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Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

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Abstract

The verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq’s logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples.

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Notes

  1. http://lipforge.ens-lyon.fr/www/crlibm/.

  2. Binary64 is the name of the IEEE 754–2008 floating-point format that was formerly known as the “double precision” format.

  3. https://code.google.com/p/flyspeck/.

  4. The author of NLCertify is considering relying on CoqInterval to check the quadratic forms that bound elementary functions. This would be a step further in getting completely verified results with NLCertify.

  5. An interval function \(\mathbf {f}\) is isotone if, for any pair of intervals \((\mathbf {x},\mathbf {x'})\), we have \(\mathbf {x}\subseteq \mathbf {x'}\implies \mathbf {f}(\mathbf {x})\subseteq \mathbf {f}(\mathbf {x'})\) (see also [11, Definition 4.8.10]).

  6. http://www.jhauser.us/arithmetic/SoftFloat.html.

  7. http://www.mpfr.org/.

  8. The unit in the last place of a real number x is the gap between the two floating-point numbers enclosing x in a given format (see also [22, p. 32]).

  9. i.e. in the univariate case, P is considered as \(P(x)=\sum \nolimits _{i=0}^n P_i\cdot (x-x_0)^i\) for a given expansion point \(x_0\).

  10. Namely, we use this simple algorithm when computing a Taylor model for identity or constant functions, as the estimation of the Taylor–Lagrange remainder is already sharp in this case.

  11. https://code.google.com/p/hol-light/.

  12. https://code.google.com/p/flyspeck/source/browse/#svn/trunk/formal_ineqs.

  13. http://forge.ocamlcore.org/projects/nl-certify.

  14. https://metitarski.googlecode.com/hg-history/V2_4/tptp/Problems/atan-problem-1-sqrt.tptp.

References

  1. Akbarpour, B., Paulson, L.C.: MetiTarski: an automatic theorem prover for real-valued special functions. J. Autom. Reason. 44(3), 175–205 (2010). doi:10.1007/s10817-009-9149-2

    Article  MathSciNet  MATH  Google Scholar 

  2. Allamigeon, X., Gaubert, S., Magron, V., Werner, B.: Certification of bounds of non-linear functions: the templates method. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) Intelligent Computer Mathematics—MKM, Calculemus, DML, and Systems and Projects. Lecture Notes in Computer Science, vol. 7961, pp. 51–65 (2013). doi:10.1007/978-3-642-39320-4_4

  3. Boldo, S., Melquiond, G.: Flocq: a unified library for proving floating-point algorithms in Coq. In: Antelo, E., Hough, D., Ienne, P. (eds.) Proceedings of the 20th IEEE Symposium on Computer Arithmetic, pp. 243–252. Tübingen, Germany (2011). doi:10.1109/ARITH.2011.40

  4. Brisebarre, N., Joldeş, M., Martin-Dorel, É., Mayero, M., Muller, J.M., Paşca, I., Rideau, L., Théry, L.: Rigorous polynomial approximation using Taylor models in Coq. In: Goodloe, A., Person, S. (eds.) Proceedings of 4th International Symposium on NASA Formal Methods. Lecture Notes in Computer Science, vol. 7226, pp. 85–99. Springer, Norfolk (2012). doi:10.1007/978-3-642-28891-3_9

  5. Ceberio, M., Granvilliers, L.: Horner’s rule for interval evaluation revisited. Computing 69(1), 51–81 (2002). doi:10.1007/s00607-002-1448-y

    Article  MathSciNet  MATH  Google Scholar 

  6. Chevillard, S., Harrison, J., Joldeş, M., Lauter, C.: Efficient and accurate computation of upper bounds of approximation errors. J. Theor. Comput. Sci. 412(16), 1523–1543 (2011). doi:10.1016/j.tcs.2010.11.052

    Article  MathSciNet  MATH  Google Scholar 

  7. Chevillard, S., Joldeş, M., Lauter, C.: Sollya: an environment for the development of numerical codes. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) Proceedings of the 3rd International Congress on Mathematical Software, Lecture Notes in Computer Science, vol. 6327, pp. 28–31. Heidelberg (2010)

  8. Daumas, M., Lester, D., Muñoz, C.: Verified real number calculations: a library for interval arithmetic. IEEE Trans. Comput. 58(2), 226–237 (2009)

    Article  MathSciNet  Google Scholar 

  9. Daumas, M., Melquiond, G., Muñoz, C.: Guaranteed proofs using interval arithmetic. In: Montuschi, P., Schwarz, E. (eds.) Proceedings of the 17th IEEE Symposium on Computer Arithmetic, pp. 188–195. Cape Cod, MA (2005). doi:10.1109/ARITH.2005.25

  10. Denman, W., Muñoz, C.: Automated real proving in PVS via MetiTarski. In: Jones, C.B., Pihlajasaari, P., Sun, J. (eds.) FM, Lecture Notes in Computer Science, vol. 8442, pp. 194–199. Springer (2014). doi:10.1007/978-3-319-06410-9_14

