3. arénaire - computer arithmetic · initially, the team was mainly focused on hardware-oriented...

3 Areacutenaire - Computer Arithmetic

Team AREacuteNAIREScientific leaders Florent DE DINECHIN and Gilles VILLARD

Evaluation 2006-2009

Web site httpwwwens-lyonfrLIPArenaireParent Organizations CNRS Eacutecole Normale Supeacuterieure de Lyon INRIA Universiteacute de LyonTeam History This group started in 1998 as a joint CNRS-ENS Lyon-INRIA project led by J-M Muller with three permanentmembers In 2004 Gilles Villard became head of the teamInitially the team was mainly focused on hardware-oriented algorithms for computer arithmetic and on elementary function evaluationBeyond several arrivals and departures (the most recent ones are listed in Section 1) many events progressively influenced the scientificorientation of the team

bull The team developed an expertise on floating-point software that goes beyond elementary functions addressing more generallythe issue of efficient high accuracy computing

bull Interval arithmetic became both a tool used pervasively by the team and a subject of research

bull A fruitful collaboration with the Compilation Expertise Center of STMicroelectronics brought us new problems (integer-basedfloating-point arithmetic division by a constant etc)

bull With the increasing complexity of the arithmetic algorithms and the target technologies the research focus shifted from designingoperators to automating or assisting this design

bull On the hardware side the target technology evolved from integrated circuits to field-programmable gate arrays with new chal-lenges and new opportunities

bull The team acquired strong expertise in euclidean lattices and linear algebra on one side because operations on these objects arethe next frontier for our arithmetic expertise and on the other side because they provide solutions to existing arithmetic problems(correct rounding polynomial approximation)

bull Emerging arithmetic problems such as the generation of polynomial approximations with constrained coefficients or the designof specific operators for cryptographic hardware clearly required skills in number theory

31 Team Composition

Current team members

Permanent ResearchersName First name Function Institution Arrival date

BRISEBARRE Nicolas Research Scientist CNRS 01092007DE DINECHIN Florent Associate Professor EacuteNS Lyon 01091998HANROT Guillaume Professor EacuteNS Lyon 01092009JEANNEROD Claude-Pierre Research Scientist INRIA 01022002LEFEgraveVRE Vincent Research Scientist INRIA 01092006LOUVET Nicolas Associate Professor UCBL 01112008MULLER Jean-Michel Senior Research Scientist CNRS 15031989REVOL Nathalie Research Scientist INRIA 01092002VILLARD Gilles Senior Research Scientist CNRS 01122000

Post-docs engineers and visitorsName First name Function and of time Institution Arrival dateFAGBOHOUN Christman Engineer INRIA 20042009MAYERO Micaela Research Scientist Univ Paris 13 secondment to INRIA 01092009NOVOCIN Andrew Engineer INRIA 01092009TAKEUGMING Honoreacute Engineer EacuteNS LyonANR TCHATER 01102008THEacuteVENY Philippe Engineer INRIA 01102009

27

28 Part 3 Areacutenaire

TORRES Serge Engineer (40) MESR 01092001VAacuteZQUEZ AacuteLVAREZ Aacutelvaro Post-doc INRIA 01102009

Doctoral Students

Name First name InstitutionDate of first registrationas doctoral student

JOLDES Mioara Reacutegion Rhocircne-Alpes 01092008JOURDAN LU Jingyan CIFRE 01032009MARTIN-DOREL Eacuterik MESR 01092009MOREL Ivan EacuteNS Lyon 01092007MOUILLERON Christophe EacuteNS Lyon 01092008NGUYEN Hong Diep INRIA CORDI 01102007PANHALEUX Adrien EacuteNS Lyon 01092008PASCA Bogdan MESR 01092008PUJOL Xavier EacuteNS Lyon 01092009REVY Guillaume MESR 01102006

Past team members

Past Members Oct 2005-Oct 2009

Name First name PositionParentInstitution Arrival Departure Current position

DAUMAS Marc Research Scientist CNRS 01101998 30112005 Professor Perpignan University

STEHLEacute Damien Research Scientist CNRS 01102006 15072008CNRS scientist seconded tothe Universities of Sydney andMacquarie until 15072010

Past Doctoral students

Name First nameFirstregistration

Departure InstitutionDoctoralschool

SupervisorCurrentposition

CHAacuteVESALONSO

FranciscoJoseacute

01112004 31102007 EacuteNS Lyon (MarieCurie grant)

MathIFM DaumasN Revol

Prof U LosAndes BogotaColombia

CHEVILLARD Sylvain 01092006 01092009 EacuteNS Lyon MathIFN BrisebarreJ-M Muller

Post-DocINRIA Lorraine

DETREY Jeacutereacutemie 01092003 15012007 EacuteNS Lyon MathIFF de DinechinJ-M Muller

CR2 INRIA

LAUTER Christoph 01092005 17102008 EacuteNS Lyon MathIFF de DinechinJ-M Muller

Intel PortlandOr

MELQUIOND Guillaume 01092003 21112006 EacuteNS Lyon MathIF M Daumas CR2 INRIA

MICHARD Romain 01092004 25062008 EacuteNS LyonINRIA MathIFJ-M MullerA Tisserand

PostDoc INSALyon

RAINASaurabhKumar

01102003 30092006 EacuteNS Lyon (ReacutegionRhocircne-Alpes grant)

MathIFC-P JeannerodJ-M MullerA Tisserand

Ass Pr at ITSEng College(Greater NoidaIndia)

VEYRAT-CHARVILLON

Nicolas 01092004 28062007 EacuteNS Lyon MathIFJ-M MullerA Tisserand

PostDoc at UC Louvain

Past post-doctoral researchers engineers and visitors (greater than or equal to 3 months)

Name First name Function Arrival Departure Home Institution(if appropriate)

Current position

BECHETOILLE Eacutedouard Engineer 01102005 30112006 INRIACNRS Research Engineer

(IPNL)

BEUCHAT Jean-Luc Post-doc 01112001 30112005Swiss National ScienceFoundation

Associate Professor atTsukuba University (Japan)

GIESBRECHT Mark Visitor 01122005 31062006University of WaterlooCanada

Ass Professor (WaterlooUniversity)

JOURDAN Nicolas Engineer 23072007 29022008 INRIAIn charge of ind relations

and techno transfer at IN-RIA Rhocircne-Alpes

sect32 Computer Arithmetic 29

LOUVET Nicolas Post-doc 01122007 30102008 INRIA Associate Professor UCBL

TISSERAND Arnaud Visitor 01102005 31082006 CNRSCR1 CNRS IRISA (Lan-

nion)

TOLI Ilia Post-doc 01052005 30042006 INRIASenior Lecturer at SuffolkUniversity (USA)

Evolution of the teamIn October 2005 there were 7 permanent members 2 post-docs 2 engineers and 7 PhD students in Areacutenaire We are now 9 permanentmembers 1 associate professor seconded to INRIA 44 engineers 1 post-doc and 10 PhD students There has been a significantturn-over among the researchers departure of Marc Daumas and Jean-Luc Beuchat Arrival then (hopefully momentarily) departure ofDamien Stehleacute arrival of Vincent Lefegravevre recruitment of Nicolas Brisebarre and Nicolas LouvetWe are proud to notice a good success of our former members

bull Eacutedouard Bechetoille (former software development engineer) is now a CNRS research engineer at IPNL (Institute for NuclearPhysics Lyon)

bull Jean-Luc Beuchat (former postdoctoral fellow) is now associate professor at Tsukuba University (Japan)

bull Sylvie Boldo Jeacutereacutemie Detrey and Guillaume Melquiond (former PhD students) are now ldquochargeacutes de rechercherdquo (that is fulltime researchers) at INRIA (in Orsay Nancy and Orsay respectively)

bull Francisco Joseacute Chaacuteves Alonso (former PhD student) is now professor at U Los Andes Bogota Colombia

bull Marc Daumas (former research scientist ndash ldquoCRrdquo at CNRS) is now full professor at Perpignan University

bull Pascal Giorgi (former PhD student) is now ldquomaicirctre de confeacuterencesrdquo (associate professor) in Montpellier University

bull Nicolas Jourdan (former software development engineer) is now in charge of industrial relations and technology transfer atINRIA Rhocircne-Alpes

bull Saurabh Raina (former PhD student) is now associate professor at the ITS Engineering College Greater Nodia India

bull Christoph Lauter (former PhD student) is now engineer at INTEL Portland (US) He is designing software support for floating-point arithmetic

