Scalable yet Rigorous Floating-Point Error Analysis

Das, Arnab; Briggs, Ian; Gopalakrishnan, Ganesh; Krishnamoorthy, Sriram; Panchekha, Pavel

doi:10.1109/sc41405.2020.00055

Cited by 24 publications

(10 citation statements)

References 31 publications

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“…Worst-case analysis of roundoff errors has been an active research area with numerous published approaches [12][13][14][15][16]18,22,33,35,37,38,46,47,50]. Our symbolic affine arithmetic used in PAF (Sect.…”

Section: Related Workmentioning

confidence: 99%

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

Constantinides

Dahlqvist

Rakamarić

et al. 2021

Computer Aided Verification

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We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are generally close to being uncorrelated with their generating distribution. 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. Our algorithm provides rigorous bounds to the output and error distributions of arithmetic expressions over random variables, evaluated in the presence of roundoff errors. It keeps track of complex dependencies between random variables using an SMT solver, and is capable of providing sound but tight probabilistic bounds to roundoff errors using symbolic affine arithmetic. We implemented the algorithm in the PAF tool, and evaluated it on FPBench, a standard benchmark suite for the analysis of roundoff errors. Our evaluation shows that PAF computes tighter bounds than current state-of-the-art on almost all benchmarks.

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Section: Related Workmentioning

confidence: 99%

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

Constantinides

Dahlqvist

Rakamarić

et al. 2021

Computer Aided Verification

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show abstract

“…This approach can be extended to and intraprocedural, modular abstract interpretation [17,36,48]. Error Taylor forms can also bound floating-point error [14,19,46] and generally providing tighter bounds than interval-based bounds, as does semidefinite programming [34]. Both error Taylor forms and semidefinite characterize error at a fixed input point and then apply sound global optimization, such as Gelpia [11], which uses interval arithmetic to compute sound bounds.…”

Section: Related Workmentioning

confidence: 99%

“…Rounding error has been responsible for scientific retractions [2,3], mispriced financial indices [20,38], miscounted votes [51], and wartime casualties [50]. Floating-point code thus needs careful validation; sound upper bounds [19,27,46], semidefinite optimization [34], input generation [15,26], and statistical methods [47] have all been proposed for such validation. Sampling-based error estimation, which can estimate typical, not worst-case, error is especially widely used [9,42,44,45].…”

Section: Introductionmentioning

confidence: 99%

An Interval Arithmetic for Robust Error Estimation

Flatt¹,

Panchekha²

2021

Preprint

Self Cite

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Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause domain errors. Others cause the algorithm to enter an infinite loop and fail to compute a ground truth. Plus, finding valid inputs is itself a challenge when invalid and unsamplable points make up the vast majority of the input space. These issues can make interval arithmetic brittle and temperamental.This paper introduces three extensions to interval arithmetic to address these challenges. Error intervals express rich notions of input validity and indicate whether all or some points in an interval violate implicit or explicit preconditions. Movability flags detect futile recomputations and prevent timeouts by indicating whether a higher-precision recomputation will yield a more accurate result. And input search restricts sampling to valid, samplable points, so they are easier to find. We compare these extensions to the state-of-the-art technical computing software Mathematica, and demonstrate that our extensions are able to resolve 60.3% more challenging inputs, return 10.2× fewer completely indeterminate results, and avoid 64 cases of fatal error.CCS Concepts: • Mathematics of computing → Arbitrary-precision arithmetic; • Computing methodologies → Representation of exact numbers.

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“…3 Error Taylor series thus provide an elegant, rigorous, and relatively accurate way to bound the error of an arbitrary floating-point expression. Recent work focuses on automating this idea [Solovyev et al 2018] and scaling it larger programs [Das et al 2020].…”

Section: Error Taylor Seriesmentioning

confidence: 99%

Faster Math Functions, Soundly

Briggs,

Panchekha

2021

Preprint

Self Cite

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Standard library implementations of functions like sin and exp optimize for accuracy, not speed, because they are intended for general-purpose use. But applications tolerate inaccuracy from cancellation, rounding error, and singularities-sometimes even very high error-and many application could tolerate error in function implementations as well. This raises an intriguing possibility: speeding up numerical code by tuning standard function implementations.This paper thus introduces OpTuner, an automatic method for selecting the best implementation of mathematical functions at each use site. OpTuner assembles dozens of implementations for the standard mathematical functions from across the speed-accuracy spectrum. OpTuner then uses error Taylor series and integer linear programming to compute optimal assignments of function implementation to use site and presents the user with a speed-accuracy Pareto curve they can use to speed up their code. In a case study on the POV-Ray ray tracer, OpTuner speeds up a critical computation, leading to a whole program speedup of 9% with no change in the program output (whereas human efforts result in slower code and lower-quality output). On a broader study of 37 standard benchmarks, OpTuner matches 216 implementations to 89 use sites and demonstrates speed-ups of 107% for negligible decreases in accuracy and of up to 438% for error-tolerant applications.

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Scalable yet Rigorous Floating-Point Error Analysis

Cited by 24 publications

References 31 publications

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations

An Interval Arithmetic for Robust Error Estimation

Faster Math Functions, Soundly

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