research-article

Finding root causes of floating point error

Authors:

Alex Sanchez-Stern,

Pavel Panchekha,

Zachary TatlockAuthors Info & Claims

PLDI 2018: Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation

Pages 256 - 269

https://doi.org/10.1145/3192366.3192411

Published: 11 June 2018 Publication History

Abstract

Floating-point arithmetic plays a central role in science, engineering, and finance by enabling developers to approximate real arithmetic. To address numerical issues in large floating-point applications, developers must identify root causes, which is difficult because floating-point errors are generally non-local, non-compositional, and non-uniform.

This paper presents Herbgrind, a tool to help developers identify and address root causes in numerical code written in low-level languages like C/C++ and Fortran. Herbgrind dynamically tracks dependencies between operations and program outputs to avoid false positives and abstracts erroneous computations to simplified program fragments whose improvement can reduce output error. We perform several case studies applying Herbgrind to large, expert-crafted numerical programs and show that it scales to applications spanning hundreds of thousands of lines, correctly handling the low-level details of modern floating point hardware and mathematical libraries and tracking error across function boundaries and through the heap.

Supplementary Material

WEBM File (p256-sanchez-stern.webm)

Download
98.19 MB

References

[1]

Micah Altman, Jeff Gill, and Michael P. McDonald. 2003. Numerical Issues in Statistical Computing for the Social Scientist. Springer-Verlag. 1–11 pages.

Digital Library

[2]

Micah Altman and Michael P. McDonald. 2003. Replication with attention to numerical accuracy. Political Analysis 11, 3 (2003), 302– 307. http://pan.oxfordjournals.org/content/11/3/302.abstract

[3]

Tao Bao and Xiangyu Zhang. 2013. On-the-fly Detection of Instability Problems in Floating-point Program Execution. SIGPLAN Not. 48, 10 (Oct. 2013), 817–832.

Digital Library

[4]

Florian Benz, Andreas Hildebrandt, and Sebastian Hack. 2012. A Dynamic Program Analysis to Find Floating-point Accuracy Problems (PLDI ’12). ACM, New York, NY, USA, 453–462.

Digital Library

[5]

Hans-J. Boehm. 2004. The constructive reals as a Java Library. J. Log. Algebr. Program 64 (2004), 3–11.

[6]

Wei-Fan Chiang, Mark Baranowski, Ian Briggs, Alexey Solovyev, Ganesh Gopalakrishnan, and Zvonimir Rakamarić. 2017. Rigorous Floating-point Mixed-precision Tuning. In Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages (POPL 2017). ACM, New York, NY, USA, 300–315.

Digital Library

[7]

Wei-Fan Chiang, Ganesh Gopalakrishnan, Zvonimir Rakamarić, and Alexey Solovyev. 2014. Efficient Search for Inputs Causing High Floating-point Errors. ACM, 43–52.

Digital Library

[8]

Nasrine Damouche, Matthieu Martel, and Alexandre Chapoutot. 2015. Formal Methods for Industrial Critical Systems: 20th International Workshop, FMICS 2015 Oslo, Norway, June 22-23, 2015 Proceedings. (2015), 31–46.

[9]

N. Damouche, M. Martel, and A. Chapoutot. 2015. Transformation of a {PID} Controller for Numerical Accuracy. Electronic Notes in Theoretical Computer Science 317 (2015), 47 – 54.

Digital Library

[10]

Nasrine Damouche, Matthieu Martel, Pavel Panchekha, Jason Qiu, Alex Sanchez-Stern, and Zachary Tatlock. 2016. Toward a Standard Benchmark Format and Suite for Floating-Point Analysis. (July 2016).

[11]

Eva Darulova and Viktor Kuncak. 2014. Sound Compilation of Reals (POPL ’14). ACM, New York, NY, USA, 235–248.

