## Section: New Results

Keywords : elementary function, libm, double precision arithmetic, double-extended precision, correct rounding.

### Correct Rounding of Elementary Functions

Participants : N. Brisebarre, F. de Dinechin, C. Lauter, G. Melquiond, J.-M. Muller.

#### Double Precision Correctly Rounded Elementary Functions

F. de Dinechin, A. Ershov (Intel Corporation) and N. Gast (ÉNS) demonstrated that correct rounding of elementary function in double precision entailed no overhead in term of average case speed, worst-case speed, and memory consumption for processors with double-extended hardware [2] . C. Lauter then extended this result to all processors with double precision hardware, thanks to a redundant triple-double format [52] . A range of implementations of correctly-rounded logarithm functions will be published in a special issue of the Journal of Theoretical Informatics [18] .

As the main difficulty in such work is the proof of the correct rounding property, F. de Dinechin, G. Melquiond and C. Lauter developed and used the Gappa tool, a high-level proof assistant which helps building machine-checkable proofs of numerical properties [32] .

Meanwhile, the CRlibm library was developed further, with the addition of
`log2` ,
`log10` and
`asin` functions, a complete rewrite of
`exp` , a version of
`log` optimized for IA32 and IA64 instruction sets, the addition of a self-test, bits of Gappa proofs for various functions, experimental code for interval functions, and many other improvements. The stable version 0.8 was released in April, and the next stable version (0.11) is
scheduled for early 2006.

`crlibm` is used by the LHC@Home project at CERN, where it allows to manage the distribution of the computation on a network of heterogeneous computers. An article on the subject was submitted by F. de Dinechin, E. McIntosh (CERN) and F. Schmidt (CERN).

F. de Dinechin and G. Villard were invited by nuclear physicists to present a survey on the subject of quadruple precision at the Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT'05) [19] .

#### Correct Rounding of Algebraic Functions

In [15] , N. Brisebarre and J.-M. Muller explicit the link between the computer arithmetic problem of providing correctly rounded algebraic functions and some diophantine approximation issues. This allows to get bounds on the accuracy with which intermediate calculations must be performed to correctly round these functions.

#### Publication of Previous Works

A former work on range reduction has been published in [14] . The proposed algorithm is fast for most cases and accurate over the full range. Furthermore, the statistical distribution of these cases has been determined.

In [30] , S. Chevillard and N. Revol present an algorithm for the evaluation of the error functions erf and erfc in arbitrary precision with correct rounding.