Rigorous Polynomial Approximation Using Taylor Models in Coq

  • Nicolas Brisebarre
  • Mioara Joldeş
  • Érik Martin-Dorel
  • Micaela Mayero
  • Jean-Michel Muller
  • Ioana Paşca
  • Laurence Rideau
  • Laurent Théry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7226)


One of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameterized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with interval coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models.


certified error bounds Taylor models Coq proof assistant rigorous polynomial approximation 


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  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C (1964)Google Scholar
  2. 2.
    Armand, M., Grégoire, B., Spiwack, A., Théry, L.: Extending Coq with Imperative Features and Its Application to SAT Verification. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 83–98. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., Salvy, B.: The Dynamic Dictionary of Mathematical Functions (DDMF). In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 35–41. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  5. 5.
    Berz, M., Makino, K.: Rigorous global search using Taylor models. In: SNC 2009: Proceedings of the 2009 Conference on Symbolic Numeric Computation, pp. 11–20. ACM, New York (2009)CrossRefGoogle Scholar
  6. 6.
    Berz, M., Makino, K., Kim, Y.K.: Long-term stability of the tevatron by verified global optimization. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 558(1), 1–10 (2006); Proceedings of the 8th International Computational Accelerator Physics Conference - ICAP 2004Google Scholar
  7. 7.
    Boespflug, M., Dénès, M., Grégoire, B.: Full Reduction at Full Throttle. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 362–377. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Boldo, S., Melquiond, G.: Flocq: A Unified Library for Proving Floating-point Algorithms in Coq. In: Proceedings of the 20th IEEE Symposium on Computer Arithmetic, Tübingen, Germany, pp. 243–252 (2011)Google Scholar
  9. 9.
    Brisebarre, N., Chevillard, S.: Efficient polynomial L  ∞ -approximations. In: Kornerup, P., Muller, J.M. (eds.) 18th IEEE Symposium on Computer Arithmetic, pp. 169–176. IEEE Computer Society, Los Alamitos (2007)CrossRefGoogle Scholar
  10. 10.
    Brisebarre, N., Muller, J.M., Tisserand, A.: Computing Machine-efficient Polynomial Approximations. ACM Trans. Math. Software 32(2), 236–256 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cháves, F.: Utilisation et certification de l’arithmétique d’intervalles dans un assistant de preuves. Thèse, École normale supérieure de Lyon - ENS LYON (September 2007),
  12. 12.
    Chevillard, S.: Évaluation efficace de fonctions numériques. Outils et exemples. Ph.D. thesis, École Normale Supérieure de Lyon, Lyon, France (2009),
  13. 13.
    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.) ICMS 2010. LNCS, vol. 6327, pp. 28–31. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Chevillard, S., Harrison, J., Joldeş, M., Lauter, C.: Efficient and accurate computation of upper bounds of approximation errors. Theoretical Computer Science 16(412), 1523–1543 (2011)CrossRefGoogle Scholar
  15. 15.
    Collins, P., Niqui, M., Revol, N.: A Taylor Function Calculus for Hybrid System Analysis: Validation in Coq. In: NSV-3: Third International Workshop on Numerical Software Verification (2010)Google Scholar
  16. 16.
    de Dinechin, F., Lauter, C., Melquiond, G.: Assisted verification of elementary functions using Gappa. In: Proceedings of the 2006 ACM Symposium on Applied Computing, Dijon, France, pp. 1318–1322 (2006),
  17. 17.
    von zur Gathen, J., Gerhard, J.: Modern computer algebra, 2nd edn. Cambridge University Press, New York (2003)zbMATHGoogle Scholar
  18. 18.
    Geuvers, H., Niqui, M.: Constructive Reals in Coq: Axioms and Categoricity. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 79–95. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Gonthier, G., Mahboubi, A., Tassi, E.: A Small Scale Reflection Extension for the Coq system. Rapport de recherche RR-6455, INRIA (2008)Google Scholar
  20. 20.
    Grégoire, B., Théry, L.: A Purely Functional Library for Modular Arithmetic and Its Application to Certifying Large Prime Numbers. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 423–437. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Griewank, A.: Evaluating Derivatives - Principles and Techniques of Algorithmic Differentiation. SIAM (2000)Google Scholar
  22. 22.
    IEEE Computer Society: IEEE Standard for Floating-Point Arithmetic. IEEE Std 754TM-2008 (August 2008)Google Scholar
  23. 23.
    Joldeş, M.: Rigourous Polynomial Approximations and Applications. Ph.D. dissertation, École Normale Supérieure de Lyon, Lyon, France (2011),
  24. 24.
    Krebbers, R., Spitters, B.: Computer Certified Efficient Exact Reals in Coq. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Calculemus/MKM 2011. LNCS, vol. 6824, pp. 90–106. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  25. 25.
    Lefèvre, V., Muller, J.M.: Worst cases for correct rounding of the elementary functions in double precision. In: Burgess, N., Ciminiera, L. (eds.) Proceedings of the 15th IEEE Symposium on Computer Arithmetic (ARITH-16), Vail, CO (June 2001)Google Scholar
  26. 26.
    Lizia, P.D.: Robust Space Trajectory and Space System Design using Differential Algebra. Ph.D. thesis, Politecnico di Milano, Milano, Italy (2008)Google Scholar
  27. 27.
    Makino, K.: Rigorous Analysis of Nonlinear Motion in Particle Accelerators. Ph.D. thesis, Michigan State University, East Lansing, Michigan, USA (1998)Google Scholar
  28. 28.
    Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. International Journal of Pure and Applied Mathematics 4(4), 379–456 (2003), MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mayero, M.: Formalisation et automatisation de preuves en analyses réelle et numérique. Ph.D. thesis, Université Paris VI (2001)Google Scholar
  30. 30.
    Melquiond, G.: Proving Bounds on Real-Valued Functions with Computations. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 2–17. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  31. 31.
    Moore, R.E.: Methods and Applications of Interval Analysis. Society for Industrial and Applied Mathematics (1979)Google Scholar
  32. 32.
    Muller, J.M.: Projet ANR TaMaDi – Dilemme du fabricant de tables – Table Maker’s Dilemma (ref. ANR 2010 BLAN 0203 01),
  33. 33.
    Muller, J.M.: Elementary Functions, Algorithms and Implementation, 2nd edn. Birkhäuser, Boston (2006)zbMATHGoogle Scholar
  34. 34.
    Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45, 236–262 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Neumaier, A.: Taylor forms – use and limits. Reliable Computing 9(1), 43–79 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    O’Connor, R.: Certified Exact Transcendental Real Number Computation in Coq. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 246–261. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  37. 37.
    The Arénaire Project: CRlibm, Correctly Rounded mathematical library (July 2006),
  38. 38.
    Remez, E.: Sur un procédé convergent d’approximations successives pour déterminer les polynômes d’approximation. C.R. Académie des Sciences 198, 2063–2065 (1934) (in French)Google Scholar
  39. 39.
    Salvy, B., Zimmermann, P.: Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Software 20(2), 163–177 (1994)zbMATHCrossRefGoogle Scholar
  40. 40.
    Stanley, R.P.: Differentiably finite power series. European Journal of Combinatorics 1(2), 175–188 (1980)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Ziv, A.: Fast evaluation of elementary mathematical functions with correctly rounded last bit. ACM Trans. Math. Software 17(3), 410–423 (1991)zbMATHCrossRefGoogle Scholar
  42. 42.
    Zumkeller, R.: Formal Global Optimisation with Taylor Models. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 408–422. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nicolas Brisebarre
    • 1
  • Mioara Joldeş
    • 4
  • Érik Martin-Dorel
    • 1
  • Micaela Mayero
    • 1
    • 2
  • Jean-Michel Muller
    • 1
  • Ioana Paşca
    • 1
  • Laurence Rideau
    • 3
  • Laurent Théry
    • 3
  1. 1.LIPCNRS UMR 5668, ENS de Lyon, INRIA Grenoble - Rhône-Alpes, UCBL, ArénaireLyonFrance
  2. 2.LIPNUMR 7030, Université Paris 13, LCRVilletaneuseFrance
  3. 3.Marelle, INRIA Sophia Antipolis - MéditerranéeSophia AntipolisFrance
  4. 4.CAPA, Dpt. of MathematicsUppsala Univ.UppsalaSweden

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