What every computer scientist should know about floating-point arithmetic
ACM Computing Surveys (CSUR)
Introduction to HOL: a theorem proving environment for higher order logic
Introduction to HOL: a theorem proving environment for higher order logic
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Convergence Estimates for the Distribution of Trailing Digits
Journal of the ACM (JACM)
Improved Trailing Digits Estimates Applied to Optimal Computer Arithmetic
Journal of the ACM (JACM)
Computer-Aided Reasoning: An Approach
Computer-Aided Reasoning: An Approach
Formal Verification of Floating Point Trigonometric Functions
FMCAD '00 Proceedings of the Third International Conference on Formal Methods in Computer-Aided Design
PVS: A Prototype Verification System
CADE-11 Proceedings of the 11th International Conference on Automated Deduction: Automated Deduction
Representable Correcting Terms for Possibly Underflowing Floating Point Operations
ARITH '03 Proceedings of the 16th IEEE Symposium on Computer Arithmetic (ARITH-16'03)
Provably faithful evaluation of polynomials
Proceedings of the 2006 ACM symposium on Applied computing
Stochastic Formal Methods: An Application to Accuracy of Numeric Software
HICSS '07 Proceedings of the 40th Annual Hawaii International Conference on System Sciences
Verified Real Number Calculations: A Library for Interval Arithmetic
IEEE Transactions on Computers
Certification of bounds on expressions involving rounded operators
ACM Transactions on Mathematical Software (TOMS)
Proofs of randomized algorithms in CoQ
MPC'06 Proceedings of the 8th international conference on Mathematics of Program Construction
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We provide bounds on the probability that accumulated errors were never above a given threshold on numerical algorithms. Such algorithms are used, for example, in aircraft and nuclear power plants. This report contains simple formulas based on Lévy's, Markov's and Hoeffding's inequalities and it presents a formal theory of random variables with a special focus on producing concrete results. We select three very common applications that cover the common practices of systems that evolve for a long time. We compute the number of bits that remain continuously significant in the first two applications with a probability of failure around one out of a billion, where worst case analysis considers that no significant bit remains. We are using PVS as such formal tools force explicit statement of all hypotheses and prevent incorrect uses of theorems.