Our former PhD students Sylvain Chevillard Romain Michard and Nicolas Veyrat-Charvillon currently have post-doc positions re-spectively with Projet CACAO of the LORIA Lab in Nancy with Projet Ares of the CITI Lab at INSA Lyon and with the Crypto Groupof Universiteacute Catholique de LouvainWe also had good success in recruiting people in the team

bull Damien Stehleacute (research scientist mdash CR2 CNRS) was recruited in 2006 He was the first ranked candidate in the national CNRSrecruitment for computer scientists

bull Nicolas Brisebarre (research scientist mdash CR1 CNRS) was recruited in 2007

bull Nicolas Louvet (associate professor mdashldquomaicirctre de confeacuterencesrdquo Universiteacute Lyon 1) was recruited in 2008

bull a newly-recruited ENS Lyon full professor Guillaume Hanrot will join the team by October 1 2009

Vincent Lefegravevre (research scientist mdash CR1 INRIA) moved from projet CACAO (LORIA Nancy) in September 2006

32 Executive summary

Keywords computer arithmetic floating-point arithmetic IEEE 754 arithmetic fixed-point arithmetic interval arithmetic finite fieldarithmetic integer arithmetic multiple precision arbitrary precision reliable computation rounding-error analysis numericalvalidation code certification formal proof software synthesis approximation theory elementary function global optimizationlinear algebra euclidean lattice cryptology pairing computer algebra digital signal processing VLSI circuit FPGA circuitVLIW processor embedded system

Research area The Areacutenaire project-team aims at elaborating and consolidating knowledge and tools in the field of computer arith-metic Computer arithmetic studies how to represent numbers in a computer and how to implement and manipulate such rep-resentations efficiently and accurately in hardware or in software it studies also the question of the adequation between suchnumber representations and numerical algorithmsWithin Areacutenaire we cover all of these aspects First we study basic arithmetic operators in various number systems from theiralgorithmic design to their optimized implementation (in hardware or software and on general-purpose or application-specifictargets) Second we study the interplay between arithmetic operators and the algorithmics of several application domainslike function evaluation polynomial approximation linear algebra lattice basis reduction cryptology and signal processingHere the question is twofold How may arithmetic choices impact such application domains in terms of complexity practicalefficiency accuracy or reliability Conversely to which extent improving algorithms and implementations within those domainsand within scientific computing in general can be useful for enhancing arithmetic that is for making it faster andor morereliable andor more accurate etc Finally those two research areas are becoming increasingly tied to a third one that ofcertifying numerical computations Within this third field we work not only on traditional paper-and-pencil error analysis butalso and more and more on automatic error analysis and formal methods


Main goals Through computer arithmetic we strive to improve the quality of computation at large both in hardware and in softwareWe look for improvements in performance (frequency or throughput for hardware cycle count for software) and cost (area orpower consumption for hardware code size for software memory consumption for both) In addition we look for improvementsin accuracy which is more specific to computer arithmetic Third we strive to increase confidence in the computed resultconfidence that each operator is well specified and behaves as expected confidence that the composition of such operators in aprogram does not entail unexpected deviations from the result to be computed Last but not least by demonstrating the practicalfeasibility of getting high-quality numerical answers we aim at fostering and being part of standardization actions in the fieldsof computer arithmetic and numerical computing

Methodology A specificity of computer arithmetic is that its relatively wide spectrum requires many areas of expertise from hardwaredesign and expert programming to formal proving and number theory This is why Areacutenaire gathers people from rather differentdomains whose common skills are arithmetic algorithmsOur targeted numerical applications benefit from any progress we make at the basic operator level In turn our optimized designsand implementations of basic operators are nowadays made possible only through our expertise and state-of-the-art softwaredevelopments in domains like linear algebra or lattice basis reduction (which offer appropriate modeling tools) Consequentlywe also study such domains for themselves with an emphasis on algorithmic complexity and the design of high-performanceand reliable softwareTo achieve our general goal of high numerical quality either at the basic operator level or at the application level we need acombination of rigorous specifications and certification tools A first approach to this is a careful study of validated arithmeticsand especially interval arithmetic as well as the possibility of computing exact error bounds or even exact results in floating-pointarithmetic Another complementary approach is to rely more systematically on formal verification and to use proof assistantslike Coq in a way that is well-suited for our purposeFinally a very significant change occurred during the recent years while a few years ago our main realizations besides algo-rithms were expert handwritten libraries (in hardware or software and offering operators ranging from the most basic ones toelementary functions and linear algebra kernels) we are now more and more interested in automating the development of suchlibraries We now produce software tools that assist us in designing and proving optimized operator libraries This requires newmathematical and algorithmic expertiseThis evolution from hand writing to synthesis is also pushing us towards the compilation community If an operator can bedesigned optimized and proven automatically it is natural that this task eventually ends up being part of the compilationprocess This explains why our major industrial partner so far is STMicroelectronicsrsquo Compilation Expertise Center in Grenoble

Highlights (1 page max)

bull Standardization of arithmetics

ndash The new revision of the IEEE Standard for floating-point arithmetic (IEEE 754-2008 available at httpieeexploreieeeorgservletopacpunumber=4610933) recommends that some elementary func-tions (sine exponential ) should be correctly rounded This is a direct consequence of the effort led by Areacutenairesince 1995 on correct rounding of elementary algebraic and transcendental functions Our ldquoworst casesrdquo for correctrounding as well as our CRlibm library played the most important role in that decision and among the 22 biblio-graphic references cited by the standard 7 are by Areacutenaire members CRlibm is cited by members of the IEEE 754revision committee as the proof that elementary functions can be correctly rounded at reasonable cost

ndash Interval arithmetic has now reached a stability and maturity that should make its standardization possible NathalieRevol has prepared the creation of and chairs the IEEE Interval Standard Working Group - P1788

bull Reference books

ndash Jean-Michel Muller published a new edition of his reference book Elementary functions algorithms and implemen-tation

ndash Our Handbook of Floating-Point Arithmetic (Birkhaumluser Boston November 2009 600 pages ISBN 78-0-8176-4704-9) gathers years of common work on IEEE floating-point arithmetic

bull Reference software

ndash Our algorithms for emulating IEEE binary floating-point arithmetic on integer processors (FLIP library) are includedin the STMicroelectronics C compilation toolset of the ST200 processor family

ndash The FPLibrary project was first to demonstrate floating-point elementary functions for FPGAs offering significantspeed-ups with respect to processor implementations These functions are still without equivalent and have beenwidely used worldwide for high-performance computing

ndash Gappa a state-of-the-art tool for certifying error bounds in rounded arithmetics originated in Areacutenaire and is stillbeing developed in close collaboration with us

ndash Sollya is a reference tool for manipulating numerical functions safely It has been the first environment that offersboth numerical safety and automation for the design of elementary function libraries It possesses unique features forpolynomial approximation and validated supremum norms

ndash The fplll library is now the reference for LLL reduction it is used or variants of it are used in PariGP Magmaand Sage the next release of NTL could be inspired from it It is also used to generate polynomial approximants inSollya


bull Other key results

ndash Ivan Morel Xavier Pujol Damien Stehleacute and Gilles Villard obtained key results on the use of floating-point arith-metic within lattice reduction algorithms including the Lenstra-Lenstra-Lovaacutesz algorithm and the enumeration ofthe shortest lattice vectors of a lattice This involves a precise understanding of the numerical properties of theGram-Schmidt orthogonalization algorithms and allows one to obtain guaranteed practical algorithms with betterasymptotic complexities

ndash Guillaume Hanrot and Damien Stehleacute completed a full worst-case analysis of Kannanrsquos algorithm for deterministi-cally computing a shortest non-zero lattice vector They lowered the best previously known complexity upper boundand built inputs for which that complexity bound is essentially reached

ndash The COSY software (developed at Michigan State University at East Lansing) implements Taylor models arithmeticusing floating-point arithmetic The proof that roundoff errors are correctly accounted for is given in [43] putting anend to a controversy in the interval arithmetic community

33 Research activities

331 Number Systems

List of participants Florent de Dinechin (MCF) Jeacutereacutemie Detrey (PhD) Jean-Michel Muller (DR)

Keywords number system redundant number systems carry propagations floating-point logarithmic number system (LNS)

Scientific issues goals and positioning of the team The study of the number systems and more generally of numerical data rep-resentations is an important topic in the project Typical examples are the redundant number systems used inside multipliersand dividers alternatives to floating-point representation for special purpose systems finite field representations with a strongimpact on cryptographic hardware circuits the performance of an interval arithmetic that heavily depends on the underlying realarithmetic

Major results Jan 2006-Dec 2009 P Kornerup (Southern Danish University Odense) and J-M Muller have shown that for normalredundant digit sets with radix greater than two a single guard digit is sufficient to determine the value of a prefix of leadingnonzero digits of arbitrary length [35] With S Collange (then a student at EacuteNS Lyon) F de Dinechin and J Detrey have arguedthat the respective merits of the floating-point and logarithmic number systems can only be assessed on a per-application basis[76] using a library of hardware operators supporting both systems [20 22]

Self-assessment That topic used to be very important in computer arithmetic Now partly due to early work by our group in thelate 90rsquos the adequate number representations for integer and real numbers depending on the context are well determinedHowever research remains active in number representations for finite-field arithmetic

332 Efficient Floating-Point Arithmetic and Applications

List of participants Nicolas Brisebarre (CR) Claude-Pierre Jeannerod (CR) Jingyan Jourdan Lu (PhD) Christoph Lauter (PhD)Vincent Lefegravevre (CR) Nicolas Louvet (MCF) Jean-Michel Muller (DR) Guillaume Revy (PhD) Gilles Villard (DR)

Keywords floating-point arithmetic multiplication FMA division square root correct rounding of algebraic functions extendedprecision error-free transforms compensated algorithms polynomial evaluation software implementation VLIW embeddedprocessor code generation and certification

Scientific issues goals and positioning of the team Efficient hardware for floating-point arithmetic (as specified by the IEEE-754standards) is becoming available on an increasing number of chips A first issue thus is to exploit the specification of suchhardware operators in order to enhance the accuracy of numerical computing in general without sacrificing efficiency Forprocessors without floating-point hardware another issue is to emulate IEEE-754 arithmetic as efficiently as possible in software

Major results Jan 2006-Dec 2009 Assuming that a fused multiply-add (FMA) floating-point instruction is available (which is thecase on Itanium and PowerPC architectures) we proposed several new algorithms in [71 10] N Brisebarre and J-M Mullershowed how to perform multiplication by an arbitrary-precision constant very fast and how to check whether the result iscorrectly rounded in [33] P Kornerup (Southern Danish University Odense) C Lauter V Lefegravevre N Louvet and J-M Mullergave algorithms that use FMAs for quickly computing correctly-rounded positive integer powersP Kornerup V Lefegravevre N Louvet and J-M Muller studied properties of summation algorithms in [107] They showed inparticular that Knuthrsquos 2Sum algorithm is minimal (in terms of number of operations and depth of the dependency graph) amongall branch-free algorithms using only rounded-to-nearest floating-point additionssubtractions S Boldo (former Areacutenaire PhDstudent at that time in lab LRI and EPI Proval of INRIA Saclay) and G Melquiond (then in Areacutenaire) showed how to emulatean FMA [158] They recently (after G Melquiond left Areacutenaire) published the final version of their work [5]Error-free transforms (EFT) such as the above 2Sum algorithm have further been studied as a way of emulating extended-precision efficiently C Lauter gave algorithms for multi-double arithmetic in [181] N Louvet (jointly with Ph Langlois fromUniversiteacute de Perpignan) showed in [110] how EFTs can be used to design a compensated Horner algorithm that simulates ktimes the precision of native floating-point arithmetic when evaluating polynomialsConcerning software emulation of floating-point operators using integer instructions C-P Jeannerod and G Revy (jointly withH Knochel and C Monat from STMicroelectronics) introduced in [166] a faster algorithm for correctly-rounded square roots


based on highly-parallel bivariate polynomial evaluation and well suited to VLIW architectures With G Villard they extendedthe approach to correctly-rounded division [105] the main difficulty here being the automatic numerical certification of thealgorithm Both algorithms were implemented in the FLIP library [195] (partly by hand partly by a code generator for fast andaccurate polynomial evaluation) With the ST200 VLIW compiler our new codes for radic and divide are respectively 24 and 18times faster than previously best ones

Key reference Our ldquoHandbook of Floating-Point Arithmeticrdquo [150]

Self-assessment Our expertise in floating-point arithmetic is one of the strengths of Areacutenaire and we propose state-of-the-art algorithmsand software implementations However we have to find a way of securing the long-term existence of our software

333 Efficient Polynomial and Rational Approximations

List of participants Nicolas Brisebarre (CR) Sylvain Chevillard (PhD) Mioara Joldes (PhD) Jean-Michel Muller (DR) ArnaudTisserand (CR) Serge Torres (IR)

Keywords floating-point arithmetic polynomial approximation rational approximation minimax approximation E-method latticebasis reduction closest vector problem linear programming

Scientific issues goals and positioning of the team Since the basic operations that are available in hardware are addi-tionssubtractions multiplications and divisions the only functions of one variable one can compute are piecewise polynomials(if division is not allowed) and rational functions (otherwise) Hence it is natural to approximate the functions we wish to eval-uate by polynomials or rational functions The classical methods for computing approximations to functions cannot be usedwithout losing accuracy since they generate approximations with ldquoinfinitely preciserdquo coefficients that must be rounded to someformat when we evaluate the polynomial or rational function the corresponding ldquorounded approximationrdquo may be significantlyless accurate than the best approximation taken among the ldquorounded polynomialsrdquo (or fractions)

Major results Jan 2006-Dec 2009 We have developed methods for generating the best or at least very good polynomial approxi-mations (with respect to the supremum norm or the L2 norm) to functions among polynomials whose coefficients follow sizeconstraints (eg the degree-i coefficient has at most mi fractional bits or it has a given fixed value) Regarding to the supre-mum norm on a compact interval one of these methods due to N Brisebarre J-M Muller and A Tisserand reduces to scanningthe integer points in a polytope [11] another one due to N Brisebarre and S Chevillard uses the LLL lattice reduction algorithm[66] This method implemented in Sollya also makes it possible to accelerate the process described in [11] About the L2 normN Brisebarre and G Hanrot (LORIA INRIA Lorraine) started a complete study of the problem [70] and proposed theoreticaland algorithmic results that make it possible to get optimal approximants Potential applications to hardwired operators havebeen studied [12] by N Brisebarre J-M Muller A Tisserand and S TorresIn [67] a joint work with M Ercegovac (University of California at Los Angeles) N Brisebarre S Chevillard J-M Mullerand S Torres extend the domain of applicability of the E-method (due to M Ercegovac) as a hardware-oriented method forevaluating elementary functions using polynomial and rational function approximations Such an approach based on linearprogramming and lattice basis reduction is presented A design is introduced and discussed

Self-assessment Until the papers published by the team there was no systematic approach to obtain good approximations to a functionf over an interval [a b] by polynomial or rational functions satisfying additional constraints (eg on the size in bits of thecoefficients or on the values of some coefficients) The leadership in this topic should be confirmed by our future works on theseveral variable case and the enhancements to come in the one variable case

334 Linear Algebra and Lattice Basis Reduction

List of participants Claude-Pierre Jeannerod (CR) Nicolas Louvet (MCF) Ivan Morel (PhD) Christophe Mouilleron (PhD) HongDiep Nguyen (PhD) Nathalie Revol (CR) Damien Stehleacute (CR) Gilles Villard (DR)

Keywords real matrix integer matrix polynomial matrix structured matrix approximant basis euclidean lattice certified computa-tion verification algorithm automatic differentiation matrix error bound algebraic complexity reduction to matrix multiplica-tion LLL lattice reduction strong lattice reduction asymptotically fast algorithm

Scientific issues goals and positioning of the team The techniques for exact linear algebra evolved quickly and significantly in thelast few years substantially reducing the complexity of several algorithms (see for instance [30] for an essentially optimalresult) A main focus is on matrices whose entries are integers or univariate polynomials over a field for which we aim atrelating the size of the data (integer bit lengths or polynomial degrees) to the cost of solving the problem exactly A first goalis to design asymptotically faster algorithms to reduce problems to matrix multiplication in a systematic way and to relatebit complexity to algebraic complexity Another direction is to make these algorithms fast in practice as well especially sinceapplications yield very large matrices that are either sparse or structured Within the LinBox international project we workon a software library that corresponds to our algorithmic research on matrices LinBox is a generic library devoted to sparseor structured exact linear algebra and its applications that allows to plug external components in a plug-and-play fashionMore recently we have also started a direction around lattice basis reduction Euclidean lattices provide powerful tools invarious algorithmic domains In particular we investigate applications in computer arithmetic cryptology algorithmic numbertheory and communication theory We work on improving the complexity estimates of lattice basis reduction algorithms andproviding better implementations of them and on obtaining more reduced bases The above recent progress in linear algebra mayprovide new insights Finally following the Areacutenaire objective of specification and certification (in an approximate arithmetic


framework like floating-point) we have also started studies around the complexity of computing certified error bounds andcondition numbers