Digital Library

[12]

François Févotte and Bruno Lathuilière. 2016. VERROU: Assessing Floating-Point Accuracy Without Recompiling. (Oct. 2016). https: //hal.archives-ouvertes.fr/hal-01383417 working paper or preprint.

[13]

Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul Zimmermann. 2007. MPFR: A Multiple-Precision Binary Floating-Point Library with Correct Rounding. ACM Trans. Math. Software 33, 2 (June 2007), 13:1–13:15.

Digital Library

[14]

Eric Goubault and Sylvie Putot. 2011. Static Analysis of Finite Precision Computations (VMCAI’11). Springer-Verlag, Berlin, Heidelberg, 232– 247. http://dl.acm.org/citation.cfm?id=1946284.1946301

[15]

Nicholas J. Higham. 2002. Accuracy and Stability of Numerical Algorithms (2nd ed.). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.

[16]

Andreas Jaeger. 2016. OpenLibm. http://openlibm.org/

[17]

W. Kahan. 1965. Pracniques: Further Remarks on Reducing Truncation Errors. Commun. ACM 8, 1 (Jan. 1965), 40–.

Digital Library

[18]

William Kahan. 1971. A Survey of Error Analysis. In IFIP Congress (2). 1214–1239. http://dblp.uni-trier.de/db/conf/ifip/ifip71-2.html# Kahan71

[19]

W. Kahan. 1987. Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit. In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.). Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9. Oxford Univ. Press, New York, 165âĂŞ211.

[20]

W. Kahan. 1998. The Improbability of Probabilistic Error Analyses for Numerical Computations. Technical Report. 34 pages. http://www.cs. berkeley.edu/~wkahan/improber.pdf

[21]

William Kahan. 2005. Floating-Point Arithmetic Besieged by “Business Decisions”. World-Wide Web lecture notes., 28 pages. http://www.cs. berkeley.edu/~wkahan/ARITH_17.pdf

[22]

C. L. Lawson, R. J. Hanson, D. R. Kincaid, and F. T. Krogh. 1979. Basic Linear Algebra Subprograms for Fortran Usage. ACM Trans. Math. Softw. 5, 3 (Sept. 1979), 308–323.

Digital Library

[23]

Vernon A. Lee, Jr. and Hans-J. Boehm. 1990. Optimizing Programs over the Constructive Reals. In Proceedings of the ACM SIGPLAN 1990 Conference on Programming Language Design and Implementation (PLDI ’90). ACM, New York, NY, USA, 102–111.

Digital Library

[24]

Matthieu Martel. 2009. Program Transformation for Numerical Precision (PEPM ’09). ACM, New York, NY, USA, 101–110.

Digital Library

[25]

B. D. McCullough and H. D. Vinod. 1999. The Numerical Reliability of Econometric Software. Journal of Economic Literature 37, 2 (1999), 633–665.

[26]

Marvin L. Minsky. 1967. Computation: Finite and Infinite Machines. Prentice-Hall, Inc., Upper Saddle River, NJ, USA.

Digital Library

[27]

Nethercote and Seward. 2007. Valgrind: A Framework for Heavyweight Dynamic Binary Instrumentation. (June 2007).

[28]

Dr. K-C Ng. 1993. FDLIBM. http://www.netlib.org/fdlibm/readme

[29]

Pavel Panchekha, Alex Sanchez-Stern, James R. Wilcox, and Zachary Tatlock. 2015. Automatically Improving Accuracy for Floating Point Expressions. In Proceedings of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI ’15). ACM.

Digital Library

[30]

Gordon D. Plotkin. 1970. A note on inductive generalization. Machine Intelligence 5 (1970), 153–163.

[31]

Kevin Quinn. 1983. Ever Had Problems Rounding Off Figures? This Stock Exchange Has. The Wall Street Journal (November 8, 1983), 37.

