## Section: New Results

Keywords : interval arithmetic, Taylor models, formal proof, Coq, PVS.

### Certified Numerical Codes, Interval Arithmetic and Taylor Models

Participants : F. Cháves, M. Daumas, G. Melquiond, N. Revol.

#### PVS-Guaranteed Proofs using Interval Arithmetic

With C. Muñoz (National Institute of Aerospace), we have designed a set of tools [31] for mechanical reasoning using interval arithmetic in the PVS proof assistant [63] . The tools implement two techniques for reducing variable dependency: interval subdivisions and Taylor expansions. Although the tools are designed for the proof assistant system PVS, expertise on PVS is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal human-theorem prover interaction.

#### Formal Certification of Arithmetic Filters for Geometric Predicates

Floating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results [65] . Taking into account all the potential problems is a tedious and error-prone work if done by hand. In collaboration with S. Pion (Géométrica team), we have studied a floating-point implementation of the 2D orientation predicate, and we have put in evidence how a formal and partially automatized verification of this algorithm avoided many pitfalls [39] . The presented method is not limited to this particular filter, though; it can easily be used to produce correct semi-static floating-point filters of other geometric predicates [22] . These filters have been added to the latest release of the CGAL software http://www.cgal.org/ .

#### Formal Proofs on Taylor Models Arithmetic

Computing with a Taylor model amounts to determine a Taylor expansion of arbitrary order, often high, along with an interval which encloses Lagrange remainder, truncation error etc. The advantage of Taylor models, compared to usual interval arithmetic, is to reduce the decorrelation of variables.

Defining operations in a proof assistant is usually very simple. However, work is needed since we have to prove that the operators implement what they are supposed to implement. We have proven the arithmetic operations and a few elementary functions on Taylor models using the proof assistant system PVS.

#### Efficient and Accurate Computations on Taylor Models with Floating-Point Arithmetic

When arithmetic on Taylor models is implemented using floating-point arithmetic for the coefficients of the Taylor models, roundoff errors due to the representation and to previous computations are also accounted for in the interval remainder [9] . Using the properties of the IEEE-754 floating-point arithmetic and algorithms proposed by Rump [62] , accurate algorithms have been proposed for the arithmetic operations on Taylor models using floating-point arithmetic [45] .