Major results Jan 2006-Dec 2009 I Morel D Stehleacute and G Villard proposed a new floating-point LLL algorithm relying on theHouseholder QR factorization algorithm [120] This algorithm is simpler to describe than the previous ones and requires lessfloating-point precision G Hanrot and D Stehleacute investigated Kannanrsquos algorithm for the computation of a shortest lattice vectorThey improved its analysis by providing a lower upper complexity bound [103] and showed that the new bound is tight [164]X Pujol and D Stehleacute showed that this algorithm can be implemented safely with floating-point arithmetic [123]C-P Jeannerod and G Villard showed in [31] that minimal basis computations and polynomial matrix fractions allow to reducethe complexity of various problems on polynomial matrices to essentially that of multiplication The computation of rationalsolutions of sparse linear systems has been improved by G Villard (jointly with W Eberly M Giesbrecht P Giorgi and AStorjohann) in [94] where an asymptotically faster algorithm is given together with a LinBox-based implementation Thecorrectness of this new algorithm then subject to the conjectured existence of suitable block projections of matrices has thenbeen established in [95] C-P Jeannerod jointly with A Bostan and Eacute Schost gave a new upper bound on the algebraiccomplexity of linear system solving for several families of structured matrices [6] A consequence of this detailed in [6] as wellis asymptotically faster algorithms for Hermite-Padeacute approximation and bivariate interpolation In [137] C-P Jeannerod CMouilleron and G Villard proposed a new simpler algorithm for Vandermonde-like and Cauchy-like linear systems showingthat the elimination-based compression steps used by classical algorithms are in fact unnecessaryBased on componentwise perturbation analyses G Villard showed in [126] how to compute certified and effective error boundsfor the R factor of the matrix QR factorization This approach has been used to certify the LLL reducedness of output basesof floating point and heuristic LLL variants In order to compute a guaranteed enclosure of the solution of a linear systemHD Nguyen and N Revol have devised an efficient implementation that makes use of fast existing methods for linear systemsolving [142] C-P Jeannerod N Louvet N Revol and G Villard showed in [135] how automatic differentiation can be usedto efficiently compute the condition number of various matrix problems

Self-assessment The strong points here are state-of-the-art complexity results and implementations in the fields of algebraic computa-tion and lattice reduction as well as new insights into the effective numerical certification of linear algebra Note that the resultsaround floating-point LLL are being used by other members of the team eg for producing polynomial approximations (seesect333)

335 Correct Rounding of Functions

List of participants Nicolas Brisebarre (CR) Sylvain Chevillard (PhD) Florent de Dinechin (MCF) Mioara Joldes (PhD) ChristophLauter (PhD) Vincent Lefegravevre (CR) Jean-Michel Muller (DR) Nathalie Revol (CR) Damien Stehleacute (CR)

Keywords elementary functions libm correct rounding double precision interval arithmetic machine-assisted proofs power func-tion

Scientific issues goals and positioning of the team The original IEEE Standard for Floating-Point arithmetic (1985) required thatthe four arithmetic operations and the square root should be correctly rounded the returned result should always be the floating-point number nearest the exact result This requirement has many useful properties Among them the fact that the operationsare fully specified makes it possible to design algorithms and proofs that use these specifications The 1985 standard did notrequire anything concerning the ldquoelementary functionsrdquo (sine exponential logarithm etc) because of a problem called theldquoTable Makerrsquos Dilemmardquo an elementary function is evaluated to some internal accuracy (higher than the target precision)and then rounded to the target precision What is the minimum accuracy necessary to ensure that rounding this evaluation isequivalent to rounding the exact result for all possible inputs This question cannot be answered in a simple manner meaningthat in the general case (an arbitrary function) correctly rounding functions requires arbitrary precision which may be very slowand resource-consuming We have focused in the previous years on computing bounds on the intermediate precision requiredfor correctly rounding some elementary functions in IEEE-754 double precision This allows us to offer correct rounding withan acceptable overhead we have experimental code where the cost of correct rounding is negligible in average and less than afactor 10 in the worst case

Major results Jan 2006-Dec 2009 G Hanrot V Lefegravevre D Stehleacute and P Zimmermann [102] studied the problem of finding hard-to-round cases of a periodic function for large floating-point inputs C Lauter and V Lefegravevre designed and proved a newalgorithm to detect the rounding boundaries for the power (xy) function in double precision [36] N Brisebarre and J-M Mullergave bounds that allow to round algebraic functions correctly [9] V Lefegravevre D Stehleacute and P Zimmermann showed that thesearch for worst (ie hardest to round) cases started by Lefegravevre and Muller in 1999 can be extended to the decimal64 formatby determining the worst cases of the decimal64 exponential function [112] In addition the programs searching for worst casesin double precision were improved and many new worst-case results were foundF de Dinechin C Lauter and J-M Muller published an algorithm for evaluating correctly rounded logarithms in double preci-sion [17] C Lauter and F de Dinechin worked on automating the generation of a machine-optimized polynomial approximationto a function [111] The resulting polynomial is implemented as a C program resorting to a minimum amount of double-doubleor triple-double arithmetic if needed and provided with a formal proofS Chevillard and N Revol designed an algorithm for computing the special function erf in arbitrary precision and with correctrounding [75]

Key references JM Muller Elementary functions algorithms and implementation 2nd Edition Birkhaumluser 2006 [149]


Self-assessment Our work on correct rounding of functions persuaded the IEEE 754 revision committee to recommend that someelementary functions should be correctly rounded and this is probably the biggest success of Areacutenaire in the recent yearsThe new standard was adopted in August 2008 CRlibm functions should now move from academic demonstrators to defaultfunctions in the GNU libc but this requires considerable development effort

336 Certified Computing

List of participants Francisco Joseacute Chaacuteves Alonso (PhD) Sylvain Chevillard (PhD) Marc Daumas (CR) Claude-Pierre Jeannerod(CR) Mioara Joldes (PhD) Nicolas Jourdan (Eng) Christoph Lauter (PhD) Vincent Lefegravevre (CR) Nicolas Louvet (MCF)Guillaume Melquiond (PhD) Hong Diep Nguyen (PhD) Nathalie Revol (CR) Guillaume Revy (PhD) Gilles Villard (DR)

Keywords floating-point arithmetic roundoff errors interval arithmetic arbitrary precision Taylor models certified linear systemssolving global optimization supremum norm standardization formal proof theorem prover Coq PVS

Scientific issues goals and positioning of the team The certification of floating-point computations gives guarantees on the qualityof the computed results Typically this is a bound for the error between the exact mathematical result and the floating-pointcomputed result Several levels of confidence can be reached The first level consists in establishing an error bound usingproperties of floating-point arithmetic or interval arithmetic Another level of confidence consists in generating a certificate thatcan be checked using formal proof using a theorem prover such as Coq or PVS

Major results Jan 2006-Dec 2009 Areacutenaire is a leader for the standardization of interval arithmetic within the IEEE P1788 workinggroup [106] and within the STL of C++ [14 130] Interval arithmetic has been used to certify efficiently (in terms of time andaccuracy) the floating-point solution of a linear system [142] Arbitrary precision interval arithmetic has been implemented inthe C library MPFI Taylor models based on this arithmetic have been developed and used to get guaranteed bounds on the errorbetween a function and its approximating polynomial [74 73]Formal proof can now make use of floating-point arithmetic and interval arithmetic interval arithmetic (and more specificallyTaylor models) has been added to PVS [72 131 177] both arithmetics have been added to Coq [77] This is part of the Gappasoftware [182] that establishes tight bounds on roundoff errors and keeps tracks of the computations paths and transforms themso they can be checked by a theorem prover (Coq) Gappa has been successfully applied to codes of geometric computing tocheck geometric certificates [128 37] and to certify the code produced by Sollya for the CRlibm library [81 163] and alsosignificant parts of the code of the FLIP library [166 105]LEMA un Langage pour les Expressions Matheacutematiques Annoteacutees Finally a study has begun to establish a language thatenables the programmer to represent not only the floating-point code but also the intended semantics and the programmerrsquosknowledge This language is based on MathML

