Fast evaluation of elementary mathematical functions with correctly rounded last bit

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Fast evaluation of elementary mathematical functions with correctly rounded last bit

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ACM Transactions on Mathematical Software (TOMS) archive
Volume 17 , Issue 3 (September 1991) table of contents

Pages: 410 - 423

Year of Publication: 1991

ISSN:0098-3500

Author

Abraham Ziv

IBM Israel, Haifa, Israel

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 7, Downloads (12 Months): 58, Citation Count: 11

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REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	Cheryl M. Black , Robert Burton and Thomas H. Miller, The Need for an Industry Standard of Accuracy for Elementary-Function Programs, ACM Transactions on Mathematical Software (TOMS), v.10 n.4, p.361-366, Dec. 1984 [doi>10.1145/2701.356101]
	2	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]
	3	Richard P. Brent, Algorithm 524: MP, A Fortran Multiple-Precision Arithmetic Package [A1], ACM Transactions on Mathematical Software (TOMS), v.4 n.1, p.71-81, March 1978 [doi>10.1145/355769.355776]
	4	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]
	5	CLENSHAW, C. W., AND OLVER, F. W. J. An unrestricted algorithm for the exponential function. SIAM J. Numer. Anal., 17, 2 (Apr. 1980).
	6	Shmuel Gal, Computing Elementary Functions: A New Approach for Achieving High Accuracy and Good Performance, Proceedings of the Symposium on Accurate Scientific Computations, p.1-16, March 12-14, 1985
	7	Shmuel Gal, An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (TOMS), v.17 n.1, p.26-45, March 1991 [doi>10.1145/103147.103151]
	8	HmL, I. D. Procedures for the basic arithmetical operators in multiple-length working. Comput. J. 11 (1968), 232-235.
	9	IBM CORPORATION. Elementary math library: Programming RPQ (5799-BTB), Program Reference and Operations Manual, 1984.
	10	IEEE Standard for Binary Floating Potnt Arithmetic. ANSI/ IEEE Std 754-1985, 1985.
	11	IEEE Standard For Radix-lndependent Floating Point Arithmetic. ANSI/IEEE Std 854-1987, 1987.
	12	Ulrich W. Kulisch , Willard L. Miranker, Computer Arithmetic in Theory and Practice, Academic Press, Inc., Orlando, FL, 1981
	13	OLVER, F. W.J. A new approach to error arithmetic. SIAM J. Numer. Anal., 15, 2 (1978), 368-393.
	14	OLVER, F. W.J. Unrestricted algorithms for generating elementary functions. Computing, Supplement 2. Springer-Verlag, New York, 1980, pp. 131-140.
	15	STERBENZ, P.H. Floating Poi~t Computation. Prentice-Hall, Englewood Cliffs, N. J., 1974.
	16	James H. Wilkinson, Rounding Errors in Algebraic Processes, Dover Publications, Incorporated, 1994
	17	W. T. Wyatt, Jr. , D. W. Lozier , D. J. Orser, A Portable Extended Precision Arithmetic Package and Library with Fortran Precompiler, ACM Transactions on Mathematical Software (TOMS), v.2 n.3, p.209-231, Sept. 1976 [doi>10.1145/355694.355695]
	18	Zw, A. Relative distance--An error measure in round-off error analysis. Math. Comput.. 39, 160 (1982), 563-569.

CITED BY 11

	Florent de Dinechin , Christoph Quirin Lauter , Guillaume Melquiond, Assisted verification of elementary functions using Gappa, Proceedings of the 2006 ACM symposium on Applied computing, April 23-27, 2006, Dijon, France
	M. J. Schulte , E. E. Swartzlander, Jr., Hardware Designs for Exactly Rounded Elementary Functions, IEEE Transactions on Computers, v.43 n.8, p.964-973, August 1994
	Walter Krämer, A Priori Worst Case Error Bounds for Floating-Point Computations, IEEE Transactions on Computers, v.47 n.7, p.750-756, July 1998
	David W. Matula , Lee D. McFearin, A formal model and efficient traversal algorithm for generating testbenches for verification of IEEE standard floating point division, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany
	Damien Stehlé , Vincent Lefèvre , Paul Zimmermann, Searching Worst Cases of a One-Variable Function Using Lattice Reduction, IEEE Transactions on Computers, v.54 n.3, p.340-346, March 2005
	Álvaro Vázquez , Elisardo Antelo, Implementation of the Exponential Function in a Floating-Point Unit, Journal of VLSI Signal Processing Systems, v.33 n.1, p.125-21, January-February 2003
	Alexey Ershov , Andrey Naraikin , Sergey Maidanov, Quad and correctly rounded double precision math functions: portable and optimized for Intel architectures, Proceedings of the 2006 ACM symposium on Applied computing, April 23-27, 2006, Dijon, France
	Vincent Lefèvre , Jean-Michel Muller , Arnaud Tisserand, Toward Correctly Rounded Transcendentals, IEEE Transactions on Computers, v.47 n.11, p.1235-1243, November 1998
	Yves Nievergelt, Analysis and applications of Priest's distillation, ACM Transactions on Mathematical Software (TOMS), v.30 n.4, p.402-433, December 2004
	Laurent Fousse , Guillaume Hanrot , Vincent Lefèvre , Patrick Pélissier , Paul Zimmermann, MPFR: A multiple-precision binary floating-point library with correct rounding, ACM Transactions on Mathematical Software (TOMS), v.33 n.2, p.13-es, June 2007
	V. V. Kuliamin, Standardization and testing of implementations of mathematical functions in floating point numbers, Programming and Computing Software, v.33 n.3, p.154-173, May 2007

INDEX TERMS

Primary Classification:
G. Mathematics of Computing
G.1 NUMERICAL ANALYSIS
G.1.0 General
Subjects: Numerical algorithms

Additional Classification:
G. Mathematics of Computing
G.1 NUMERICAL ANALYSIS
G.1.2 Approximation
Subjects: Elementary function approximation
G.4 MATHEMATICAL SOFTWARE
Subjects: Efficiency

General Terms:
Algorithms, Standardization, Theory

Keywords:
compatibility, correct rounding, mathematical library

REVIEW

"Heinrich W. Guggenheimer : Reviewer"

The variety of rounding rules used for elementary mathematical function routines makes it difficult to use different routines and obtain the same values up to the last bit. The technique proposed here to obtain correct function values (given a more...

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Abraham Ziv: colleagues

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