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Midpoints and Exact Points of Some Algebraic Functions in Floating-Point Arithmetic
February 2011 (vol. 60 no. 2)
pp. 228-241
Claude-Pierre Jeannerod, INRIA Grenoble - Rhône-Alpes, Universit de Lyon, Lyon
Nicolas Louvet, Université Claude Bernard Lyon 1, Universit de Lyon, Lyon
Jean-Michel Muller, CNRS, Universit de Lyon, Lyon
Adrien Panhaleux, École Normale Supérieure de Lyon, Universit de Lyon, Lyon
When implementing a function f in floating-point arithmetic, if we wish correct rounding and good performance, it is important to know if there are input floating-point values x such that f(x) is either the middle of two consecutive floating-point numbers (assuming rounded-to-nearest arithmetic), or a floating-point number (assuming rounded toward \pm \infty or toward 0 arithmetic). In the first case, we say that f(x) is a midpoint, and in the second case, we say that f(x) is an exact point. For some usual algebraic functions and various floating-point formats, we prove whether or not there exist midpoints or exact points. When there exist midpoints or exact points, we characterize them or list all of them (if there are not too many). The results and the techniques presented in this paper can be used in particular to deal with both the binary and the decimal formats defined in the IEEE 754-2008 standard for floating-point arithmetic.

[1] American National Standards Institute and Institute of Electrical and Electronic Engineers, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754-1985, 1985.
[2] A. Baker, Transcendental Number Theory. Cambridge Univ. Press, 1975.
[3] G. Everest and T. Ward, "An Introduction to Number Theory," Graduate Texts in Mathematics, Springer-Verlag, 2005.
[4] R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics, second ed. Addison-Wesley, 1994.
[5] IEEE Computer Society, IEEE Standard for Floating-Point Arithmetic, IEEE Standard 754-2008, http://ieeexplore.ieee.org/ servletopac?punumber=4610933 , Aug. 2008.
[6] C. Iordache and D.W. Matula, "On Infinitely Precise Rounding for Division, Square Root, Reciprocal and Square Root Reciprocal," Proc. 14th IEEE Symp. Computer Arithmetic, I. Koren and P. Kornerup, eds., pp. 233-240, Apr. 1999.
[7] W. Kahan, "A Logarithm Too Clever by Half," http://http.cs.berkeley.edu/~wkahanLOG10HAF.TXT , 2004.
[8] C.Q. Lauter and V. Lefèvre, "An Efficient Rounding Boundary Test for ${\rm Pow}(x,y)$ in Double Precision," IEEE Trans. Computers, vol. 58, no. 2, pp. 197-207, Feb. 2009.
[9] P. Markstein, IA-64 and Elementary Functions: Speed and Precision Prentice-Hall, 2000.
[10] J.-M. Muller, Elementary Functions, Algorithms and Implementation, second ed. Birkhäuser, 2006.
[11] M. Parks, "Inexact Quotients and Square Roots," http://www.geocities.com/ieee754/papersparks-inexact.ps , 2010.
[12] E.M. Schwarz, R.M. Smith, and C.A. Krygowski, "The S/390 G5 Floating-Point Unit Supporting Hex and Binary Architectures," Proc. 14th IEEE Symp. Computer Arithmetic (ARITH-14), pp. 258-265, Apr. 1999.
[13] S. Wagon, "The Euclidean Algorithm Strikes Again," The Am. Math. Monthly, vol. 97, no. 2, pp. 125-129, Feb. 1990.
[14] M. Waldschmidt, "Transcendence of Periods: The State of the Art," Pure and Applied Math. Quarterly, vol. 2, no. 2, pp. 435-463, 2006.
[15] 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.

Index Terms:
Floating-point arithmetic, correct rounding, algebraic function.
Citation:
Claude-Pierre Jeannerod, Nicolas Louvet, Jean-Michel Muller, Adrien Panhaleux, "Midpoints and Exact Points of Some Algebraic Functions in Floating-Point Arithmetic," IEEE Transactions on Computers, vol. 60, no. 2, pp. 228-241, Feb. 2011, doi:10.1109/TC.2010.144
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