Overall Objectives
Research Program
Application Domains
Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
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Section: Research Program

Function evaluation

Towards automatic design of function programs or circuits

Concerning function evaluation, what we have successfully automated so far is program generation for statically defined elementary functions of one real variable. These techniques will certainly need refining as we try to apply them to more functions, in particular to special functions, or to compound functions. To apply these techniques to arbitrary code at compile time further leads to several challenges. The first one is to identify a relevant function in a program, along with the useful information that will allow to implement it efficiently: range of the input values, needed output accuracy and/or rounding mode, etc. It requires interaction with compilation people working on classical compilers.

A second challenge then is to analyze such a function automatically, which typically implies the following tasks:

This general automation process will be progressively set up and refined by working on concrete implementations of a significant set of operators, hopefully driven by applications and through industrial collaborations: all the C99 elementary functions, other functions such as the inverse cumulative distribution functions used in random number generators, variations around the Euclidean norm such as x/x2+y2, complex arithmetic, interval operators, FFTs, etc.

Most of the software we design brings in some floating-point functionality. We currently target two main types of processors:

Mathematical tools for function evaluation

Challenges in function approximation

The algorithms currently implemented in the Sollya toolbox (see Section  5.1 ) provide, for functions of one variable, an end-to-end solution for finding near-optimal polynomial approximations whose coefficients are machine numbers. This includes a validated tight bound of the approximation error. We now want to generalize them in three main directions:

In these three directions, we eventually want to constraint coefficients to be machine numbers.

Approximation for digital filters

A digital filter implements a given transfer function, either as a polynomial (finite impulse response, or FIR filter) or as a rational function (infinite impulse response, or IIR filter). Classical techniques and toolboxes exist for computing such filters, but they amount to computing infinite precision solutions and rounding them. This rounding turns out to be numerically unstable in some situations. We intend to study to what extent an improved rounding procedure might improve the filter, in terms of efficiency (software implementation) or size (hardware implementation).

Collaborations on this subject have begun with researchers from the signal processing community.

Challenges in the search for hard-to-round cases

For a given function f and a given floating-point format, the “hardest to round” (HR) points are the floating-point numbers x such that f(x) is nearest to a value where the rounding function changes. Knowing these HR points makes it possible to design efficient programs that, given a floating-point number y, always return the floating-point number nearest f(y). Such programs are called “correctly rounded” implementations of f.

We have obtained and published HR points in the binary64 (“double precision”) and decimal64 formats of the IEEE 754-2008 standard for the most important functions of the standard mathematical library. However, in this line of research, we now have to tackle difficult challenges. First, our current methods for finding HR-points cannot be used in big precisions such as the binary128 (“quad precision”) and decimal128 (128-bit decimal) formats of the IEEE 754-2008 standard. Also, the processes that generate our HR cases are based on complex and very long calculations (years of cumulated CPU time) that inevitably cast some doubt on the correctness on their results. Hence, we reconsider the methods used to get HR points, and mainly focus on three aspects:

These three aspects have been at the core of our TaMaDi ANR project (see Section  8.1.2 ): the project brought significant progress, but there is still much to be done.