Self-assessment Areacutenaire is playing a leading role in the standardization of interval arithmetic gained through its joint expertise ininterval arithmetic and floating-point arithmeticWe have developed a state-of-the-art implementation in interval arithmetic linear system solving Our work is pioneering inthe field of certification and formal proof for floating-point computations In particular a main asset of the CRlibm and FLIPlibraries on top of their execution time is that significant parts of their codes are certifiedOur future directions are still more automation and the completion of a standard for interval arithmeticThe weakness here is that expertise in formal proof is fluctuating some members specialists in formal proof have now leftthe team new persons arrive with temporary positions to reinforce this direction We have to secure this field either throughestablished collaborations with experts from other teams or through the recruitment of an expert in formal proof in link withcomputer arithmetic

337 Hardware arithmetic operators

List of participants Jean-Luc Beuchat (postdoc) Nicolas Brisebarre (CR) Jeacutereacutemie Detrey (PhD) Florent de Dinechin (MCF) Ro-main Michard (PhD) Jean-Michel Muller (DR) Bogdan Pasca (PhD) Arnaud Tisserand (CR) Nicolas Veyrat-Charvillon (PhD)

Keywords ASIC FPGA circuit generator fixed-point floating-point arithmetic operators function evaluation integrated circuitdigit-recurrence algorithms division square-root exponential logarithm trigonometric sine cosine tridiagonal systems mod-ular multipliers

Scientific issues goals and positioning of the team Hardware arithmetic operators are at the core of any computing system largeand small This research domain has shifted from integer to floating-point operators with renewed interest in FMA and dec-imal architectures due to the IEEE-754 standard revision The team produces novel hardware-oriented algorithms and alsoestablished arithmetic libraries for reconfigurable circuits (FPGAs) We also investigate operators for cryptography

Major results Jan 2006-Dec 2009 M D Ercegovac (UCLA) and J-M Muller have introduced a digit-recurrence algorithm forevaluating complex square-roots [24] They have adapted Ercegovacrsquos E-method to the digit-recurrence solution of some tridi-agonal linear systems [96] and to the evaluation of complex polynomials [98 25] They have designed operators for complexarithmetic [97 93]J Detrey and F de Dinechin have explored hardware algorithms specifically designed for elementary function evaluation onFPGAs They have introduced table-based methods for exponential and logarithm [21] then sine and cosine [90 91] up tosingle-precision For larger precisions they have introduced a recurrence algorithm well suited to the fine-grain look-up table(LUT) structure of FPGAs [92] These operators are available as part of the FPLibrary project (see 381)


F de Dinechin has worked with O Cret I Trestian R Tudoran L Darabant and L Vacariu all from UT Cluj-Napoca on theacceleration using FPGAs of a large floating-point physics simulation for a biomedical application [125 15] A conclusionwas that efficient floating-point on FPGAs should not use the same operators as in processors but explore original application-specific ones [162] This is now the subject of Bogdan Pascarsquos PhD and the FloPoCo project which supersedes the FPLibrary(see 381) In addition to the previous elementary functions operators studied so far include multipliers by a constant [69 68]application-specific floating-point accumulators and dot-products [85] squarers and other optimized multipliers [84] squareroot [79]In addition FloPoCo is an architecture generator framework with several novel features such as target-specific operator special-ization automatic frequency-directed pipeline and test-bench generation [80]With A Tisserand (LIRMM) N Veyrat-Charvillon used special polynomial approximations to optimize arithmetic opera-tors [38] R Michard N Veyrat-Charvillon and A Tisserand also studied hardware operators for evaluating power func-tions [118]J-L Beuchat and J-M Muller have exhibited a family of modular multipliers suited for FPGA applications [4] In collaborationwith T Miyoshi and E Okamoto (Tsukuba Univ) they have written a survey on multiplication in GF(p) and GF(pn)In collaboration with E Okamoto (Univ Tsukuba Japan) J-L Beuchat N Brisebarre and J Detrey have introduced newarithmetic operators for pairing-based cryptography [60 58 59 2] Their paper [58] won the best paper award at CHESrsquo2007In collaboration with R Glabb (Calgary University) L Imbert and A Tisserand (LIRMM Montpellier) and G Jullien (CalgaryUniversity) N Veyrat-Charvillon designed a multi-Mode operator for SHA-2 hash functions [27]

Self-assessment The strong points here is the shared Areacutenaire expertise in optimization automation and validation enabling open-source projects such as FPLibraryFloPoCo which are widely used worldwide The main weak points is a lack of hand-onexpertise in VLSI technology For instance power consumption which is more and more important in deep sub-micron tech-nologies should probably be considered as a first-class objective along with cost performance and accuracy This weaknessshould be solved either by recruiting a VLSI expert or by building stronger collaboration with the VLSI community or withcompanies designing VLSI chips

34 Application domains and social or economic impact

Keywords arithmetic operator hardware implementation dedicated circuit control validation proof numerical software certifiedcomputingOur expertise covers application domains for which the quality such as the efficiency or safety of the arithmetic operators is an issueOn the one hand it can be applied to hardware oriented developments for example to the design of arithmetic primitives which arespecifically optimized for the target application and support On the other hand it can also be applied to software programs whennumerical reliability issues arise these issues can consist in improving the numerical stability of an algorithm computing guaranteedresults (either exact results or certified enclosures) or certifying numerical programs

bull The application domains of hardware arithmetic operators are digital signal processing image processing embedded applica-tions reconfigurable computing and cryptography More generally hardware arithmetic operators are used in all numericalcalculations Software support of floating-point arithmetic is mostly applied to media processing on embedded chips that haveno FPU

bull Developments of correctly rounded elementary functions is critical to the reproducibility of floating-point computations Ex-ponentials and logarithms for instance are routinely used in accounting systems for interest calculation where roundoff errorshave a financial meaning Our current focus is on bounding the worst-case time for such computations which is required toallow their use in safety critical applications and in proving the correct rounding property for a complete implementation

bull Certifying a numerical application usually requires bounds on rounding errors and ranges of variables Some of the tools wedevelop compute or verify such bounds For increased confidence in the numerical applications they may also generate formalproofs of the arithmetic properties These proofs can then be machine-checked by proof assistants like Coq

bull Arbitrary precision interval arithmetic can be used in two ways to validate a numerical result To quickly check the accuracy of aresult one can replace the floating-point arithmetic of the numerical software that computed this result by high-precision intervalarithmetic and measure the width of the interval result a tight result corresponds to good accuracy When getting a guaranteedenclosure of the solution is an issue then more sophisticated procedures such as those we develop must be employed this isthe case of global optimization problems

bull The design of faster algorithms for matrix polynomials provides faster solutions to various problems in control theory especiallythose involving multivariable linear systems

bull Lattice reduction algorithms have direct applications in public-key cryptography They also naturally arise in computer algebrain the research for good constrained approximations to functions and of worst cases for correct rounding A new and promisingfield of applications is communication theory

35 Visibility and Attractivity

351 Prizes and Awards

People Awarded


bull D Stehleacute second prize of the PhD award from the SPECIF association in 2006

bull D Stehleacute second prize of the PhD award from the ASTI association in 2006

Awarded communications

bull N Brisebarre and J Detrey CHES 2007 best paper award (joint work with J-L Beuchat and E Okamoto (Tsukuba UnivJapan))

352 Contribution to the Scientific Community

Management of Scientific Organisations

bull J-M Muller is ldquochargeacute de Missionrdquo at the ldquoSciences et technologies de lrsquoinformation et de lrsquoingeacutenieacuterierdquo institute of the FrenchCNRS since April 2006

bull N Revol is chair of the IEEE P1788 working group for the standardization of interval arithmetic launched in June 2008

bull G Villard is member of the board of the CNRS-GDR Informatique Matheacutematique (headed by B Valleacutee)

Administration of Professional Societies

Editorial Boards

bull IEEE Transactions on Computers J-M Muller (guest editor special section on Computer Arithmetic Feb 2009)

bull JUCS (Journal of Universal Computer Science) J-M Muller

bull Journal of Symbolic Computation G Villard

bull TCS (Theoretical Computer Science) M Daumas and N Revol (guest editors of a special issue on Real Numbers and ComputersFeb 2006)

bull LNCS (Lecture Notes in Computer Science) vol 5045 P Hertling C Hoffmann W Luther and N Revol special issue onReliable Implementation of Real Number Algorithms Theory and Practice 2009

