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Dandelion: Certified Approximations of Elementary Functions

Authors Heiko Becker, Mohit Tekriwal, Eva Darulova, Anastasia Volkova, Jean-Baptiste Jeannin

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Author Details

Heiko Becker
  • MPI-SWS, Saarland Informatics Campus (SIC), Germany
Mohit Tekriwal
  • University of Michigan, Ann Arbor, MI, USA
Eva Darulova
  • Uppsala University, Sweden
Anastasia Volkova
  • Nantes Université, France
Jean-Baptiste Jeannin
  • University of Michigan, Ann Arbor, MI, USA

Funding

This work was supported in part by the University of Michigan.

Acknowledgements

The authors would like to thank John Harrison for the insightful discussion and for providing the source code for his paper that inspired the Dandelion work. Further, we thank Magnus Myreen and Michael Norrish for their help with improving the HOL4 implementation of Dandelion. We also thank Samuel Coward for helping us with the MetiTarski evaluation. Finally, we thank the anonymous ITP reviewers for their feedback on the paper. {}

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Heiko Becker, Mohit Tekriwal, Eva Darulova, Anastasia Volkova, and Jean-Baptiste Jeannin. Dandelion: Certified Approximations of Elementary Functions. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITP.2022.6

Abstract

Elementary function operations such as sin and exp cannot in general be computed exactly on today’s digital computers, and thus have to be approximated. The standard approximations in library functions typically provide only a limited set of precisions, and are too inefficient for many applications. Polynomial approximations that are customized to a limited input domain and output accuracy can provide superior performance. In fact, the Remez algorithm computes the best possible approximation for a given polynomial degree, but has so far not been formally verified.
This paper presents Dandelion, an automated certificate checker for polynomial approximations of elementary functions computed with Remez-like algorithms that is fully verified in the HOL4 theorem prover. Dandelion checks whether the difference between a polynomial approximation and its target reference elementary function remains below a given error bound for all inputs in a given constraint. By extracting a verified binary with the CakeML compiler, Dandelion can validate certificates within a reasonable time, fully automating previous manually verified approximations.

Subject Classification

ACM Subject Classification
  • Software and its engineering → Formal software verification
Keywords
  • elementary functions
  • approximation
  • certificate checking

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Supplementary Materials

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