## Section: New Results

Keywords : elementary functions, libm, correct rounding, double precision, interval arithmetic, machine-assisted proofs, power function.

### Correct Rounding of Elementary Functions

Participants : N. Brisebarre, S. Chevillard, F. de Dinechin, C. Lauter, J.-M. Muller, N. Revol.

#### Bounds for correct rounding of the algebraic functions

In [13] , 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 them to get bounds on the accuracy with which intermediate calculations must be performed to correctly round these functions.

#### Exact cases of the power function

C. Lauter has described and implemented an efficient algorithm to
filter floating-point numbers (x, y) such that x^{y} is a
floating-point number or a mid-point between such numbers. It is
important to detect these cases, which would otherwise lead to an
infinite loop in a multilevel correctly rounded implementation of the
power function. This new algorithm is much faster in average than the
previous ones [56] .

#### Machine-assisted proofs

The proof of the correct rounding property is the main difficulty of the implementation of a correctly rounded function. F. de Dinechin, C. Lauter and G. Melquiond have published the methodology currently used in the CRlibm library, which uses the Gappa tool [34] . This methodology was lacking a certified bound on the approximation error of a function by a polynomial. As no existing tool could provide such a validated infinite norm, S. Chevillard and C. Lauter have developed such a tool [30] .

#### Interval elementary functions

Using CRlibm as a starting point, it is possible to derive elementary functions for double-precision interval arithmetic which 1/ are fully validated, 2/ return the smallest possible floating-point interval, and 3/ have a performance within a factor two of the scalar function, thanks to parallel evaluation of endpoints [33] . In collaboration with S. Maidanov from Intel, F. de Dinechin studied the implementation of such interval functions for the Itanium-2 processor [35] .

#### Applications of CRlibm

F. de Dinechin collaborated with Eric McIntosh and Franck Schmidt at CERN, to ensure portability of a chaotic computation distributed over an untrusted network of heterogeneous machines. Among other techniques, this application uses CRlibm [17] . It has been published in a conference on computational physics [36] . As these issues interest a large community of physicists, F. de Dinechin and G. Villard were invited to publish a survey on the subject [18] , following a presentation at the 2005 Workshop on Advanced Computing and Analysis Techniques in Physics Research.

#### Computation of the error functions in arbitrary precision with correct rounding

We continued our work on the evaluation of the error functions erf and erfc in arbitrary precision with correct rounding. We focused on the optimal strategy to adapt automatically the computing precision when the current precision does not suffice to round correctly the result [31] .