Variable precision exponential function

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.
• 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.
• 3: Richard P. Brent, Fast Multiple-Precision Evaluation of Elementary Functions, Journal of the ACM (JACM), v.23 n.2, p.242-251, April 1976 [doi>10.1145/321941.321944]
• 4: Richard P. Brent, A Fortran Multiple-Precision Arithmetic Package, ACM Transactions on Mathematical Software (TOMS), v.4 n.1, p.57-70, March 1978 [doi>10.1145/355769.355775]
• 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.
• 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.
• 7: William James Cody, Software Manual for the Elementary Functions (Prentice-Hall series in computational mathematics), Prentice-Hall, Inc., Upper Saddle River, NJ, 1980
• 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.
• 9: R C. Holt , J N.P Hume, Introduction to computer science using the TURING programming language, Reston Publishing Co., Reston, VA, 1984
• 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.
• 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.
• 12: T. E. Hull , A. Abrham, Properly rounded variable precision square root, ACM Transactions on Mathematical Software (TOMS), v.11 n.3, p.229-237, Sept. 1985 [doi>10.1145/214408.214413]
• 13: T. E. Hull , A. Abrham , M. S. Cohen , A. F. X., Curley , C. B. Hall , D. A. Penny , J. T. M., Sawchuk, Numerical Turing, ACM SIGNUM Newsletter, v.20 n.3, p.26-34, July 1985 [doi>10.1145/1057947.1057949]

INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Elementary function approximation
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Verification**; Algorithm design and analysis; Certification and testing

General Terms:
Algorithms, Theory, Verification