Fast evaluation of elementary mathematical functions with correctly rounded last bit
ACM Transactions on Mathematical Software (TOMS)
GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable
ACM Transactions on Mathematical Software (TOMS)
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
Constructive Reals in Coq: Axioms and Categoricity
TYPES '00 Selected papers from the International Workshop on Types for Proofs and Programs
Worst Cases for Correct Rounding of the Elementary Functions in Double Precision
ARITH '01 Proceedings of the 15th IEEE Symposium on Computer Arithmetic
Modern Computer Algebra
Interactive Theorem Proving and Program Development
Interactive Theorem Proving and Program Development
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
Assisted verification of elementary functions using Gappa
Proceedings of the 2006 ACM symposium on Applied computing
Computing machine-efficient polynomial approximations
ACM Transactions on Mathematical Software (TOMS)
Elementary Functions: Algorithms and Implementation
Elementary Functions: Algorithms and Implementation
Efficient polynomial L-approximations
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
On Taylor Model Based Integration of ODEs
SIAM Journal on Numerical Analysis
Proving Bounds on Real-Valued Functions with Computations
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
Certified Exact Transcendental Real Number Computation in Coq
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
Rigorous global search using taylor models
Proceedings of the 2009 conference on Symbolic numeric computation
Sollya: an environment for the development of numerical codes
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
The dynamic dictionary of mathematical functions (DDMF)
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Efficient and accurate computation of upper bounds of approximation errors
Theoretical Computer Science
Computer certified efficient exact reals in Coq
MKM'11 Proceedings of the 18th Calculemus and 10th international conference on Intelligent computer mathematics
Flocq: A Unified Library for Proving Floating-Point Algorithms in Coq
ARITH '11 Proceedings of the 2011 IEEE 20th Symposium on Computer Arithmetic
Formal global optimisation with taylor models
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Extending coq with imperative features and its application to SAT verification
ITP'10 Proceedings of the First international conference on Interactive Theorem Proving
Full reduction at full throttle
CPP'11 Proceedings of the First international conference on Certified Programs and Proofs
Formalization of Bernstein Polynomials and Applications to Global Optimization
Journal of Automated Reasoning
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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.