research-article

Open access

An approach to generate correctly rounded math libraries for new floating point variants

Authors:

Mridul Aanjaneya,

John Gustafson,

Santosh NagarakatteAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 5, Issue POPL

Article No.: 29, Pages 1 - 30

https://doi.org/10.1145/3434310

Published: 04 January 2021 Publication History

Abstract

Given the importance of floating point (FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and posits). These representations do not have correctly rounded math libraries. Further, the use of existing FP libraries for these new representations can produce incorrect results. This paper proposes a novel approach for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods generate polynomials that approximate the real value of an elementary function 𝑓 (𝑥) and produce wrong results due to approximation errors and rounding errors in the implementation. In contrast, our approach generates polynomials that approximate the correctly rounded value of 𝑓 (𝑥) (i.e., the value of 𝑓 (𝑥) rounded to the target representation). It provides more margin to identify efficient polynomials that produce correctly rounded results for all inputs. We frame the problem of generating efficient polynomials that produce correctly rounded results as a linear programming problem. Using our approach, we have developed correctly rounded, yet faster, implementations of elementary functions for multiple target representations.

References

[1]

Denis Arzelier, Florent Bréhard, and Mioara Joldes. 2019. Exchange Algorithm for Evaluation and Approximation ErrorOptimized Polynomials. In 2019 IEEE 26th Symposium on Computer Arithmetic (ARITH). 30-37. https://doi.org/10.1109/ ARITH. 2019.00014

[2]

Florian Benz, Andreas Hildebrandt, and Sebastian Hack. 2012. A Dynamic Program Analysis to Find Floating-point Accuracy Problems. In Proceedings of the 33rd ACM SIGPLAN Conference on Programming Language Design and Implementation (Beijing, China) ( PLDI '12). ACM, New York, NY, USA, 453-462. https://doi.org/10.1145/2345156.2254118

Digital Library

[3]

Jeremy Bernstein, Jiawei Zhao, Markus Meister, Ming-Yu Liu, Anima Anandkumar, and Yisong Yue. 2020. Learning compositional functions via multiplicative weight updates. arXiv: 2006. 14560 [cs.NE]

[4]

Sylvie Boldo, Marc Daumas, and Ren-Cang Li. 2009. Formally Verified Argument Reduction with a Fused Multiply-Add. In IEEE Transactions on Computers, Vol. 58. 1139-1145. https://doi.org/10.1109/TC. 2008.216

Digital Library

[5]

Peter Borwein and Tamas Erdelyi. 1995. Polynomials and Polynomial Inequalities. Springer New York. https://doi.org/10. 1007/978-1-4612-0793-1

[6]

Nicolas Brisebarre and Sylvvain Chevillard. 2007. Eficient polynomial L-approximations. In 18th IEEE Symposium on Computer Arithmetic (ARITH '07). https://doi.org/10.1109/ARITH. 2007.17

Digital Library

[7]

Nicolas Brisebarre, Jean-Michel Muller, and Arnaud Tisserand. 2006. Computing Machine-Eficient Polynomial Approximations. In ACM ACM Transactions on Mathematical Software, Vol. 32. Association for Computing Machinery, New York, NY, USA, 236-256. https://doi.org/10.1145/1141885.1141890

Digital Library

[8]

Nicolas Brunie, Florent de Dinechin, Olga Kupriianova, and Christoph Lauter. 2015. Code Generators for Mathematical Functions. In 2015 IEEE 22nd Symposium on Computer Arithmetic. 66-73. https://doi.org/10.1109/ARITH. 2015.22

Digital Library

[9]

Hung Tien Bui and Sofiene Tahar. 1999. Design and synthesis of an IEEE-754 exponential function. In Engineering Solutions for the Next Millennium. 1999 IEEE Canadian Conference on Electrical and Computer Engineering, Vol. 1. 450-455 vol. 1. https://doi.org/10.1109/CCECE. 1999.807240

[10]

Sylvain Chevillard, John Harrison, Mioara Joldes, and Christoph Lauter. 2011. Eficient and accurate computation of upper bounds of approximation errors. Theoretical Computer Science 412. https://doi.org/10.1016/j.tcs. 2010. 11.052

Digital Library

[11]

