Section: New Results

Correct Rounding of Elementary Functions

Participants : Florent de Dinechin, Vincent Lefèvre, Jean-Michel Muller, Bogdan Pasca, Serge Torres.

FPGA Acceleration of the Search For Hardest-to-Round Cases

The IEEE 754-2008 standard for floating-point arithmetic recommends (yet does not dictate) that some elementary functions should be correctly rounded. That is, given a rounding function $\circ$, (e.g., round to nearest even, or round to $\pm \infty$), when evaluating function $f$ at the floating-point number $x$, the system should always return $\circ \left(f\right(x\left)\right)$.

Building a fast correctly rounded library for some target floating-point (FP) format requires preliminarily solving a problem called the table maker's dilemma. This requires very large computations which may use environments and formats totally different from the target environment and format. F. de Dinechin, V. Lefèvre, J.-M. Muller, B. Pasca and A. Plesco suggest performing these computations on an FPGA. Their paper [45] won the best paper award at the ASAP2011 conference.

Hierarchical Polynomial Approximation of a Function by Polynomials

Algorithms used to search for the hardest-to-round cases of a function requires the approximation of the function by small-degree polynomials on small intervals. This can be done efficiently by a hierarchical polynomial approximation. Work is being done to improve this method by replacing interval arithmetic (as partly used in the current tools) by static error bounds. This will allow us to better control the precision needed to compute the coefficient of the polynomials. The implementation will also be simpler.