[32]

Jonathan Richard Shewchuk. 1996. Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In Applied Computational Geometry: Towards Geometric Engineering, Ming C. Lin and Dinesh Manocha (Eds.). Lecture Notes in Computer Science, Vol. 1148. Springer-Verlag, 203–222. From the First ACM Workshop on Applied Computational Geometry.

Digital Library

[33]

Hang Si. 2015. TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator. ACM Trans. Math. Softw. 41, 2, Article 11 (Feb. 2015), 36 pages.

Digital Library

[34]

Alexey Solovyev, Charlie Jacobsen, Zvonimir Rakamaric, and Ganesh Gopalakrishnan. 2015. Rigorous Estimation of Floating-Point Roundoff Errors with Symbolic Taylor Expansions (FM’15). Springer.

[35]

O. Tange. 2011. GNU Parallel - The Command-Line Power Tool. ;login: The USENIX Magazine 36, 1 (Feb 2011), 42–47. http://www.gnu.org/s/ parallel

[36]

U.S. General Accounting Office. 1992. Patriot Missile Defense: Software Problem Led to System Failure at Dhahran, Saudi Arabia. http://www. gao.gov/products/IMTEC-92-26

[37]

Ran Wang, Daming Zou, Xinrui He, Yingfei Xiong, Lu Zhang, and Gang Huang. 2015. Detecting and Fixing Precision-Specific Operations for Measuring Floating-Point Errors (FSE’15).

Digital Library

[38]

Debora Weber-Wulff. 1992. Rounding error changes Parliament makeup. http://catless.ncl.ac.uk/Risks/13.37.html#subj4

Cited By

Yi XYu HChen LMao XWang J(2024)FPCC: Detecting Floating-Point Errors via Chain ConditionsProceedings of the ACM on Programming Languages10.1145/36897648:OOPSLA2(1504-1531)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689764
Wang XYi XYu HHuang CPeng L(2024)Parallel Optimization for Accelerating the Generation of Correctly Rounded Elementary FunctionsProceedings of the 53rd International Conference on Parallel Processing10.1145/3673038.3673125(21-31)Online publication date: 12-Aug-2024
https://dl.acm.org/doi/10.1145/3673038.3673125
Xu JCui MLi FZhang ZYang HZhou BZhao JChristakis MPradel M(2024)Arfa: An Agile Regime-Based Floating-Point Optimization Approach for Rounding ErrorsProceedings of the 33rd ACM SIGSOFT International Symposium on Software Testing and Analysis10.1145/3650212.3680378(1516-1528)Online publication date: 11-Sep-2024
https://dl.acm.org/doi/10.1145/3650212.3680378
Show More Cited By

Index Terms

Finding root causes of floating point error
1. Software and its engineering
  1. Software notations and tools
    1. Software maintenance tools

Recommendations

Finding root causes of floating point error
PLDI '18

Floating-point arithmetic plays a central role in science, engineering, and finance by enabling developers to approximate real arithmetic. To address numerical issues in large floating-point applications, developers must identify root causes, which is ...
Decimal Floating-Point Multiplication

Decimal multiplication is important in many commercial applications including financial analysis, banking, tax calculation, currency conversion, insurance, and accounting. This paper presents the design of two decimal floating-point multipliers: one ...
Low-Cost Binary128 Floating-Point FMA Unit Design with SIMD Support

Binary64 arithmetic is rapidly becoming inadequate to cope with today's large-scale computations due to an accumulation of errors. Therefore, binary128 arithmetic is now required to increase the accuracy and reliability of these computations. At the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

PLDI 2018: Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation

June 2018

825 pages

ISBN:9781450356985

DOI:10.1145/3192366

General Chair:
Jeffrey S. Foster
University of Maryland at College Park, USA
,
Program Chair:
Dan Grossman
University of Washington, USA

ACM SIGPLAN Notices Volume 53, Issue 4
PLDI '18
April 2018
834 pages
ISSN:0362-1340
EISSN:1558-1160
DOI:10.1145/3296979
Editor:
Matthew Fluet
Rodchester Institude of Technology
Issue’s Table of Contents

