Abstract
Among other objectives, rewriting programs serves as a useful technique to improve numerical accuracy. However, this optimization is not intuitive and this is why we switch to automatic transformation techniques. We are interested in the optimization of numerical programs relying on the IEEE754 floating-point arithmetic. In this article, our main contribution is to study the impact of optimizing the numerical accuracy of programs on the time required by numerical iterative methods to converge. To emphasize the usefulness of our tool, we make it optimize several examples of numerical methods such as Jacobi’s method, Newton-Raphson’s method, etc. We show that significant speedups are obtained in terms of number of iterations, time and flops.
This work was supported by the ANR Project ANR-12-INSE-0007 “CAFEIN”.
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References
Abdelmalek, N.: Roundoff error analysis for gram-schmidt method and solution of linear least squares problem. BIT 11, 345–368 (1971)
ANSI/IEEE. IEEE Standard for Binary Floating-point Arithmetic, std 754–2008 (second edn) (2008)
Benz, F., Hildebrandt, A., Hack, S.: A dynamic program analysis to find floating-point accuracy problems. In: PLDI 2012, pp. 453–462. ACM (2012)
Consel, C., Danvy, O.: Partial evaluation of pattern matching in strings. Inf. Proc. Lett. 30(2), 79–86 (1989)
Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction of approximations of fixed points. In: POPL 1977, pp. 238–252. ACM (1977)
Cousot, P., Cousot, R.: Systematic design of program transformation frameworks by abstract interpretation. In: POPL 2002, pp. 178–190. ACM (2002)
Cytron, R., Gershbein, R.: Efficient accomodation of may-alias information in SSA form. In: PLDI 1993, pp. 36–45. ACM (1993)
Damouche, N., Martel, M., Chapoutot, A.: Intra-procedural optimization of the numerical accuracy of programs. In: Núñez, M., Güdemann, M. (eds.) Formal Methods for Industrial Critical Systems. LNCS, vol. 9128, pp. 31–46. Springer, Heidelberg (2015)
Damouche, N., Martel, M., Chapoutot, A.: Optimizing the accuracy of a rocket trajectory simulation by program transformation. In: Computing Frontiers, pp. 40:1–40:2. ACM (2015)
Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Goubault, E., Putot, S.: Static analysis of finite precision computations. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 232–247. Springer, Heidelberg (2011)
Hankin, E.: Lambda Calculi A Guide For Computer Scientists. Clarendon Press, Oxford (1994)
Hernandez, V., Roman, J.E., Tomas, A., Vidal, V.: Orthogonalization routine in SLEPc Technical Report STR-1. In: Polytechnic University of Valencia. STR1 (2007)
Hunt, S., Sands, D.: Binding time analysis: a new perspective. In: PEPM 1991, pp. 154–165 (1991)
Ioualalen, A., Martel, M.: A new abstract domain for the representation of mathematically equivalent expressions. In: Miné, A., Schmidt, D. (eds.) SAS 2012. LNCS, vol. 7460, pp. 75–93. Springer, Heidelberg (2012)
Jones, N.D., Gomard, C.K., Sestoft, P.: Partial Evaluation and Automatic Program Generation. Prentice Hall International, Englewood Cliffs (1993). ISBN 0-13-020249-5
Kendall, A.: An Introduction to Numerical Analysis. John Wiley & Sons, New York (1989)
Langlois, Ph., Louvet, N.: How to ensure a faithful polynomial evaluation with the compensated horner algorithm. In: ARITH-18, pp. 141–149. IEEE Computer Society (2007)
Martel, M.: Semantics of roundoff error propagation in finite precision calculations. Higher-Order Symbolic Comput. 19(1), 7–30 (2006)
Martel, M.: Accurate evaluation of arithmetic expressions (invited talk). Electr. Notes Theor. Comput. Sci. 287, 3–16 (2012)
Mouilleron, C.: Efficient computation with structured matrices and arithmetic expressions. Ph.D. thesis, Université de Lyon-ENS de Lyon, November 2011
Muller, G., Volanschi, E.-N., Marlet, R.: Scaling up partial evaluation for optimizing the sun commercial RPC protocol. In: PEPM 1997, pp. 116–126. ACM (1997)
Muller, J.-M., Brisebarre, N., de Dinechin, F., Jeannerod, C.-P., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.: Handbook of Floating-Point Arithmetic, Birkhäuser Boston (2010)
Tate, R., Stepp, M., Tatlock, Z., Lerner, S.: Equality saturation: a new approach to optimization. In: POPL 2009, pp. 264–276. ACM (2009)
Tate, R., Stepp, M., Tatlock, Z., Lerner, S.: Equality saturation: a new approach to optimization. Log. Meth. Comput. Sci. 7(1), 1–37 (2011)
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Damouche, N., Martel, M., Chapoutot, A. (2015). Impact of Accuracy Optimization on the Convergence of Numerical Iterative Methods. In: Falaschi, M. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2015. Lecture Notes in Computer Science(), vol 9527. Springer, Cham. https://doi.org/10.1007/978-3-319-27436-2_9
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DOI: https://doi.org/10.1007/978-3-319-27436-2_9
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