Organisation of Conferences and Workshops

bull J-M Muller was co-program chair of the 18th IEEE Symposium on Computer Arithmetic 2009

bull G Villard was chair of the steering committee for the International Symposium on Symbolic and Algebraic Computation (IS-SAC) 2006

bull F de Dinechin C-P Jeannerod V Lefegravevre N Louvet and N Revol are the organizers of RAIM 2009 (Rencontres ldquoArithmeacute-tique de lrsquoInformatique Matheacutematiquerdquo du GDR IM - October 26-28 2009 - LIP)

bull C-P Jeannerod co-organized (jointly with F Rastello) two one-day workshops at LIP on the theme of Compiler Optimizationfor Embedded Systems (Sept 12 2007 and June 15 2009)

bull J-M Muller is member of the steering committee of the RNC (Real Numbers and Computers) and ARITH (IEEE Symposiumon Computer Arithmetic) series of conferences He co-organized the MSRrsquo2007 conference in Lyon

bull D Stehleacute co-organized LLL+25 an international conference held in honour of the 25th anniversary of the LLL algorithm Theconference took place in Caen on JuneJuly 2007

bull N Revol co-organized the Dagstuhl seminar no 06021 on Reliable Implementation of Real Number Algorithms Theory andPractice 2006 and the 3rd MathLogAps Workshop 2007

Program committee members

bull International Symposium on Symbolic and Algebraic Computation (ISSACrsquo06) C-P Jeannerod 2006

bull 18th IEEE Symposium on Computer Arithmetic J-M Muller (co-program chair) 2007

bull 19th IEEE Symposium on Computer Arithmetic J-M Muller 2009

bull 17th 18th 19th and 20th IEEE International Conference on Application-specific Systems Architectures and Processors (ASAP)J-M Muller 2006 2007 2008 2009

bull International Conference on Field-Programmable Logic and Applications (FPL) F de Dinechin 2007 2008 2009

bull International Conference on Field-Programmable Technologies (FPT) F de Dinechin 2006 2009

bull Symposium en Architectures de Machines F de Dinechin steering committee member 2008 2009

bull 7th and 8th Conference on Real Numbers and Computers (RNC) N Brisebarre 2006 and 2008

bull Computability and Complexity in Analysis (CCArsquo08) N Revol 2008

bull First International Conference on Symbolic Computation and Cryptography (SCCrsquo08) D Stehleacute 2008

bull International Symposium on Symbolic and Algebraic Computation (ISSACrsquo07) Parallel Symbolic Computation (PASCOrsquo07)Computer Algebra in Scientific Computing (CASCrsquo06) Transgressive Computing 2006 G Villard


International expertise

bull Board of examiners for foreign PhDs J-M Muller was reviewer of the PhD dissertation of Aacute Vaacutezquez (Univ Santiago deCompostela 2009)

bull Contributions to Standardization Bodies

ndash F de Dinechin C Lauter V Lefegravevre G Melquiond and J-M Muller have participated in the balloting process for therevision of the IEEE-754 standard for floating-point arithmetic

ndash V Lefegravevre participates in the Austin Common Standards Revision Group (for the revision of POSIX IEEE Std 10031)

ndash N Revol prepared the project approval request for an IEEE working group for the standardization of interval arithmeticapproved in June 2008 as IEEE P1788 project and she has been elected as chair of this Working Group

National expertise

bull INRIArsquos Commission drsquoEvaluation J-M Muller 2006

bull ANR Architectures du futur J-M Muller (member of evaluation committee 2006)

bull ANR Calcul intensif et simulation J-M Muller (member of steering committee 2007)

bull steering committee of Cluster Isle (Informatique Signal et Logiciels Embarqueacutes) reacutegion Rhocircne Alpes J-M Muller (2006)

bull Conseil Scientifique Grenoble INP J-M Muller (member) since 2008

bull G Villard was a member of the evaluation committee of the ANR program Deacutefis

bull Board of examiners for French PhDs and Habilitations

ndash C-P Jeannerod was in the PhD committee of S-K Raina (EacuteNS Lyon 2006)

ndash J-M Muller participated to the board of examiners for 11 French PhD dissertations since 2006 (Melquiond Raina DetreyVeyrat-Charvillon Rivaille Fournel Louvet Grosse Michard Lauter Chevillard) and 2 French ldquohabilitationsrdquo (Messinede Dinechin)

ndash N Revol was in the PhD committees of M Charikhi (U Paris 6 2005) L Lamarque (U Bourgogne 2006) and FJ ChaacutevesAlonso (EacuteNS Lyon 2007)

ndash G Villard was in the habilitation committee of L Imbert P Loidreau member of the board of examiners for the PhDdefense of O Bouissou F Chaacutevez N Meacuteloni C Pernet

ndash N Brisebarre was in the PhD committee of S Chevillard

bull National recruiting committees J-M Muller was a member of the final recruiting committee (ldquojury drsquoadmissionrdquo) CR1 andCR2 of CNRS in 2006

bull Local recruiting committee

ndash N Brisebarre belonged to the hiring committee (ldquocommission de speacutecialistesrdquo) of U St-Eacutetienne (pure mathematics 2001-2007)

ndash C-P Jeannerod belonged to the hiring committees for junior software-development engineers (ldquoingeacutenieurs associeacutesrdquo) atINRIA Rhocircne-Alpes in 2006 and 2007

ndash From 2006 to 2008 J-M Muller belonged to the ldquocommission de speacutecialistesrdquo (permanent recruiting committee) of ENSLyon After 2009 the rules changed so that there is a new committee for each position J-M Muller belonged to arecruiting committee for a position of ldquoMaicirctre de Confeacuterencesrdquo (senior lecturer) in ENS Lyon in April-May 2009 for aposition of ldquoMaicirctre de Confeacuterencesrdquo in Bordeaux University in April-May 2009 and to the hiring committee for juniorscientists (CR2) at INRIA Rennes - Bretagne Atlantique in 2008

ndash N Revol belonged to the ldquocommission de speacutecialistesrdquo of U Joseph Fourier Grenoble (applied math 2001-2006) and EacuteNSLyon (computer science 2004-2008) N Revol belonged to the hiring committees for junior scientists at INRIA Grenoble- Rhocircne-Alpes in 2007 and 2009 She belongs to the hiring committee for postdoc at INRIA Grenoble - Rhocircne-Alpessince 2007

ndash G Villard organized the evaluation seminar of INRIA SymB theme November 14-15 2006 Paris He belonged to theldquocommissions de speacutecialistesrdquo of U Lille U Perpignan UJF Grenoble and to a hiring committee at U Perpignan (2009)

bull Evaluation committees in the context of his ldquochargeacute de missionrdquo position in CNRS J-M Muller represented the CNRS (asan observer) at the evaluation committees of the following laboratories LSIS (Marseille December 2006) LIMOS (Clermont-Ferrand December 2006) LIF (Marseille January 2007) LINA (Nantes January 2007) LIFC (Besanccedilon February 2007) LITours (Tours June 2007) GSCOP (Grenoble January 2008) LIP6 (Paris January 2008) LIPN (Villetaneuse January 2008)LIX (Palaiseau February 2008) LORIA (Nancy February 2008) LSV (Cachan December 2008) LIFL (Lille December2008)


353 Public Dissemination

bull C-P Jeannerod and N Revol presented research careers to high-school students during the 10th ldquoMondial des Meacutetiersrdquo (Febru-ary 2006)

bull J-M Muller gave a popular science conference ldquoRepreacutesentation des nombres et calcul sur ordinateurrdquo at Maison du Livre delrsquoImage et du Son Villeurbanne Nov 21 2008 V Lefegravevre and J-M Muller wrote a popular science article for Images desMatheacutematiques (httpimagesmathcnrsfr)

bull N Revol regularly visits high-schools (3 to 5 visits per year)

36 Contracts and grants

361 External contracts and grants (Industry European National)

Ministry Grant ACI ldquoSecurity in Computer Sciencerdquo OCAM Opeacuterateurs Cryptographiques et Arithmeacutetique Mateacuterielle 3 years(2003-2006) headed by N Sendrier (SECRET INRIA Rocquencourt) 43 kCKeywords digital signature arithmetic operator FPGAParticipants J-L Beuchat A Tisserand G VillardThe OCAM project was done in collaboration with the Codes - now called SECRET - team (Inria Rocquencourt) and the teamArithmeacutetique Informatique of the Lirmm laboratory at Montpellier (see httpwww-rocqinriafrcodesOCAM)The goal of OCAM was the development of hardware operators for cryptographic applications based on the algebraic theory ofcodes The FPGA implementation of a new digital signature algorithm is used as a first target application (see sect 61)