  11. Hansen, E., Walster, G.: Global Optimization Using Interval Analysis: Revised and Expanded. Monographs and Textbooks in Pure and Applied Mathematics. CRC Press, Boca Raton (2003)

    Google Scholar 

  12. Harrison, J.: Verifying the accuracy of polynomial approximations in HOL. In: Gunter, E.L., Felty, A.P. (eds.) Proceedings of the 10th International Conference on Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science, vol. 1275, pp. 137–152. Murray Hill, NJ, USA (1997). doi:10.1007/BFb0028391

  13. Harrison, J.: Verifying nonlinear real formulas via sums of squares. In: Schneider, K., Brandt, J. (eds.) Proceedings of the 20th International Conference on Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science, vol. 4732, pp. 102–118. Kaiserslautern, Germany (2007)

  14. Joldeş, M.: Rigorous Polynomial Approximations and Applications. Ph.D. thesis, ENS de Lyon, France (2011). http://tel.archives-ouvertes.fr/tel-00657843/en/

  15. Makino, K.: Rigorous Analysis of Nonlinear Motion in Particle Accelerators. Ph.D. thesis, Michigan State University, East Lansing, Michigan, USA (1998)

  16. Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. Int. J. Pure Appl. Math. 4(4), 379–456 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Martin-Dorel, É., Mayero, M., Paşca, I., Rideau, L., Théry, L.: Certified, efficient and sharp univariate Taylor models in Coq. In: IEEE, SYNASC 2013, pp. 193–200. Timişoara, Romania (2013). doi:10.1109/SYNASC.2013.33

  18. Melquiond, G.: Floating-point arithmetic in the Coq system. In: Proceedings of the 8th Conference on Real Numbers and Computers, pp. 93–102. Santiago de Compostela, Spain (2008)

  19. Melquiond, G.: Proving bounds on real-valued functions with computations. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) Proceedings of the 4th International Joint Conference on Automated Reasoning, Lecture Notes in Artificial Intelligence, vol. 5195, pp. 2–17. Sydney, Australia (2008). doi:10.1007/978-3-540-71070-7_2

  20. Melquiond, G.: Floating-point arithmetic in the Coq system. Inf. Comput. 216, 14–23 (2012). doi:10.1016/j.ic.2011.09.005

    Article  MathSciNet  MATH  Google Scholar 

  21. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  22. 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). doi:10.1007/978-0-8176-4705-6

    Book  MATH  Google Scholar 

  23. Muñoz, C., Narkawicz, A.: Formalization of a representation of Bernstein polynomials and applications to global optimization. J. Autom. Reason. 51(2), 151–196 (2013). doi:10.1007/s10817-012-9256-3

    Article  MATH  Google Scholar 

  24. Narkawicz, A., Muñoz, C.: A formally verified generic branching algorithm for global optimization. In: Cohen, E., Rybalchenko, A. (eds.) Proceedings of the 5th International Conference on Verified Software: Theories, Tools, Experiments. Lecture Notes in Computer Science, vol. 8164, pp. 326–343. Menlo Park, CA, USA (2013). doi:10.1007/978-3-642-54108-7_17

  25. Solovyev, A., Hales, T.C.: Formal verification of nonlinear inequalities with Taylor interval approximations. In: Brat, G., Rungta, N., Venet, A. (eds.) Proceedings of the 5th International Symposium on NASA Formal Methods. Lecture Notes in Computer Science, vol. 7871, pp. 383–397. Moffett Field, CA, USA (2013). doi:10.1007/978-3-642-38088-4_26

  26. Tang, P.T.P.: Table-driven implementation of the exponential function in IEEE floating-point arithmetic. ACM Trans. Math. Softw. 15(2), 144–157 (1989). doi:10.1145/63522.214389

    Article  MATH  Google Scholar 

  27. Ziv, A.: Fast evaluation of elementary mathematical functions with correctly rounded last bit. ACM Trans. Math. Softw. 17(3), 410–423 (1991). doi:10.1145/114697.116813

    Article  MATH  Google Scholar 

  28. Zumkeller, R.: Global Optimization in Type Theory. Ph.D. thesis, École polytechnique, France (2008). http://alacave.net/~roland/FormalGlobalOpt.pdf

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Acknowledgments

We would like to thank the people from the ANR TaMaDi project for initiating and greatly contributing to the CoqApprox project.

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Authors and Affiliations

  1. Université Toulouse 3, Institut de Recherche en Informatique de Toulouse, UMR 5505 CNRS, IRIT, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Érik Martin-Dorel

  2. Inria Saclay–Île-de-France, LRI, UMR 8623 CNRS Inria Univ. Paris-Sud, Univ. Paris Saclay, PCRI, bât 650, Université Paris-Sud, 91405, Orsay Cedex, France

    Guillaume Melquiond

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  1. Érik Martin-Dorel
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Correspondence to Érik Martin-Dorel.

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This work was funded by the Verasco ANR project (ref. ANR-11-INSE-003). It was partly done while the first author was with Inria Saclay–Île-de-France, in the LRI research laboratory.

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Martin-Dorel, É., Melquiond, G. Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq. J Autom Reasoning 57, 187–217 (2016). https://doi.org/10.1007/s10817-015-9350-4

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