Publication 2008 Issue No. 2 - February Abstract - Correctly Rounded Multiplication by Arbitrary Precision Constants
Correctly Rounded Multiplication by Arbitrary Precision Constants
February 2008 (vol. 57 no. 2)
pp. 165-174
 ASCII Text x Nicolas Brisebarre, Jean-Michel Muller, "Correctly Rounded Multiplication by Arbitrary Precision Constants," IEEE Transactions on Computers, vol. 57, no. 2, pp. 165-174, February, 2008.
 BibTex x @article{ 10.1109/TC.2007.70813,author = {Nicolas Brisebarre and Jean-Michel Muller},title = {Correctly Rounded Multiplication by Arbitrary Precision Constants},journal ={IEEE Transactions on Computers},volume = {57},number = {2},issn = {0018-9340},year = {2008},pages = {165-174},doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2007.70813},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - Correctly Rounded Multiplication by Arbitrary Precision ConstantsIS - 2SN - 0018-9340SP165EP174EPD - 165-174A1 - Nicolas Brisebarre, A1 - Jean-Michel Muller, PY - 2008KW - computer arithmeticKW - floating-point arithmeticVL - 57JA - IEEE Transactions on ComputersER -
We introduce an algorithm for multiplying a floating-point number $x$ by a constant $C$ that is not exactly representable in floating-point arithmetic. Our algorithm uses a multiplication and a fused multiply and add instruction. Such instructions are available in some modern processors such as the IBM Power PC and the Intel/HP Itanium. We give three methods for checking whether, for a given value of $C$ and a given floating-point format, our algorithm returns a correctly rounded result for any $x$. When it does not, some of our methods return all the values $x$ for which the algorithm fails. We generalize our study to the case where a wider internal format is used for the intermediate calculations, which gives a fourth method. Our programs and some additional information (such as the case where an arbitrary nonbinary even radix is used), as well as examples of runs of our programs can be downloaded from \\url{http://perso.ens-lyon.fr/jean-michel.muller/MultConstant.html}

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Applied Math. Series 55, US Nat'l Bureau of Standards, 1964.
[2] N. Brisebarre, J.-M. Muller, and S. Raina, “Accelerating Correctly Rounded Floating-Point Division When the Divisor Is Known in Advance,” IEEE Trans. Computers, vol. 53, no. 8, pp. 1069-1072, Aug. 2004.
[3] M.A. Cornea-Hasegan, R.A. Golliver, and P. Markstein, “Correctness Proofs Outline for Newton-Raphson Based Floating-Point Divide and Square Root Algorithms,” Proc. 14th IEEE Symp. Computer Arithmetic, I. Koren and P. Kornerup, eds., pp. 96-105, Apr. 1999.
[4] B.P. Flannery, W.H. Press, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C, second ed. Cambridge Univ. Press, 1992.
[5] D. Goldberg, “What Every Computer Scientist Should Know about Floating-Point Arithmetic,” ACM Computing Surveys, vol. 23, no. 1, pp. 5-47, Mar. 1991.
[6] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers. Oxford Univ. Press, 1979.
[7] J. Harrison, “A Machine-Checked Theory of Floating-Point Arithmetic,” Proc. 12th Int'l Conf. Theorem Proving in Higher Order Logics, Y. Bertot, G. Dowek, A. Hirschowitz, C. Paulin, and L.Théry, eds., pp. 113-130, Sept. 1999.
[8] W. Kahan, “A Logarithm Too Clever by Half,” http://http.cs.berkeley.edu/ wkahanLOG10HAF.TXT , 2004.
[9] A.Y. Khinchin, Continued Fractions. Dover, 1997.
[10] V. Lefèvre, “An Algorithm that Computes a Lower Bound on the Distance between a Segment and $Z^{2}$ ,” Developments in Reliable Computing, pp. 203-212, Kluwer Academic, 1999.
[11] R.-C. Li, S. Boldo, and M. Daumas, “Theorems on Efficient Argument Reductions,” Proc. 16th IEEE Symp. Computer Arithmetic, 2003.
[12] P. Markstein, IA-64 and Elementary Functions: Speed and Precision, Hewlett-Packard Professional Books. Prentice Hall, 2000.
[13] P.W. Markstein, “Computation of Elementary Functions on the IBM Risc System/6000 Processor,” IBM J. Research and Development, vol. 34, no. 1, pp. 111-119, Jan. 1990.
[14] J.-M. Muller, “On the Definition of ${\rm ulp}(x)$ ,” Technical Report 2005-09, LIP Laboratory, ENS Lyon, ftp://ftp.ens-lyon.fr/pub/LIP/Rapports/RR/ RR2005RR2005-09.pdf, 2005.
[15] M.A. Overton, Numerical Computing with IEEE Floating-Point Arithmetic. SIAM, 2001.
[16] O. Perron, Die Lehre von den Kettenbrüchen, 3. verb. und erweiterte Aufl., pp. 1954-1957. Teubner,
[17] H.M. Stark, An Introduction to Number Theory. MIT Press, 1981.

Index Terms:
computer arithmetic, floating-point arithmetic
Citation:
Nicolas Brisebarre, Jean-Michel Muller, "Correctly Rounded Multiplication by Arbitrary Precision Constants," IEEE Transactions on Computers, vol. 57, no. 2, pp. 165-174, Feb. 2008, doi:10.1109/TC.2007.70813