Sylvain Chevillard, Mioara Joldes, and Christoph Lauter. 2010. Sollya: An Environment for the Development of Numerical Codes. In Mathematical Software-ICMS 2010 (Lecture Notes in Computer Science, Vol. 6327 ). Springer, Heidelberg, Germany, 28-31. https://doi.org/10.1007/978-3-642-15582-6_5

[12]

Sylvain Chevillard and Christopher Lauter. 2007. A Certified Infinite Norm for the Implementation of Elementary Functions. In Seventh International Conference on Quality Software (QSIC 2007 ). 153-160. https://doi.org/10.1109/QSIC. 2007.4385491

[13]

Sangeeta Chowdhary, Jay P. Lim, and Santosh Nagarakatte. 2020. Debugging and Detecting Numerical Errors in Computation with Posits. In 41st ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI'20). https: //doi.org/10.1145/3385412.3386004

Digital Library

[14]

William J Cody and William M Waite. 1980. Software manual for the elementary functions. Prentice-Hall, Englewood Clifs, NJ.

[15]

Mike Cowlishaw. 2008. IEEE Standard for Floating-Point Arithmetic. IEEE 754-2008. IEEE Computer Society. 1-70 pages. https://doi.org/10.1109/IEEESTD. 2008.4610935

[16]

Nasrine Damouche and Matthieu Martel. 2018. Salsa: An Automatic Tool to Improve the Numerical Accuracy of Programs. In Automated Formal Methods (Kalpa Publications in Computing, Vol. 5 ), Natarajan Shankar and Bruno Dutertre (Eds.). 63-76. https://doi.org/10.29007/j2fd

[17]

Catherine Daramy, David Defour, Florent Dinechin, and Jean-Michel Muller. 2003. CR-LIBM: A correctly rounded elementary function library. In Proceedings of SPIE Vol. 5205 : Advanced Signal Processing Algorithms, Architectures, and Implementations XIII, Vol. 5205. https://doi.org/10.1117/12.505591

[18]

Marc Daumas, Guillaume Melquiond, and Cesar Munoz. 2005. Guaranteed proofs using interval arithmetic. In 17th IEEE Symposium on Computer Arithmetic (ARITH'05). 188-195. https://doi.org/10.1109/ARITH. 2005.25

Digital Library

[19]

Florent de Dinechin, Christopher Lauter, and Guillaume Melquiond. 2011. Certifying the Floating-Point Implementation of an Elementary Function Using Gappa. In IEEE Transactions on Computers, Vol. 60. 242-253. https://doi.org/10.1109/TC. 2010.128

Digital Library

[20]

Florent de Dinechin, Christoph Quirin Lauter, and Guillaume Melquiond. 2006. Assisted Verification of Elementary Functions Using Gappa. In Proceedings of the 2006 ACM Symposium on Applied Computing (Dijon, France) (SAC '06). Association for Computing Machinery, New York, NY, USA, 1318-1322. https://doi.org/10.1145/1141277.1141584

Digital Library

[21]

Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul Zimmermann. 2007. MPFR: A Multipleprecision Binary Floating-point Library with Correct Rounding. ACM Trans. Math. Software 33, 2, Article 13 ( June 2007 ). https://doi.org/10.1145/1236463.1236468

Digital Library

[22]

Zhoulai Fu and Zhendong Su. 2019. Efective Floating-point Analysis via Weak-distance Minimization. In Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation (Phoenix, AZ, USA) ( PLDI 2019). ACM, New York, NY, USA, 439-452. https://doi.org/10.1145/3314221.3314632

Digital Library

[23]

Ambros Gleixner, Daniel E. Stefy, and Kati Wolter. 2015. Iterative Refinement for Linear Programming. Technical Report 15-15. ZIB, Takustr. 7, 14195 Berlin. https://doi.org/10.1287/ijoc. 2016.0692

[24]

Ambros M. Gleixner, Daniel E. Stefy, and Kati Wolter. 2012. Improving the Accuracy of Linear Programming Solvers with Iterative Refinement. In Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (Grenoble, France) (ISSAC '12). Association for Computing Machinery, New York, NY, USA, 187-194. https://doi.org/10. 1145/2442829.2442858

Digital Library

[25]

Eric Goubault. 2001. Static Analyses of the Precision of Floating-Point Operations. In Proceedings of the 8th International Symposium on Static Analysis (SAS). Springer, 234-259. https://doi.org/10.1007/3-540-47764-0_14

[26]

John Gustafson. 2017. Posit Arithmetic. https://posithub.org/docs/Posits4.pdf

[27]

John Gustafson. 2020. The Minefield Method: A Uniformly Fast Solution to the Table-Maker's Dilemma. https://bit.ly/2ZP4kHj

[28]

John Gustafson and Isaac Yonemoto. 2017. Beating Floating Point at Its Own Game: Posit Arithmetic. Supercomputing Frontiers and Innovations: an International Journal 4, 2 ( June 2017 ), 71-86. https://doi.org/10.14529/jsfi170206

Digital Library

[29]

John Harrison. 1997a. Floating point verification in HOL light: The exponential function. In Algebraic Methodology and Software Technology, Michael Johnson (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 246-260. https: //doi.org/10.1007/BFb0000475

[30]

John Harrison. 1997b. Verifying the Accuracy of Polynomial Approximations in HOL. In International Conference on Theorem Proving in Higher Order Logics. https://doi.org/10.1007/BFb0028391

[31]

John Harrison. 2009. HOL Light: An Overview. In Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2009 (Lecture Notes in Computer Science, Vol. 5674 ), Stefan Berghofer, Tobias Nipkow, Christian Urban, and Makarius Wenzel (Eds.). Springer-Verlag, Munich, Germany, 60-66. https://doi.org/10.1007/978-3-642-03359-9_4

Digital Library

[32]

IBM. 2008. Accurate Portable MathLib. http://oss.software.ibm.com/mathlib/

[33]

Intel. 2019. Delivering a New Intelligence with AI at Scale. https://www.intel.com/content/www/us/en/artificial-intelligence/ posts/nnp-aisummit.html

[34]

Claude-Pierre Jeannerod, Hervé Knochel, Christophe Monat, and Guillaume Revy. 2011. Computing Floating-Point Square Roots via Bivariate Polynomial Evaluation. IEEE Trans. Comput. 60. https://doi.org/10.1109/TC. 2010.152

Digital Library

[35]

Jef Johnson. 2018. Rethinking floating point for deep learning. http://export.arxiv.org/abs/ 1811.01721

[36]

William Kahan. 2004. A Logarithm Too Clever by Half. https://people.eecs.berkeley.edu/~wkahan/LOG10HAF.TXT

[37]

Dhiraj D. Kalamkar, Dheevatsa Mudigere, Naveen Mellempudi, Dipankar Das, Kunal Banerjee, Sasikanth Avancha, Dharma Teja Vooturi, Nataraj Jammalamadaka, Jianyu Huang, Hector Yuen, Jiyan Yang, Jongsoo Park, Alexander Heinecke, Evangelos Georganas, Sudarshan Srinivasan, Abhisek Kundu, Misha Smelyanskiy, Bharat Kaul, and Pradeep Dubey. 2019. A Study of BFLOAT16 for Deep Learning Training. arXiv: 1905.12322

[38]

Olga Kupriianova and Christoph Lauter. 2014. Metalibm: A Mathematical Functions Code Generator. In 4th International Congress on Mathematical Software. https://doi.org/10.1007/978-3-662-44199-2_106

[39]

Wonyeol Lee, Rahul Sharma, and Alex Aiken. 2017. On Automatically Proving the Correctness of Math.h Implementations. Proceedings of the ACM on Programming Languages 2, POPL, Article 47 ( Dec. 2017 ), 32 pages. https://doi.org/10.1145/ 3158135

Digital Library

[40]

Vincent Lefèvre and Jean-Michel Muller. 2001. Worst Cases for Correct Rounding of the Elementary Functions in Double Precision. In 15th IEEE Symposium on Computer Arithmetic (Arith '01). 111-118. https://doi.org/10.1109/ARITH. 2001. 930110

[41]

Vincent Lefèvre, Jean-Michel Muller, and Arnaud Tisserand. 1998. Toward correctly rounded transcendentals. IEEE Trans. Comput. 47, 11 ( 1998 ), 1235-1243. https://doi.org/10.1109/12.736435

Digital Library

[42]

Cerlane Leong. 2019. SoftPosit-Math. https://gitlab.com/cerlane/softposit-math

[43]

Jay P. Lim, Mridul Aanjaneya, John Gustafson, and Santosh Nagarakatte. 2020a. A Novel Approach to Generate Correctly Rounded Math Libraries for New Floating Point Representations. arXiv: 2007. 05344 [cs.MS]

[44]

Jay P. Lim and Santosh Nagarakatte. 2020a. RLibm. https://github.com/rutgers-apl/rlibm

[45]

Jay P. Lim and Santosh Nagarakatte. 2020b. RLibm-generator. https://github.com/rutgers-apl/rlibm-generator

[46]

Jay P. Lim, Matan Shachnai, and Santosh Nagarakatte. 2020b. Approximating Trigonometric Functions for Posits Using the CORDIC Method. In Proceedings of the 17th ACM International Conference on Computing Frontiers (Catania, Sicily, Italy) ( CF '20). Association for Computing Machinery, New York, NY, USA, 19-28. https://doi.org/10.1145/3387902.3392632

Digital Library

[47]

Guillaume Melquiond. 2019. Gappa. http://gappa.gforge.inria.fr

[48]

Sun Microsystems. 2008. LIBMCR 3 " 16 February 2008 " "libmcr-0.9". http://www.math.utah.edu/cgi-bin/man2html.cgi?/usr/ local/man/man3/libmcr.3

[49]

Jean-Michel Muller. 2005. Elementary Functions: Algorithms and Implementation. Birkhauser. https://doi.org/10.1007/978-1-4899-7983-4

[50]

NVIDIA. 2020. TensorFloat-32 in the A100 GPU Accelerates AI Training, HPC up to 20x. https://blogs.nvidia.com/blog/2020/ 05/14/tensorfloat-32-precision-format/

[51]

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, Vol. 50. Association for Computing Machinery, New York, NY, USA, 1-11. https://doi.org/10.1145/ 2813885.2737959

Digital Library

[52]

Eugene Remes. 1934. Sur un procédé convergent d'approximations successives pour déterminer les polynômes d'approximation. Comptes rendus de l' Académie des Sciences 198 ( 1934 ), 2063-2065.

[53]

Alex Sanchez-Stern, Pavel Panchekha, Sorin Lerner, and Zachary Tatlock. 2018. Finding Root Causes of Floating Point Error. In Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation (Philadelphia, PA, USA) ( PLDI 2018). ACM, New York, NY, USA, 256-269. https://doi.org/10.1145/3296979.3192411

Digital Library

[54]

Joe Sawada. 2002. Formal verification of divide and square root algorithms using series calculation. In 3rd International Workshop on the ACL2 Theorem Prover and its Applications.

[55]

Giuseppe Tagliavini, Stefan Mach, Davide Rossi, Andrea Marongiu, and Luca Benin. 2018. A transprecision floating-point platform for ultra-low power computing. In 2018 Design, Automation Test in Europe Conference Exhibition (DATE). 1051-1056. https://doi.org/10.23919/DATE. 2018.8342167

[56]

Ping-Tak Peter Tang. 1990. Table-Driven Implementation of the Logarithm Function in IEEE Floating-Point Arithmetic. ACM Trans. Math. Software 16, 4 (Dec. 1990 ), 378-400. https://doi.org/10.1145/98267.98294

Digital Library

[57]

Lloyd N. Trefethen. 2012. Approximation Theory and Approximation Practice (Other Titles in Applied Mathematics). Society for Industrial and Applied Mathematics, USA.

[58]

Shibo Wang and Pankaj Kanwar. 2019. BFloat16: The secret to high performance on Cloud TPUs. https://cloud.google.com/ blog/products/ai-machine-learning/bfloat16-the-secret-to-high-performance-on-cloud-tpus

[59]

Xin Yi, Liqian Chen, Xiaoguang Mao, and Tao Ji. 2019. Eficient Automated Repair of High Floating-Point Errors in Numerical Libraries. Proceedings of the ACM on Programming Languages 3, POPL, Article 56 ( Jan. 2019 ), 29 pages. https://doi.org/10.1145/3290369

Digital Library

[60]

Abraham Ziv. 1991. Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded Last Bit. ACM Trans. Math. Software 17, 3 (Sept. 1991 ), 410-423. https://doi.org/10.1145/114697.116813

Digital Library

[61]

Daming Zou, Muhan Zeng, Yingfei Xiong, Zhoulai Fu, Lu Zhang, and Zhendong Su. 2019. Detecting Floating-Point Errors via Atomic Conditions. Proceedings of the ACM on Programming Languages 4, POPL, Article 60 ( Dec. 2019 ), 27 pages. https://doi.org/10.1145/3371128

Digital Library

Cited By

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
Aanjaneya MNagarakatte S(2024)Maximum Consensus Floating Point Solutions for Infeasible Low-Dimensional Linear Programs with Convex Hull as the Intermediate RepresentationProceedings of the ACM on Programming Languages10.1145/36564278:PLDI(1239-1263)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656427
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

An approach to generate correctly rounded math libraries for new floating point variants
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
  2. Mathematical software
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis
      1. Numeric approximation algorithms

Recommendations

One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modes

Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such ...
High performance correctly rounded math libraries for 32-bit floating point representations
PLDI 2021: Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLIBM approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the ...
Progressive polynomial approximations for fast correctly rounded math libraries
PLDI 2022: Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation

This paper presents a novel method for generating a single polynomial approximation that produces correctly rounded results for all inputs of an elementary function for multiple representations. The generated polynomial approximation has the nice ...

Comments

Information & Contributors

Information

Published In

cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 5, Issue POPL

January 2021

1789 pages

EISSN:2475-1421

DOI:10.1145/3445980

Issue’s Table of Contents

Copyright © 2021 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 04 January 2021

Published in PACMPL Volume 5, Issue POPL

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Research-article

Funding Sources

National Science Foundation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

14
Total Citations
View Citations
992
Total Downloads

Downloads (Last 12 months)211
Downloads (Last 6 weeks)25

Reflects downloads up to 06 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

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
Aanjaneya MNagarakatte S(2024)Maximum Consensus Floating Point Solutions for Infeasible Low-Dimensional Linear Programs with Convex Hull as the Intermediate RepresentationProceedings of the ACM on Programming Languages10.1145/36564278:PLDI(1239-1263)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656427
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
Briggs ILad YPanchekha P(2024)Implementation and Synthesis of Math Library FunctionsProceedings of the ACM on Programming Languages10.1145/36328748:POPL(942-969)Online publication date: 5-Jan-2024
https://dl.acm.org/doi/10.1145/3632874
Aanjaneya MNagarakatte SDubach CBruening DHardekopf B(2023)Fast Polynomial Evaluation for Correctly Rounded Elementary Functions using the RLIBM ApproachProceedings of the 21st ACM/IEEE International Symposium on Code Generation and Optimization10.1145/3579990.3580022(95-107)Online publication date: 17-Feb-2023
https://dl.acm.org/doi/10.1145/3579990.3580022
Qu YXu JZhou BHao JLi FZhang Z(2023)SCR-LIBM: A Correctly Rounded Elementary Function Library in Double-PrecisionInternational Journal of Software Engineering and Knowledge Engineering10.1142/S021819402350067534:04(675-703)Online publication date: 7-Dec-2023
https://doi.org/10.1142/S0218194023500675
Brisebarre NFilip S(2023) Towards Machine-Efficient Rational L ∞ -Approximations of Mathematical Functions 2023 IEEE 30th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH58626.2023.00029(119-126)Online publication date: 4-Sep-2023
https://doi.org/10.1109/ARITH58626.2023.00029
Chowdhary SNagarakatte S(2022)Fast shadow execution for debugging numerical errors using error free transformationsProceedings of the ACM on Programming Languages10.1145/35633536:OOPSLA2(1845-1872)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563353
Zou DGu YShi YWang MXiong YSu Z(2022)Oracle-free repair synthesis for floating-point programsProceedings of the ACM on Programming Languages10.1145/35633226:OOPSLA2(957-985)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563322
Aanjaneya MLim JNagarakatte SJhala RDillig I(2022)Progressive polynomial approximations for fast correctly rounded math librariesProceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation10.1145/3519939.3523447(552-565)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519939.3523447
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

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 Article

Figures

Tables

Media

View Issue’s Table of Contents