Abstract
When a floating-point computation is done, it is most of the time incorrect. The rounding error can be bounded by folklore formulas, such as \(\varepsilon |x|\) or \(\varepsilon |{\circ }(x)|\). This gets more complicated when underflow is taken into account as an absolute term must be considered. Now, let us compute this error bound in practice. A common method is to use a directed rounding in order to be sure to get an over-approximation of this error bound. This article describes an algorithm that computes a correct bound using only rounding to nearest, therefore without requiring a costly change of the rounding mode. This is formally proved using the Coq formal proof assistant to increase the trust in this algorithm.
S. Boldo—This work was supported by the FastRelax (ANR-14-CE25-0018-01) project of the French National Agency for Research (ANR).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
References
Boldo, S.: Deductive formal verification: how to make your floating-point programs behave. Thèse d’habilitation, Université Paris-Sud, October 2014
Boldo, S., Melquiond, G.: Flocq: a unified library for proving floating-point algorithms in Coq. In: Antelo, E., Hough, D., Ienne, P. (eds.) 20th IEEE Symposium on Computer Arithmetic, Tübingen, Germany, pp. 243–252 (2011)
Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23(1), 5–48 (1991)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM (2002)
Microprocessor Standards Committee: IEEE Standard for Floating-Point Arithmetic. IEEE Std. 754-2008, pp. 1–58, August 2008
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)
Rump, S.M.: Fast and parallel interval arithmetic. BIT Numer. Math. 39(3), 534–554 (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Boldo, S. (2017). Computing a Correct and Tight Rounding Error Bound Using Rounding-to-Nearest. In: Bogomolov, S., Martel, M., Prabhakar, P. (eds) Numerical Software Verification. NSV 2016. Lecture Notes in Computer Science(), vol 10152. Springer, Cham. https://doi.org/10.1007/978-3-319-54292-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-54292-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54291-1
Online ISBN: 978-3-319-54292-8
eBook Packages: Computer ScienceComputer Science (R0)