## Section: New Results

Keywords : interval arithmetic, Taylor models, formal proof, Isabelle, PVS, implementation, arbitrary precision.

### Interval Arithmetic, Taylor Models and Certification

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

#### Interval arithmetic, arbitrary precision, validated infinite norm

Results computed using interval arithmetic and its variants (Taylor models, affine arithmetic...) are enclosures of the real, unknown sought value. This guarantee on the result is needed in the development of the CRlibm library [17] , where an upper bound of the approximation error is needed, in order to guarantee the property of correct rounding. A. Touhami was hired as a post-doc and his task was to implement a validated infinite norm on one variable, using arbitrary precision interval arithmetic. Unfortunately (for us), he rapidly got a permanent position as assistant professor in Morocco and left us after only two months.

#### 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 variable dependency.

To get a further level of certification, we have implemented Taylor models in a proof assistant that formally checks the enclosure property of the result along with computing it. We have enriched our library of arithmetic operations on Taylor models using the proof assistant system PVS, indeed we have proved several elementary functions [29] . This implied to prove several theorems of real analysis prior to the elementary functions.

In collaboration with Martin Hofmann (Maximilian-Ludwig Universität, Munich), we have implemented interval arithmetic in another proof assistant, Isabelle. This is a preliminary to the implementation of Taylor models.

#### Accurate implementations of 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.

The properties of IEEE-754 floating-point arithmetic enables us to compute with "double-double", roughly speaking as if we had twice the computing precision. We propose to take benefit of these properties to implement very accurately Taylor models using floating-point arithmetic [47] .