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Variable precision exponential function

Editor: John R. Rice Authors:

T. E. Hull,

A. AbrhamAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 12, Issue 2

Pages 79 - 91

https://doi.org/10.1145/6497.6498

Published: 01 June 1986 Publication History

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Abstract

The exponential function presented here returns a result which differs from e^x by less than one unit in the last place, for any representable value of x which is not too close to values for which e^x would overflow or underflow. (For values of x which are not within this range, an error condition is raised.)

It is a “variable precision” function in that it returns a p-digit approximation for a p-digit argument, for any p = 0 (p-digit means p-decimal-digit). The program and analysis are valid for all p = 0, but current implementations place a restriction on p.

The program is presented in a Pascal-like programming language called Numerical Turing which has special facilities for scientific computing, including precision control, directed roundings, and built-in functions for getting and setting exponents.

References

[1]

ABRHAM, A. Variable precision elementary functions. M.Sc. thesis, Dept. of Computer Science, Univ. of Toronto, Toronto, 1985.

Google Scholar

[2]

BORWEIN, J. M., AND BORWEIN, P.B. The Arithmetic-Geometric Mean and fast computation of elementary functions. SIAM Rev. 26, 3 (July 1984), 351-366.

Google Scholar

[3]

BRENT, R.P. Fast multiple-precision evaluation of elementary functions. J. ACM 23, 2 (Apr. 1976), 242-251.

Crossref

Google Scholar

[4]

BRENT, R.P. A FORTRAN multiple-precision arithmetic package. ACM Trans. Math. Softw. 4, 1 (Mar. 1978), 57-70.

Crossref

Google Scholar

[5]

BRENT, R.P. Unrestricted algorithms for elementary and special functions. In Proceedings of the IFIP Congress 80 (Tokyo and Melbourne, Oct. 1980), Simon Lavington Ed., North-Holland, Amsterdam, 613-619.

Google Scholar

[6]

CLENSHAW, C. W., AND OLVER, F. W.J. An unrestricted algorithm for the exponential function. SIAM J. Nurner. Anal. 17, 2 (1980), 310-331.

Google Scholar

[7]

CODY, W. J., AND WAITE, W. Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs, N.J., 1980.

Crossref

Google Scholar

[8]

DAVIS, P.J. Gamma function and related functions. In Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Eds., National Bureau of Standards, Applied Mathematics Series 55, U.S. Government Printing Office, Washington D.C., June 1964, 253-293.

Google Scholar

[9]

HOLT, R. C., AND CORDY, J.R. The Turing language report. Tech. Rep. CSRI-153, Dept. of Computer Science, Univ. of Toronto (revised July 1985). (An earlier version of this report also appears as an appendix in a text by Holt, R. C. and Hume, J. N. P., An Introduction to Computer Science Using the Turing Programming Language. Reston, Reston, Va., 1984.)

Crossref

Google Scholar

[10]

HULL, T.E. Precision control, exception handling and the choice of numerical algorithms. In Proceedings of the Dundee Conference on Numerical Analysis, G. A. Watson, Ed., Springer- Verlag, New York, 1982, 169-178.

Google Scholar

[11]

HULL, T.E. The use of controlled precision, in Proceedings of the IFIP TC2 Working Conference on the Relationship between Numerical Computation and Programming Languages (Boulder, Colo., Aug. 1981), J. K. Reid, Ed., North-Holland, Amsterdam, 1982, 71-84.

Google Scholar

[12]

HULL, T. E., AND ABRHAM, A. Properly rounded variable precision square root. ACM Trans. Math. So{tw. 1I, 3 (Sept. 1985), 229-237.

Crossref

Google Scholar

[13]

HULL, T. E., ABRHAM, A., COHEN, M.S., CURLEY, A. F. X., HALL, C. B., PENNY, D. A., AND SAWCHUK, J. T.M. Numerical Turing. ACM SIGNUM Newsl. 20, 3 (July 1985), 26-34.

Crossref

Google Scholar

Cited By

View all

Changfeng YYongjuan MJin X(2011)A quick algorithms of high precision on the exponential and logarithmic functions2011 IEEE 2nd International Conference on Computing, Control and Industrial Engineering10.1109/CCIENG.2011.6007982(159-162)Online publication date: Aug-2011
https://doi.org/10.1109/CCIENG.2011.6007982
Khajah HOrtiz E(1994)Ultra-high precision computationsComputers & Mathematics with Applications10.1016/0898-1221(94)90148-127:7(41-57)Online publication date: Apr-1994
https://doi.org/10.1016/0898-1221(94)90148-1
Hull TCohen MHall C(1991)Specifications for a variable-precision arithmetic coprocessor[1991] Proceedings 10th IEEE Symposium on Computer Arithmetic10.1109/ARITH.1991.145548(127-131)Online publication date: 1991
https://doi.org/10.1109/ARITH.1991.145548

Index Terms

Variable precision exponential function
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
  2. Mathematical software
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis
  2. Formal languages and automata theory
    1. Formalisms
      1. Rewrite systems

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Reviews

Reviewer: Sven-Ake Gustafson

This paper presents a computer program which evaluates the exponential function with prescribed precision. The program is written in a Pascal-like language known as Numerical Turing, which is described in a referenced work [1]. The present paper gives a theoretical proof that the claimed accuracy and efficiency of the program for the exponential function are achieved. The analysis is detailed, but easy to follow. It is based on elementary relations and well-known results. The theoretical findings have been checked by means of extensive numerical experiments. The authors indicate that similar programs are available for the square root function as well. One may hope that they will continue their work and treat other standard functions too, giving us a library which will undoubtedly be of value for scientific calculations. The extensive list of references adds to the value of the paper.

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 12, Issue 2

June 1986

96 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/6497

Editor:
John R. Rice

Issue’s Table of Contents

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 June 1986

Published in TOMS Volume 12, Issue 2

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Changfeng YYongjuan MJin X(2011)A quick algorithms of high precision on the exponential and logarithmic functions2011 IEEE 2nd International Conference on Computing, Control and Industrial Engineering10.1109/CCIENG.2011.6007982(159-162)Online publication date: Aug-2011
https://doi.org/10.1109/CCIENG.2011.6007982
Khajah HOrtiz E(1994)Ultra-high precision computationsComputers & Mathematics with Applications10.1016/0898-1221(94)90148-127:7(41-57)Online publication date: Apr-1994
https://doi.org/10.1016/0898-1221(94)90148-1
Hull TCohen MHall C(1991)Specifications for a variable-precision arithmetic coprocessor[1991] Proceedings 10th IEEE Symposium on Computer Arithmetic10.1109/ARITH.1991.145548(127-131)Online publication date: 1991
https://doi.org/10.1109/ARITH.1991.145548

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Abstract

References

Cited By

Index Terms

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Global exponential stability of impulsive neural networks with variable delay: an LMI approach

Surgical precision JIT compilers

Surgical precision JIT compilers

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