine number andg(x i ) is very close to an exactly representable number. By a random sampling in an interval centered onx i we can even find a triplet\(\{ \hat x_i ,f(\hat x_i ),g(\hat x_i )\} \) of nearly machine numbers. The table method together with a polynomial approximation of the function near a table value provides last bit accuracy for more than 99.8% of the argument values without using extended precision calculations [3, 4, 10, 11].

Key words

Accurate table method elementary functions Bresenham's algorithm computer arithmetic 

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REFERENCES

  1. 1.
    ANSI/IEEE:IEEE Standard for Radix-Independent Floating Point Arithmetic 854–1987, New York, 1987.Google Scholar
  2. 2.
    J. E. Bresenham,Algorithm for Computer Control of a Digital Plotter, IBM Systems J., Jan. 1965, pp. 25–30.Google Scholar
  3. 3.
    S. Gal,Computing elementary functions: A new approach for achieving high accuracy and good performance, in Accurate Scientific Computations, W. L. Miranker et al. editors, Lecture Notes in Computer Science 235, Springer, New York, 1986, pp. 1–16.Google Scholar
  4. 4.
    S. Gal and B. Bachelis,An accurate elementary mathematical library for the IEEE floating point standard, ACM Trans. on Math. Software 17, 1991, pp. 26–45.Google Scholar
  5. 5.
    G. H. Hardy, and E. M. Wright,An Introduction to the Theory of Numbers, Oxford, 1938.Google Scholar
  6. 6.
    IEEE:IEEE Standard for Binary Floating-Point Arithmetic, 754–1985, New York, 1985.Google Scholar
  7. 7.
    A. Janser and W. Luther,Der Bresenham-Algorithmus und andere graphische Grundprozeduren, in Micro-Computer Forum für Bildung und Wissenschaft Bd. 5, K. Dette, D. Haupt, and C Polze editors, Springer, Berlin, 1992, pp. 255–261.Google Scholar
  8. 8.
    A. Janser and W. Luther,Der Bresenham-Algorithmus, Bericht Nr. SI-10, 1994, Schriftenreihe des Informatikinstituts der Gerhard-Mercator-Universität - GH Duisburg, ISSN: 09442-4164.Google Scholar
  9. 9.
    W. Krämer,Die Berechnung von Standardfunktionen in Rechenanlagen, Jahrbuch Überblicke Mathematik 1992, S. D. Chatterji et al. editors, Vieweg, Wiesbaden, 1992, pp. 97–115.Google Scholar
  10. 10.
    P. T. P. Tang,Table-driven implementation of the exponential function in IEEE floating-point arithmetic, ACM Trans. on Math. Software, Vol 15, No 2 (1989), pp. 144–157.Google Scholar
  11. 11.
    P. T. P. Tang,Table-driven implementation of the Expm1 function in IEEE floating-point arithmetic, ACM Trans. on Math. Software, Vol 18, No 2 (1992), pp. 211–222.Google Scholar

Copyright information

© BIT Foundation 1995

Authors and Affiliations

  • Wolfram Luther
    • 1
  1. 1.FB 11, Informatik IIGerhard-Mercator-Universität-Gesamthochschule DuisburgLotharstraße 65Duisburg

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