Abstract. Interval-based methods are commonly used for computing numerical bounds on expressions and proving inequalities on real numbers. Yet they are hardly used in proof assistants, as the large amount of numerical computations they require keeps them out of reach from deductive proof processes. However, evaluating programs inside proofs is an efficient way for reducing the size of proof terms while performing numerous computations. This work shows how programs combining automatic differentiation with floating-point and interval arithmetic can be used as efficient yet certified solvers. They have been implemented in a library for the Coq proof system. This library provides tactics for proving inequalities on real-valued expressions. 1

In modern computers, the floating point unit is the part of the processor delivering the highest computing power and getting most attention from the design team. Performance of any multiple precision application will be dramatically enhanced by adequate use of floating point expansions. We present in this work three multiplication algorithms faster and more integrated than the stepwise algorithm proposed earlier. We have tested these new algorithms on an application that computes the determinant of a matrix. In the absence of overflow or underflow, the process is error free and possibly more efficient than its integer based counterpart. 1. Introduction Some groups have developed multiple precision packages based on the error-free integer arithmetic to compute some very precise quantities on a computer. Some of the most successful packages available today for a fast accurate scientific computation are GMP (Gnu Multiple Precision), Brent's MP [3] and Bailey's package 1 [1]. Expansion...

The process of proving some mathematical theorems can be greatly reduced by relying on numericallyintensive computations with a certified arithmetic. This article presents a formalization of floatingpoint arithmetic that makes it possible to efficiently compute inside the proofs of the Coq system. This certified library is a multi-radix and multi-precision implementation free from underflow and overflow. It provides the basic arithmetic operators and a few elementary functions. 1

We provide sufficient conditions that formally guarantee that the floating-point computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint numbers and rounding modes in the Program Verification System (PVS). Our work is based on a well-known formalization of floating-point arithmetic in the proof assistant Coq, where polynomial evaluation has been already studied. However, thanks to the powerful proof automation provided by PVS, the sufficient conditions proposed in our work are more general than the original ones.

Floating-point arithmetic is defined by the IEEE-754 standard and has often been formalized. We propose a new Coq formalization based on the bit-level representation of the standard and we prove strong links between this new formalization and a previous high-level one. In this process, we have defined functions for any rounding mode described by the standard. Our library can now be applied to certify both software and hardware. Developing results in those two dramatically different directions, like no other formal development so far, guarantees that nothing was forgotten or poorly specified in our formalization. It also lets us compare our work with the existing bit-level formalizations developed with other proof assistants.