## Section: New Results

Keywords : floating-point arithmetic, polynomial approximation.

### Efficient Polynomial Approximation

Participants : N. Brisebarre, S. Chevillard, J.-M. Muller, S. Torres.

We have developed methods for generating the best, or at least very
good, polynomial approximations (with respect to the infinite norm
or the L^{2} norm)
to functions, among polynomials whose coefficients
follow size constraints (e.g., the degree-i coefficient has at
most m_{i} fractional bits, or it has a given, fixed, value).
Regarding to the infinite norm on a given compact interval,
one of these methods, due to N. Brisebarre, J.-M. Muller and
A. Tisserand
(CNRS, LIRMM, U. Montpellier) reduces to scanning the integer points in a
polytope [14] , another one, due to
N. Brisebarre and
S. Chevillard, uses the
LLL lattice reduction algorithm [49] .
About the L^{2} norm, N. Brisebarre and G. Hanrot (LORIA, INRIA
Lorraine)
started a complete study of the
problem [50] and proposed theoretical and algorithmic
results that make it possible to get optimal approximants.

Potential applications to hardwired operators have been studied [15] by N. Brisebarre, J.-M. Muller, A. Tisserand and S. Torres. We are developing a library for generating such polynomial approximations (see Section 5.5 ).