Ministry Grant ACI ldquoNew Interfaces of Mathematicsrdquo GAAP Eacutetude et outils pour la Geacuteneacuteration Automatique drsquoApproximantsPolynomiaux efficaces en machine 4 years (2004-2008) headed by N Brisebarre 14kCKeywords polynomial approximation minimax approximation floating-point arithmetic polytope linear programmingParticipants N Brisebarre S Chevillard J-M Muller A Tisserand S TorresThe GAAP project was a collaboration with the LaMUSE laboratory (U Saint-Eacutetienne) The goal was the development ofsoftware tools aimed at obtaining very good polynomial approximants under various constraints on the size in bits and thevalues of the coefficients The target applications were software and hardware implementations such as embedded systems forinstance The software output of this project is made of the C libraries Sollya and MEPLib

ANR Gecko Project Geometrical Approach to Complexity and Applications 3 years (2005-2008) headed by B Salvy (AlgorithmsINRIA Rocquencourt) 6kCKeywords geometry algorithm analysis polynomial matrix integer matrixParticipant G VillardOther teams participating have been the Eacutecole Polytechnique Univ Nice Sophia-Antipolis and Univ P Sabatier in ToulouseThe project at the meeting point of numerical analysis effective methods in algebra symbolic computation and complexitytheory had the goal to improve solution methods for algebraic or linear differential equations by taking geometry into accountAdvances in Lyon have been related to taking into account geometrical structure aspects for matrix problems

ANR EVA-Flo Project Eacutevaluation et Validation Automatiques pour le calcul Flottant 4 years (2006-2010) headed by N Revol1305kCKeywords automation floating-point computation specification and validationParticipants all Areacutenaire Dali (LP2A U Perpignan) Measi (LIST CEA Saclay) and Tropics (INRIA Sophia-Antipolis)This project focuses on the way a mathematical formula is evaluated in floating-point arithmetic The approach is threefoldstudy of algorithms for approximating and evaluating mathematical formulae validation of such algorithms and automation ofthe process

ANR LaRedA Project Lattice Reduction Algorithms 3 years (2008-2010) headed by Brigitte Valleacutee (CNRSGREYC) and ValeacuterieBertheacute (CNRSLIRMM) 10kCKeywords lattice reduction average-case analysis experimentsParticipants Ivan Morel Damien StehleacuteThe aim of the project is to finely analyze lattice reduction algorithms such as LLL by using experiments probabilistic toolsand dynamic analysis Among the major goals are the average-case analysis of LLL and its output distribution In Lyon weconcentrate on the experimental side of the project (by using fpLLL and MAGMA) and the applications of lattice reductionalgorithms

ANR TCHATER Project Terminal Coheacuterent Heacuteteacuterodyne Adaptatif TEmps Reacuteel 3 years (2008-2010) headed by Alcatel-LucentFrance 131kCKeywords optical networks 40Gbs Digital Signal Processing FPGAParticipants Florent de Dinechin Gilles VillardThe TCHATER project is a collaboration between Alcatel-Lucent France E2V Semiconductors GET-ENST and the INRIAAreacutenaire and ASPI projectteams Its purpose is to demonstrate a coherent terminal operating at 40Gbs using real-time digitalsignal processing and efficient polarization division multiplexing In Lyon we will study the FPGA implementation of specificalgorithms for polarization demultiplexing and forward error correction with soft decoding


Reacutegion Rhocircne-Alpes Grant 3 years (Oct 2003-Sept 2006) headed by C-P Jeannerod and J-M MullerKeywords emulation of floating-point arithmetic integer processorParticipants C-P Jeannerod J-M Muller S-K Raina A TisserandThis project in collaboration with STMicroelectronics was supported by both Reacutegion Rhocircne-Alpes and STMicroelectronics Thegoal was to design and implement an efficient floating-point library for some embedded processors (namely the ST200 family)that only have integer arithmetic units This project has resulted in the first version of the FLIP software library and the PhD ofS-K Raina

Reacutegion Rhocircne-Alpes Grant 3 years (Oct 2008-Sept 2011) headed by C-P Jeannerod and J-M MullerKeywords floating-point arithmetic function evaluation compilation approximationParticipants Nicolas Brisebarre Sylvain Chevillard Claude-Pierre Jeannerod Mioara Joldes Jean-Michel Muller NathalieRevol Guillaume Revy Gilles VillardSince October 2008 we have obtained a 3-year grant from Reacutegion Rhocircne-Alpes That grant funds a PhD student MioaraJoldes The project consists in automating as much as possible the generation of code for approximating functions Instead ofcalling functions from libraries we wish to elaborate approximations at compile-time in order to be able to directly approximatecompound functions or to take into account some information (typically input range information) that might be available at thattime In this project we collaborate with the STMicroelectronicsrsquo Compilation Expertise Center in Grenoble (C Bertin HKnochel and C Monat) STMicroelectronics is funding another PhD grant on these themes (see the next item)

STMicroelectronics CIFRE PhD Grant 3 years (Mar 2009-Feb 2012) headed by C-P Jeannerod and J-M MullerJingyan Jourdan Lu is supported by a CIFRE PhD Grant from STMicroelectronics (Compilation Expertise Center Grenoble)on the theme of arithmetic code generation and specialization for embedded processors Advisors C-P Jeannerod and J-MMuller (Areacutenaire) C Monat (STMicroelectronics)

MinalogicEMSOC SCEPTRE Grant 3 years (Oct 2006-Dec 2009) headed by C-P Jeannerod and J-M MullerKeywords synthesis of algorithms compilation SOCs embedded systemsParticipants C-P Jeannerod N Jourdan J-M Muller G Revy G VillardSince October 2006 we have been involved in Sceptre a project of the EMSOC cluster of the Minalogic Competitivity CentreThis project led by STMicroelectronics aims at providing new techniques for implementing software on system-on-chipsWithin Areacutenaire we are focusing on the generation of optimized code for accurate evaluation of mathematical functions ourpartner at STMicroelectronics is the Compiler Expertise Center (Grenoble)

CNRS Grant PEPS FiltrOptim Calcul efficace et fiable en traitement du signal optimisation de la synthegravese de filtres numeacuteriquesen virgules fixe et flottante one year (2008) headed by N Brisebarre 10 kCKeywords digital signal processing fixed-point floating-point constrained coefficientParticipants N Brisebarre S Chevillard F de Dinechin M Joldes J-M MullerThe project FiltrOptim is within the 2008 program Programmes Exploratoires Pluridisciplinaires (PEPS) of the ST2I departmentof the CNRS The participants belong to the CAIRN (IRISA Lannion) and Areacutenaire (LIP ENS Lyon) teams The goal is todevelop methods that allow for the efficient implementation of a numerical algorithm of a signal processing filter correspondingto some given constraints usually given by a frequential frame and the requested formats of the coefficients

CNRS Grant PEPS ELRESAS Reacuteduction de reacuteseaux solutions exactes et approcheacutees Applications agrave la cryptologie et agravelrsquoarithmeacutetique des ordinateurs one year (2009) headed by N Brisebarre 55kCKeywords euclidean lattice LLL SVP CVP cryptology computer arithmeticParticipants Nicolas Brisebarre Sylvain Chevillard Ivan Morel Damien Stehleacute Gilles VillardThe project ELRESAS is within the 2009 program Programmes Exploratoires Pluridisciplinaires (PEPS) of the ST2I departmentof the CNRS The participants belong to the Areacutenaire (LIP ENS Lyon) CACAO (LORIA Nancy) CASYS (LJK Grenoble)MOAIS (LIG Grenoble) and ldquoTheacuteorie des nombresrdquo (Inst C Jordan Univ Lyon 1) teams The general goal is the improvementof the algorithmic of euclidean lattices and some of its applications

PAI Bracircncusi (France-Romania) Reconfigurable systems for biomedical applications 2 years (2007-2008) headed by F deDinechinKeywords FPGA floating-point biomedical engineeringParticipants F de Dinechin B PascaThis PAI (programme drsquoactions inteacutegreacutees) is headed by F de Dinechin and O Cret from Technical University of Cluj-Napoca inRomania It also involves the INRIA Compsys team-project in Lyon Its purpose is to use FPGAs to accelerate the computationof the magnetic field produced by a set of coils The Cluj-Napoca team provides the biomedical expertise the initial floating-point code the computing platform and general FPGA expertise Areacutenaire contributes the arithmetic aspects in particularspecializing and fusing arithmetic operators and error analysis Compsys brings in expertise in automatic parallelization