Copyright © 2018 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGPLAN: ACM Special Interest Group on Programming Languages

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 11 June 2018

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Research-article

Conference

PLDI '18

Sponsor:

SIGPLAN

PLDI '18: ACM SIGPLAN Conference on Programming Language Design and Implementation

June 18 - 22, 2018

PA, Philadelphia, USA

Acceptance Rates

Overall Acceptance Rate 406 of 2,067 submissions, 20%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

53
Total Citations
View Citations
616
Total Downloads

Downloads (Last 12 months)78
Downloads (Last 6 weeks)8

Reflects downloads up to 23 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Yi XYu HChen LMao XWang J(2024)FPCC: Detecting Floating-Point Errors via Chain ConditionsProceedings of the ACM on Programming Languages10.1145/36897648:OOPSLA2(1504-1531)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689764
Wang XYi XYu HHuang CPeng L(2024)Parallel Optimization for Accelerating the Generation of Correctly Rounded Elementary FunctionsProceedings of the 53rd International Conference on Parallel Processing10.1145/3673038.3673125(21-31)Online publication date: 12-Aug-2024
https://dl.acm.org/doi/10.1145/3673038.3673125
Xu JCui MLi FZhang ZYang HZhou BZhao JChristakis MPradel M(2024)Arfa: An Agile Regime-Based Floating-Point Optimization Approach for Rounding ErrorsProceedings of the 33rd ACM SIGSOFT International Symposium on Software Testing and Analysis10.1145/3650212.3680378(1516-1528)Online publication date: 11-Sep-2024
https://dl.acm.org/doi/10.1145/3650212.3680378
Xu JSong GZhou BLi FHao JZhao JLee IChabbi MSteuwer M(2024)A Holistic Approach to Automatic Mixed-Precision Code Generation and Tuning for Affine ProgramsProceedings of the 29th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming10.1145/3627535.3638484(55-67)Online publication date: 2-Mar-2024
https://dl.acm.org/doi/10.1145/3627535.3638484
Sathyavathi NBeulet P(2024)Deca-Rounding Methods for Floating Point Numbers: Error Analysis & Hardware ImplementationJournal of Signal Processing Systems10.1007/s11265-023-01899-z96:1(31-50)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1007/s11265-023-01899-z
Zhang ZXu JHao JQu YHe HZhou B(2024)Hierarchical search algorithm for error detection in floating-point arithmetic expressionsThe Journal of Supercomputing10.1007/s11227-023-05523-680:1(1183-1205)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1007/s11227-023-05523-6
Ren XKawai MHoshino TKatagiri TNagai T(2024)Auto‐Tuning Mixed‐Precision Computation by Specifying Multiple RegionsConcurrency and Computation: Practice and Experience10.1002/cpe.8326Online publication date: 7-Nov-2024
https://doi.org/10.1002/cpe.8326
Doyle W(2023)Searching For Black Holes Using Auto DifferentiationEmerging Minds Journal for Student Research10.59973/emjsr.101(17-38)Online publication date: 6-Jul-2023
https://doi.org/10.59973/emjsr.10
Demeure NChevalier CDenis CDossantos-Uzarralde P(2023)Algorithm 1029: Encapsulated Error, a Direct Approach to Evaluate Floating-Point AccuracyACM Transactions on Mathematical Software10.1145/354920548:4(1-16)Online publication date: 22-Mar-2023
https://dl.acm.org/doi/10.1145/3549205
Yu HYi XYin BLi FChen ZHuang C(2023)Efficient Generation of Floating-Point Inputs for Compiler-Induced Variability2023 IEEE International Conference on Software Analysis, Evolution and Reengineering (SANER)10.1109/SANER56733.2023.00030(224-235)Online publication date: Mar-2023
https://doi.org/10.1109/SANER56733.2023.00030
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents