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On Sound Relative Error Bounds for Floating-Point Arithmetic [article]

Anastasiia Izycheva, Eva Darulova
2017 arXiv   pre-print
In this paper, we investigate how state-of-the-art static techniques for computing sound absolute error bounds can be used, extended and combined for the computation of relative errors.  ...  State-of-the-art static analysis tools for verifying finite-precision code compute worst-case absolute error bounds on numerical errors.  ...  RELATED WORK The goal of this work is an automated and sound static analysis technique for computing tight relative error bounds for floating-point arithmetic.  ... 
arXiv:1707.02121v2 fatcat:j3d26pjgdfe7fpnfx2cmokvffu

Daisy - Framework for Analysis and Optimization of Numerical Programs (Tool Paper) [chapter]

Eva Darulova, Anastasiia Izycheva, Fariha Nasir, Fabian Ritter, Heiko Becker, Robert Bastian
2018 Lecture Notes in Computer Science  
This paper presents a new framework, called Daisy, which provides in a single tool the main building blocks for accuracy analysis of floating-point and fixed-point computations which have emerged from  ...  Automated techniques for analysis and optimization of finite-precision computations have recently garnered significant interest. Most of these were, however, developed independently.  ...  For instance, the tools Fluctuat [22] , Rosa [14] , Gappa [17] , FPTaylor [41] , Real2Float [31] and PRECiSA [34] automatically provide sound error bounds on floating-point (and some also on fixed-point  ... 
doi:10.1007/978-3-319-89960-2_15 fatcat:ktavggsosfh5zihqco7rtcolou

Compositional Analysis of Floating-Point Linear Numerical Filters [chapter]

David Monniaux
2005 Lecture Notes in Computer Science  
We give a compositional, effective abstraction for digital linear filters expressed as block diagrams, yielding sound, precise bounds for fixedpoint or floating-point implementations of the filters.  ...  For safety-critical applications, it is necessary to bound all variables and outputs of all filters.  ...  We actually take the coarser inequality |x−x| ≤ ε rel |x| + ε abs . See [1] for more details on floating-point numbers and [9] for more about the affine bound on the error.  ... 
doi:10.1007/11513988_21 fatcat:xyd4asi775d2thoobsp2yji5q4

A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4 [article]

Heiko Becker, Nikita Zyuzin, Raphael Monat, Eva Darulova, Magnus O. Myreen, Anthony Fox
2018 arXiv   pre-print
We present implementations of certificate generation and checking for both Coq and HOL4 and evaluate it on a number of examples from the literature.  ...  This paper presents a formally verified and modular tool which fully automatically checks the correctness of finite-precision roundoff error bounds encoded in a certificate.  ...  Finally, note that input ranges are also necessary for computing concrete relative error bounds. b) Sound Verification of Floating-point Computations: Absence of runtime errors in floating-point computations  ... 
arXiv:1707.02115v2 fatcat:2c2jwcvlpfcsvnu4ncod2wuqdm

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations [article]

George Constantinides and Fredrik Dahlqvist and Zvonimir Rakamaric and Rocco Salvia
2021 arXiv   pre-print
Based on these theoretical advances, we propose a model of IEEE floating-point arithmetic for numerical expressions with probabilistic inputs and an algorithm for evaluating this model.  ...  We present a detailed study of roundoff errors in probabilistic floating-point computations.  ...  Conclusions and Future Work In this paper, we presented our approach to computing rigorous (i.e., sound) probabilistic roundoff error bounds for arithmetic floating-point computations involving random  ... 
arXiv:2105.13217v1 fatcat:ujwdxk2ctfasjevi3j2s5pctti

Trustworthy numerical computation in Scala

Eva Darulova, Viktor Kuncak
2011 SIGPLAN notices  
Our numerical data type library tracks a (double) floating point value, but also a guaranteed upper bound on the error between this value and the ideal value that would be computed in the real-value semantics  ...  We present a library solution for rigorous arithmetic computation.  ...  tool that provides sound guarantees on the floating-point roundoff errors committed.  ... 
doi:10.1145/2076021.2048094 fatcat:f2sg4mf2dzdfdfxhsnp6flr7je

On Sound Compilation of Reals [article]

Eva Darulova, Viktor Kuncak
2013 arXiv   pre-print
Our verification step generates verification conditions that treat different uncertainties in a unified way and encode reasoning about floating-point roundoff errors into reasoning about real numbers.  ...  We propose a new procedure that combines exact SMT solving over reals with approximate and sound affine and interval arithmetic.  ...  For such cases our approach estimates a sound upper bound on the total error of the computation.  ... 
arXiv:1309.2511v1 fatcat:riq4wqszj5emvmb7vlrqkbvlse

Sound compilation of reals

Eva Darulova, Viktor Kuncak
2014 Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14  
Our verification step generates verification conditions that treat different uncertainties in a unified way and encode reasoning about floating-point roundoff errors into reasoning about real numbers.  ...  We propose a new procedure that combines exact SMT solving over reals with approximate and sound affine and interval arithmetic.  ...  For such cases our approach estimates a sound upper bound on the total error of the computation.  ... 
doi:10.1145/2535838.2535874 dblp:conf/popl/DarulovaK14 fatcat:gsgwhva5gvbprfcf5fk3ajcyvy

Termination analysis of floating-point programs using parameterizable rational approximations

Fonenantsoa Maurica, Frédéric Mesnard, Étienne Payet
2016 Proceedings of the 31st Annual ACM Symposium on Applied Computing - SAC '16  
Our approach consists in translating the floating-point programs into rational ones by means of sound approximations. We approximate the floating-point expressions using piecewise linear functions.  ...  Analysis of floating-point programs is a topic that received an increasing attention the past few years. However, only very few works have been done regarding their termination analysis.  ...  However, the optimal bound for the absolute error has long been known and the optimal bound for the relative error has very recently been proved in [12] .  ... 
doi:10.1145/2851613.2851834 dblp:conf/sac/MauricaMP16 fatcat:pgyml6mvpneapm6bgqexe52ymy

Efficient decision procedure for non-linear arithmetic constraints using CORDIC

Malay K. Ganai, Franjo Ivancic
2009 2009 Formal Methods in Computer-Aided Design  
Research Use of floating-point library Speed is traded off with accuracy (acceptable) Verification Accuracy can not be traded off (undesirable) Use of floating-point library or precise arithmetic Introduction  ...  p = s • t can be cordic-translated as Mp = CordMult(s,Mt) p ≈ Mp °2 Et Bound on Err rel depends only on n.  ...  using CORDIC Discussed a sound and complete decision procedure based on DPLL-style interval search engine, with guiding mechanism Our formulation uses off-the-shelf SMT solvers for LRA, and therefore,  ... 
doi:10.1109/fmcad.2009.5351140 dblp:conf/fmcad/GanaiI09 fatcat:dnmmgye5abexbl6zk27tuxjr6q

Verifying bit-manipulations of floating-point

Wonyeol Lee, Rahul Sharma, Alex Aiken
2016 Proceedings of the 37th ACM SIGPLAN Conference on Programming Language Design and Implementation - PLDI 2016  
We apply our technique to Intel's implementations of transcendental functions and prove formal error bounds for these widely used routines.  ...  Reasoning about floating-point is difficult and becomes only more so if there is an interplay between floating-point and bit-level operations.  ...  Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.  ... 
doi:10.1145/2908080.2908107 dblp:conf/pldi/LeeSA16 fatcat:ug6bwa6dunf63o6smclxk4w26i

Exploiting Verified Neural Networks via Floating Point Numerical Error [article]

Kai Jia, Martin Rinard
2021 arXiv   pre-print
However, many verifiers inaccurately model floating point arithmetic but do not thoroughly discuss the consequences.  ...  We show that the negligence of floating point error leads to unsound verification that can be systematically exploited in practice.  ...  Acknowledgments We would like to thank Gagandeep Singh and Kai Xiao for providing invaluable suggestions on an early manuscript.  ... 
arXiv:2003.03021v4 fatcat:6c3gysniajdxpb4wmzumoecj7m

Verifying bit-manipulations of floating-point

Wonyeol Lee, Rahul Sharma, Alex Aiken
2016 SIGPLAN notices  
We apply our technique to Intel's implementations of transcendental functions and prove formal error bounds for these widely used routines.  ...  Reasoning about floating-point is difficult and becomes only more so if there is an interplay between floating-point and bit-level operations.  ...  Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.  ... 
doi:10.1145/2980983.2908107 fatcat:zydjmwnwmbc5ti6ntrvwe34jj4

Robustness Analysis of Floating-Point Programs by Self-Composition

Liqian Chen, Jiahong Jiang, Banghu Yin, Wei Dong, Ji Wang
2014 Journal of Applied Mathematics  
It converts floating-point expressions into linear expressions with interval coefficients in exact real arithmetic.  ...  To handle floating-point arithmetic, we employ an abstraction that encompasses the effect of rounding and that can encompass all rounding modes.  ...  First, let us consider the upper bound on rounding errors due to one floating-point operation.  ... 
doi:10.1155/2014/789213 fatcat:iqgzc23jtjdbxbsyl6nvphwbca

Towards a Compiler for Reals [article]

Eva Darulova, Viktor Kuncak
2016 arXiv   pre-print
We present a unified technique for floating-point and fixed-point arithmetic of various precisions which can handle nonlinear arithmetic, determine closed- form symbolic invariants for unbounded loops  ...  We present the source-to-source compiler Rosa which takes as input a real-valued program with error specifications and synthesizes code over an appropriate floating-point or fixed-point data type.  ...  -an approach for sound and fully automatic error estimation for nonlinear expressions for floating-point as well as fixed-point arithmetic of various precisions.  ... 
arXiv:1410.0198v3 fatcat:tuusqzdsurekzg5snpb2a5vftq
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