How do you compute the midpoint of an interval?

Goualard, Frédéric

doi:10.1145/2493882

Cited by 14 publications

(12 citation statements)

References 15 publications

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“…Unfortunately, the finitary nature of floating-point, along with its uneven distribution of representable numbers introduces round-off errors, as well as does not preserve many familiar laws (e.g., associativity of +) [22]. This mismatch often necessitates re-verification using tools that precisely compute round-off error bounds (e.g., as illustrated in [21]).…”

Section: Introductionmentioning

confidence: 99%

“…The IEEE 754 standard [28] concisely formalized in (e.g.) [22] defines a binary floating-point number as a triple of sign (0 or 1), significand, and exponent, i.e., (sgn, sig, exp), with numerical value (−1) sgn × sig × 2 exp . The standard defines three general binary formats with sizes of 32, 64, and 128 bits, varying in constraints on the sizes of sig and exp.…”

Section: Introductionmentioning

confidence: 99%

“…Common rounding operators are rounding to nearest (ties to even), toward zero, and toward ±∞. A simple model of rounding is given by the following formula [22] rnd(x) = x(1 + e) + d…”

Section: Introductionmentioning

confidence: 99%

“…For instance, if op is − or + then d = 0 [22]. Also, if op is × and one of the arguments is a nonnegative power of two then e = d = 0.…”

Section: Introductionmentioning

confidence: 99%

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Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Solovyev

Jacobsen

Rakamarić

et al. 2015

FM 2015: Formal Methods

123

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Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation. Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor ExpansionsAlexey Solovyev, Charles Jacobsen, Zvonimir Rakamarić, and Ganesh Gopalakrishnan School of Computing, University of Utah, Salt Lake City, UT 84112, USA {monad,charlesj,zvonimir,ganesh}@cs.utah.edu Abstract. Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Common rounding operators are rounding to nearest (ties to even), toward zero, and toward ±∞. A simple model of rounding is given by the following formula [22] rnd(x) = x(1 + e) + d…”

Section: Introductionmentioning

confidence: 99%

“…For instance, if op is − or + then d = 0 [22]. Also, if op is × and one of the arguments is a nonnegative power of two then e = d = 0.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Solovyev

Jacobsen

Rakamarić

et al. 2015

FM 2015: Formal Methods

123

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show abstract

“…round to the nearest, ties to even) defined in the IEEE 754 standard [1]. A rounding operator can be modeled as follows [15]:…”

Section: Quantization Error Analysis and Model Extractionmentioning

confidence: 99%

Automatic Verification of Finite Precision Implementations of Linear Controllers

Park

Pajić

Sokolsky

et al. 2017

Tools and Algorithms for the Construction and Analysis of Systems

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We consider the problem of verifying finite precision implementation of linear time-invariant controllers against mathematical specifications. A specification may have multiple correct implementations which are different from each other in controller state representation, but equivalent from a perspective of input-output behavior (e.g., due to optimization in a code generator). The implementations may use finite precision computations (e.g. floating-point arithmetic) which cause quantization (i.e., roundoff) errors. To address these challenges, we first extract a controller's mathematical model from the implementation via symbolic execution and floating-point error analysis, and then check approximate input-output equivalence between the extracted model and the specification by similarity checking. We show how to automatically verify the correctness of floating-point controller implementation in C language using the combination of techniques such as symbolic execution and convex optimization problem solving. We demonstrate the scalability of our approach through evaluation with randomly generated controller specifications of realistic size. Disciplines Computer Engineering | Computer Sciences AbstractWe consider the problem of verifying finite precision implementation of linear time-invariant controllers against mathematical specifications. A specification may have multiple correct implementations which are different from each other in controller state representation, but equivalent from a perspective of input-output behavior (e.g., due to optimization in a code generator). The implementations may use finite precision computations (e.g. floating-point arithmetic) which cause quantization (i.e., roundoff) errors. To address these challenges, we first extract a controller's mathematical model from the implementation via symbolic execution and floating-point error analysis, and then check approximate input-output equivalence between the extracted model and the specification by similarity checking. We show how to automatically verify the correctness of floating-point controller implementation in C language using the combination of techniques such as symbolic execution and convex optimization problem solving. We demonstrate the scalability of our approach through evaluation with randomly generated controller specifications of realistic size.

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A framework to test interval arithmetic libraries and their IEEE 1788‐2015 compliance

Benet,

Ferranti,

Revol

2023

Concurrency and Computation

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SummaryAs developers of libraries implementing interval arithmetic, we faced the same difficulties when it comes to testing our libraries. What must be tested? How can we devise relevant test cases for unit testing? How can we ensure a high (and possibly 100%) test coverage? Before considering these questions, we briefly recall the main features of interval arithmetic and of the IEEE 1788‐2015 standard for interval arithmetic. After listing the different aspects that, in our opinion, must be tested, we contribute a first step towards offering a test suite for an interval arithmetic library. First we define a format that enables the exchange of test cases, so that they can be read and tried easily. Then we offer a first set of test cases, for a selected set of mathematical functions. Next, we examine how the Julia interval arithmetic library, IntervalArithmetic.jl, actually performs to these tests. As this is an ongoing work, we list extra tests that we deem important to perform.

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How do you compute the midpoint of an interval?

Cited by 14 publications

References 15 publications

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Automatic Verification of Finite Precision Implementations of Linear Controllers

A framework to test interval arithmetic libraries and their IEEE 1788‐2015 compliance

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