Toward correctly rounded transcendentals

IEEE Transactions on Computers

Volumn 47, Issue 11, 1998, Pages 1235-1243

  Lefèvre, Vincent ^b   Muller, Jean Michel ^a,b   Tisserand, Arnaud ^b  

^a IEEE   (France)

^b UNIVERSITÉ DE LYON   (France)

Author keywords

Elementary functions; Floating point arithmetic; Rounding; Table Maker's Dilemma

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; DATA PROCESSING; ERROR CORRECTION; FUNCTION EVALUATION; POLYNOMIALS;

ELEMENTARY FUNCTIONS;

DIGITAL ARITHMETIC;

EID: 0032205291     PISSN: 00189340     EISSN: None     Source Type: Journal    
DOI: 10.1109/12.736435     Document Type: Article

References (25)

1. R.C. Agarwal, J.C. Cooley, F.G. Gustavson, J.B. Shearer, G. Slishman, and B. Tuckerman, "New Scalar and Vector Elementary Functions for the IBM System/ 370," IBM J. Research and Development, vol. 30, no. 2, pp. 126-144, Mar. 1986.
2. ANSI and IEEE, "IEEE Standard for Binary Floating-Point Arithmetic," ANSI/ IEEE Standard, Std 754-1985, New York, 1985.
3. G.A. Baker, Essentials of Padé Approximants. New York: Academic Press, 1975.
4. V. Berthé, "Three Distance Theorems and Combinatorics on Words," technical report, Institut de Mathématiques de Luminy, Marseille, France, 1997.
5. R.P. Brent "Fast Multiple Precision Evaluation of Elementary Functions," J. ACM, vol. 23, pp. 242-251, 1976.
6. M. Daumas and D.W. Matula, "Rounding of Floating-Point Intervals," Proc. SCAN-93, Vienna, Austria, June 1993.
7. M. Daumas, C. Mazene, X. Merrheim, and J.M. Muller, "Modular Range Reduction: A New Algorithm for Fast and Accurate Computation of the Elementary Functions," J. Universal Computer Science, vol. 1, no. 3, pp. 162-175, Mar. 1995.
8. B. DeLugish, "A Class of Algorithms for Automatic Evaluation of Functions and Computations in a Digital Computer," PhD thesis, Dept. of Computer Science, Univ. of Illinois, Urbana-Champaign, 1970.
9. C.B. Dunham, "Feasibility of 'Perfect' Function Evaluation," SIGNUM Newsletter, vol. 25, no. 4, pp. 25-26, Oct. 1990.
10. S. Gal and B. Bachelis, "An Accurate Elementary Mathematical Library for the IEEE Floating Point Standard," ACM Trans. Math. Software, vol. 17, no. 1, pp. 26-45, Mar. 1991.
11. D. Goldberg, "What Every Computer Scientist Should Know About Floating-Point Arithmetic," ACM Computing Surveys, vol. 23, no. 1, pp. 5-47, Mar. 1991.
12. W. Kahan, "Minimizing q*m-n," Text accessible electronically at http://http.cs.berkeley.edu/∼wkahan/. At the beginning of the file "nearpi.c," 1983.
13. W. Kahan, "Lecture Notes on the Status of IEEE-754," Postscript file accessible electronically through the Internet at the address http://http.cs.berkeley.edu/∼wkahan/ieee754status/ieee754.ps, 1996.
14. D. Knuth, The Art of Computer Programming, vol. 2. Reading, Mass.: Addison Wesley, 1973.
15. D. Knuth, The Art of Computer Programming, vol. 3. Reading, Mass.: Addison Wesley, 1973.
16. 2," Research Report RR97-18, LIP, École Normale Supérieure de Lyon, 1997. Available at http://www.ens-lyon.fr/LIP/research_reports.us.html.
17. J.M. Muller and A. Tisserand, "Towards Exact Rounding of the Elementary Functions," Scientific Computing and Validated Numerics (Proc. SCAN '95), G. Alefeld, A. Frommer, and B. Lang, eds., Wuppertal, Germany, 1996.
18. J.M. Muller, Elementary Functions, Algorithms and Implementation. Boston: Birkhauser, 1997.
19. Y.V. Nesterenko and M. Waldschmidt, "On the Approximation of the Values of Exponential Function and Logarithm by Algebraic Numbers (in Russian)," Mat. Zapiski, vol. 2, pp. 23-42, 1996.
20. M. Payne and R. Hanek, "Radian Reduction for Trigonometric Functions," SIGNUM Newsletter, vol. 18, pp. 19-24, 1983.
21. M. Schulte and E.E. Swartzlander, "Exact Rounding of Certain Elementary Functions," Proc. 11th IEEE Symp. Computer Arithmetic, E.E. Swartzlander, M.J. Irwin, and G. Jullien, eds., pp. 138-145, Windsor, Canada, June 1993.
22. M.J. Schulte and E.E. Swartzlander, "Hardware Designs for Exactly Rounded Elementary Functions," IEEE Trans. Computers, vol. 43, no. 8, pp. 964-973, Aug. 1994.
23. R.A. Smith, "A Continued-Fraction Analysis of Trigonometric Argument Reduction," IEEE Trans. Computers, vol. 44, no. 11, pp. 1,348-1,351, Nov. 1995.
24. A. Ziv, "Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded Last Bit," ACM Trans. Math. Software, vol. 17, no. 3, pp. 410-423, Sept. 1991.
25. D. Zuras, "More on Squaring and Multiplying Large Integers," IEEE Trans. Computers, vol. 43, no. 8, pp. 899-908, Aug. 1994.