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Documents authored by Becker, Heiko

Document
DOI: 10.4230/LIPIcs.ITP.2022.6
Dandelion: Certified Approximations of Elementary Functions

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

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

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.
Cite as

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)

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@InProceedings{becker_et_al:LIPIcs.ITP.2022.6,
  author =	{Becker, Heiko and Tekriwal, Mohit and Darulova, Eva and Volkova, Anastasia and Jeannin, Jean-Baptiste},
  title =	{{Dandelion: Certified Approximations of Elementary Functions}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.6},
  URN =		{urn:nbn:de:0030-drops-167155},
  doi =		{10.4230/LIPIcs.ITP.2022.6},
  annote =	{Keywords: elementary functions, approximation, certificate checking}
}
Document
Artifact
DOI: 10.4230/DARTS.8.2.10
Verified Compilation and Optimization of Floating-Point Programs in CakeML (Artifact)

Authors: Heiko Becker, Robert Rabe, Eva Darulova, Magnus O. Myreen, Zachary Tatlock, Ramana Kumar, Yong Kiam Tan, and Anthony Fox

Published in: DARTS, Volume 8, Issue 2, Special Issue of the 36th European Conference on Object-Oriented Programming (ECOOP 2022)

Abstract
Verified compilers such as CompCert and CakeML have become increasingly realistic over the last few years, but their support for floating-point arithmetic has thus far been limited. In particular, they lack the "fast-math-style" optimizations that unverified mainstream compilers perform. Supporting such optimizations in the setting of verified compilers is challenging because these optimizations, for the most part, do not preserve the IEEE-754 floating-point semantics. However, IEEE-754 floating-point numbers are finite approximations of the real numbers, and we argue that any compiler correctness result for fast-math optimizations should appeal to a real-valued semantics rather than the rigid IEEE-754 floating-point numbers. This document describes the artifact for RealCake, an extension of CakeML that achieves end-to-end correctness results for fast-math-style optimized compilation of floating-point arithmetic. This result is achieved by giving CakeML a flexible floating-point semantics and integrating an external proof-producing accuracy analysis. RealCake’s end-to-end theorems relate the I/O behavior of the original source program under real-number semantics to the observable I/O behavior of the compiler generated and fast-math-optimized machine code.
Cite as

Heiko Becker, Robert Rabe, Eva Darulova, Magnus O. Myreen, Zachary Tatlock, Ramana Kumar, Yong Kiam Tan, and Anthony Fox. Verified Compilation and Optimization of Floating-Point Programs in CakeML (Artifact). In Special Issue of the 36th European Conference on Object-Oriented Programming (ECOOP 2022). Dagstuhl Artifacts Series (DARTS), Volume 8, Issue 2, pp. 10:1-10:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{becker_et_al:DARTS.8.2.10,
  author =	{Becker, Heiko and Rabe, Robert and Darulova, Eva and Myreen, Magnus O. and Tatlock, Zachary and Kumar, Ramana and Tan, Yong Kiam and Fox, Anthony},
  title =	{{Verified Compilation and Optimization of Floating-Point Programs in CakeML (Artifact)}},
  pages =	{10:1--10:2},
  journal =	{Dagstuhl Artifacts Series},
  ISSN =	{2509-8195},
  year =	{2022},
  volume =	{8},
  number =	{2},
  editor =	{Becker, Heiko and Rabe, Robert and Darulova, Eva and Myreen, Magnus O. and Tatlock, Zachary and Kumar, Ramana and Tan, Yong Kiam and Fox, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DARTS.8.2.10},
  URN =		{urn:nbn:de:0030-drops-162086},
  doi =		{10.4230/DARTS.8.2.10},
  annote =	{Keywords: compiler verification, compiler optimization, floating-point arithmetic}
}
Document
DOI: 10.4230/LIPIcs.ECOOP.2022.1
Verified Compilation and Optimization of Floating-Point Programs in CakeML

Authors: Heiko Becker, Robert Rabe, Eva Darulova, Magnus O. Myreen, Zachary Tatlock, Ramana Kumar, Yong Kiam Tan, and Anthony Fox

Published in: LIPIcs, Volume 222, 36th European Conference on Object-Oriented Programming (ECOOP 2022)

Abstract
Verified compilers such as CompCert and CakeML have become increasingly realistic over the last few years, but their support for floating-point arithmetic has thus far been limited. In particular, they lack the "fast-math-style" optimizations that unverified mainstream compilers perform. Supporting such optimizations in the setting of verified compilers is challenging because these optimizations, for the most part, do not preserve the IEEE-754 floating-point semantics. However, IEEE-754 floating-point numbers are finite approximations of the real numbers, and we argue that any compiler correctness result for fast-math optimizations should appeal to a real-valued semantics rather than the rigid IEEE-754 floating-point numbers. This paper presents RealCake, an extension of CakeML that achieves end-to-end correctness results for fast-math-style optimized compilation of floating-point arithmetic. This result is achieved by giving CakeML a flexible floating-point semantics and integrating an external proof-producing accuracy analysis. RealCake’s end-to-end theorems relate the I/O behavior of the original source program under real-number semantics to the observable I/O behavior of the compiler generated and fast-math-optimized machine code.
Cite as

Heiko Becker, Robert Rabe, Eva Darulova, Magnus O. Myreen, Zachary Tatlock, Ramana Kumar, Yong Kiam Tan, and Anthony Fox. Verified Compilation and Optimization of Floating-Point Programs in CakeML. In 36th European Conference on Object-Oriented Programming (ECOOP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 222, pp. 1:1-1:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{becker_et_al:LIPIcs.ECOOP.2022.1,
  author =	{Becker, Heiko and Rabe, Robert and Darulova, Eva and Myreen, Magnus O. and Tatlock, Zachary and Kumar, Ramana and Tan, Yong Kiam and Fox, Anthony},
  title =	{{Verified Compilation and Optimization of Floating-Point Programs in CakeML}},
  booktitle =	{36th European Conference on Object-Oriented Programming (ECOOP 2022)},
  pages =	{1:1--1:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-225-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{222},
  editor =	{Ali, Karim and Vitek, Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2022.1},
  URN =		{urn:nbn:de:0030-drops-162290},
  doi =		{10.4230/LIPIcs.ECOOP.2022.1},
  annote =	{Keywords: compiler verification, compiler optimization, floating-point arithmetic}
}
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