A Purely Functional Library for Modular Arithmetic and Its Application to Certifying Large Prime Numbers

Grégoire, Benjamin; Théry, Laurent

doi:10.1007/11814771_36

Cited by 16 publications

(12 citation statements)

References 8 publications

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“…Other researchers have been using Coq for intensive numerical proofs before [1,10] and their library for big integers was integrated in the standard version. In particular, that library re-uses an algorithm for square roots that had been formally proved correct in [3].…”

Section: Using Big Integersmentioning

confidence: 99%

Fixed Precision Patterns for the Formal Verification of Mathematical Constant Approximations

Bertot

2015

Proceedings of the 2015 Conference on Certified Programs and Proofs

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We describe two approaches for the computation of mathematical constant approximations inside interactive theorem provers. These two approaches share the same basis of fixed point computation and differ only in the way the proofs of correctness of the approximations are described. The first approach performs interval computations, while the second approach relies on bounding errors, for example with the help of derivatives. As an illustration, we show how to describe good approximations of the logarithm function and we compute π to a precision of a million decimals inside the proof system, with a guarantee that all digits up to the millionth decimal are correct. All these experiments are performed with the Coq system, but most of the steps should apply to any interactive theorem prover.

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Section: Using Big Integersmentioning

confidence: 99%

Fixed Precision Patterns for the Formal Verification of Mathematical Constant Approximations

Bertot

2015

Proceedings of the 2015 Conference on Certified Programs and Proofs

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“…This program is executable but almost useless since it is based on a Peano representation of the natural numbers. Our next step was to derive an equivalent program using a more efficient representation of natural numbers, provided by the type BigN [24]. This code also receives some optimizations to implement faster operations of multiplications and divisions by powers of 2 and fast modular exponentiations.…”

Section: Doing An Iteration Is Performed Bymentioning

confidence: 99%

Distant Decimals of $$\pi $$ π : Formal Proofs of Some Algorithms Computing Them and Guarantees of Exact Computation

2017

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We describe how to compute very far decimals of π and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of π and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way. Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic-geometric means. All proofs and computations can be made inside the Coq system. We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation.

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“…This is the case for the above mentioned computable real libraries. Moreover, they are defined within Coq on top of the multiple-precision arithmetic library based on binary tree described in [20]. So only the machine modular arithmetic (32 or 64 bits depending on the machine) is used in the computations in Coq.…”

Section: The Coq Proof Assistantmentioning

confidence: 99%

Rigorous Polynomial Approximation Using Taylor Models in Coq

Brisebarre

Joldeş

Martin-Dorel

et al. 2012

Lecture Notes in Computer Science

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Abstract.One of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameterized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with interval coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models.Keywords: certified error bounds, Taylor models, Coq proof assistant, rigorous polynomial approximation Rigorous Approximation of Functions by PolynomialsIt is frequently useful to replace a given function of a real variable by a simpler function, such as a polynomial, chosen to have values close to those of the given function, since such an approximation may be more compact to represent and store but also more efficient to evaluate and manipulate. As long as evaluation is concerned, polynomial approximations are especially important. In general the basic functions that are implemented in hardware on a processor are limited to addition, subtraction, multiplication, and sometimes division. Moreover, division is significantly slower than multiplication. The only functions of one variable Brisebarre,Joldeş,Martin-Dorel,Mayero,Muller,Paşca,Rideau,Théry that one may evaluate using a bounded number of additions/subtractions, multiplications and comparisons are piecewise polynomials: hence, on such systems, polynomial approximations are not only a good choice for implementing more complex functions, they are frequently the only one that makes sense.Polynomial approximations for widely used functions used to be tabulated in handbooks [1]. Nowadays, most computer algebra systems provide routines for obtaining polynomial approximations of commonly used functions. However, when bounds for the approximation errors are available, they are not guaranteed to be accurate and are sometimes unreliable.Our goal is to provide efficient and quickly computable rigorous polynomial approximations, i.e., polynomial approximations for which (i) the provided error bound is tight and not underestimated, (ii) the framework is suitable for formal proof (indeed, the computations are done in a formal proof checker), while requiring computation times similar to those of a conventional C implementation. MotivationsMost numerical systems depend on standard functions like exp, sin, etc., which are implemented in libraries calle...

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A Purely Functional Library for Modular Arithmetic and Its Application to Certifying Large Prime Numbers

Cited by 16 publications

References 8 publications

Fixed Precision Patterns for the Formal Verification of Mathematical Constant Approximations

Fixed Precision Patterns for the Formal Verification of Mathematical Constant Approximations

Distant Decimals of $$\pi $$ π : Formal Proofs of Some Algorithms Computing Them and Guarantees of Exact Computation

Rigorous Polynomial Approximation Using Taylor Models in Coq

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