PHC Sakura - INRIA Ayame junior program (France-Japan) Software and Hardware Components for Pairing-Based Cryptogra-phy 2 years (2008-2009) headed by G Hanrot (LORIA) and E Okamoto (LCIS Univ of Tsukuba)Keywords cryptography (hyper)elliptic curves pairing finite field arithmetic hardware accelerator FPGAParticipant N Brisebarre


The participants belong to the Areacutenaire (LIP ENS Lyon) and CACAO (LORIA Nancy) teams for the French part and to the LCIS(Univ of Tsukuba Japan) and the Future Univ of Hakodate (Japan) The goal of this project is the enhancement of softwareand hardware implementations of pairings defined over algebraic curves We aim at developing a more efficient arithmetic overJacobian of (hyper)elliptic curves implementing genus 2 curve based pairings and designing algorithms and implementationsresistant to side channel attacks

CNRS Grant Eacutechange de chercheurs 2007 headed by D Stehleacute 25kCKeywords Lattice reductionParticipant D StehleacuteIn 2006 D Stehleacute obtained a grant to visit the Magma team at the University of Sydney Australia The aim of the visit whichtook place in late 2007 was to investigate strong lattice reduction

CNRS Grant Eacutechange de chercheurs 2009 headed by G Villard 5kCKeywords Lattice reductionParticipant G VillardIn 2008 G Villard obtained a grant to visit the Magma team at the University of Sydney Australia The visit will take placewithin the end of 2009

ARC Grant Australian Research Council Discovery Grant on Lattices and their Theta Series 3 years (2008-2011) headed by JCannon (University of Sydney) and R Brent (Australian National University Canberra) 10kCKeywords strong lattice reduction lattice theta series security estimate of NTRUParticipant D StehleacuteJ Cannon R Brent and D Stehleacute obtained this funding to investigate on short lattice points enumeration This is intricatelyrelated to strong lattice reduction and is thus important to assess the security of lattice-based cryptosystems such as NTRUhttpwwwntrucom The main target of the project is to improve the theory and the algorithms related to lattice thetaseries

Rhocircne-Alpes Region Grant Exploradoc 2 years (Jul 2008 - Jul 2010) headed by I MorelKeywords Fast LLL-type algorithmsParticipant I MorelI Morel obtained this funding to cover the costs of his cotutelle PhD thesis between ENS Lyon (local advisor G Villard) andthe University of Sydney (local advisor J Cannon) His research focuses on the development analysis and implementation offast LLL-type lattice reduction algorithms

XtremeData University Program 1 year (2007) headed by F de DinechinKeywords high-performance computing FPGA coprocessor XtremeData University ProgramParticipants Florent de Dinechin Bogdan PascaF de Dinechin was included in the XtremeData University program co-sponsored by Altera AMD Sun Microsystems andXtremeData inc He received a donation of an XD1000 Development System costing 15000 US$ and the associated program-ming tools

362 Research Networks (European National Regional Local)

Areacutenaire took part in the European Marie-Curie MathLogAps European Network Early Stage Research Training Site in MATHematicalLOGic and APplicationS (see httpwwwmathsmanchesteracuklogicmathlogaps) MathLogAps ran for 4years until August 31 2008 It funded 9 short term stays for students and 24 PhD theses among them the PhD thesis of FJ ChaacutevesAlonso It also subsidized 4 training workshops for students one per year one of these workshops was co-organized by AreacutenaireAreacutenaire takes part in the ldquoArithmeacutetiquerdquo and ldquoCalcul Formelrdquo groups of the GDR ldquoInformatique Matheacutematiquerdquo (IM) Areacutenaireactively participates to national events organized by this GDR

bull Gilles Villard is head of the working group ldquoCalcul formel arithmeacutetique protection de lrsquoinformation geacuteomeacutetrierdquo of GDR IM

bull we are involved in the Journeacutees RAIM (Rencontres Arithmeacutetique de lrsquoInformatique Matheacutematique) of GDR IM Nathalie Revolmanaged the track ldquoComputer arithmeticrdquo of the January 2007 session (Montpellier June 22-24 2007) Nicolas Brisebarremanaged the track ldquoComputer Arithmeticrdquo of the June 2008 session (Lille June 3-5 2008) Areacutenaire will organize the October2009 session (see httpwwwens-lyonfrLIPArenaireRAIM09)

Areacutenaire is also actively involved in the GDR ldquoArchitecture systegraveme et reacuteseaurdquo J-M Muller then F de Dinechin have been membersof the steering committee of the Symposium on Architectures and F de Dinechin has given a lecture at the Archi09 winter school

363 Internal Funding

CNRS ENS and Ministry of Research internal 10kC (2006) 13kC (2007) 11kC (2008) and 10kC (2009)INRIA internal 262kC (2006) 269kC (2007) 25kC (2008) and 30kC (2009)


Library

Program

developed using

links against

Development tools

Generalminuspurpose libraries

Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR

Figure 31 Relationships between some Areacutenaire developments

37 International collaborations resulting in joint publications

With joint publications

bull Centro de investigacioacuten y de Estudios Avanzados del IPN Mexico (F Rodriacuteguez Henriacutequez)

bull CERN Geneva Switzerland (E McIntosh F Schmidt)

bull Future University-Hakodate Japan (M Shirase et Tsuyoshi Takagi)

bull Intel Nizhniy Novgorod Laboratory Russia (S Maidanov)

bull National Institute of Aerospace USA (C Muntildeoz)

bull North Carolina State University USA (E Kaltofen)

bull Southern Danish University Odense Denmark (P Kornerup)

bull Technical University of Cluj-Napoca Romania (O Cret)

bull University of Calgary Canada (W Eberly)

bull University of California at Los Angeles USA (M Ercegovac Pouya Dormiani)

bull University of Louisiana at Lafayette USA (RB Kearfott)

bull University of Tsukuba Japan (J-L Beuchat T Miyoshi and E Okamoto)

bull University of Waterloo Canada (M Giesbrecht G Labahn A Storjohann)

bull University of Western Ontario Canada (Eacute Schost)

bull McGill University Montreal Canada (X-W Chang)

38 Software production and Research Infrastructure

Software plays an important role in Areacutenairersquos production The software designed by Areacutenaire can be downloaded at httpwwwens-lyonfrLIPArenaireWare We have designed

bull User-space software libraries of functions (such as CRlibm the first library of elementary functions with proven correct round-ing or FLIP a floating-point library for processors without hardware floating-point units)

bull user-space hardware libraries and generators such as FloPoCo a generator of arithmetic cores for FPGAs

bull tools for helping to build or to validate numerical software such as Sollya (the swiss army knife of elementary function devel-opment) or Gappa (a high-level proof assistant)

Thanks to their unique features the user-space libraries have been widely disseminated worldwide Our tools are designed for expertsand their diffusion is more restricted but they are nevertheless internationally recognizedThere are many dependencies between these tools as illustrated on Figure 31


381 Software Descriptions

CRlibm a double-precision mathematical library with correct rounding

bull Type library

bull CRlibm is an efficient mathematical library (libm) with proven correct rounding

bull It offers 21 elementary functions specified by the C99 standard

bull The availability and performance of CRlibm convinced the IEEE-754 revision committee that correct rounding is feasible atreasonable cost They recommend the correct rounding of elementary functions in the IEEE-754R standard adopted in June2008

bull Status free software LGPL

bull State stable Current effort aims at integrating CRlibm in the GNU libc

Sollya a toolbox for manipulating numerical functions

bull Type toolbox

bull Sollya is a numerical toolbox for manipulating numerical functions Its distinguishing feature is the focus on safety numericalresults are certified or a warning is produced It has been developed for CRlibm but is now being used in several other projects

bull Sollya is an interactive tool with an interface similar to Maple including a scripting language It can also be used as a library

bull Sollya offers for instance fast and certified numerical routines for safe plotting polynomial approximation and computation of1D infinite norms It is currently the only one tool that computes constrained polynomial approximations

bull Status free software CeCILL-C licence (GPL-compatible)

bull State stable

FPLibrary a library of floating-point operators for FPGAs

bull Type library of hardware components

bull FPLibrary is a library of floating-point operators essentially targeted to FPGAs

bull The operators are parameterized in exponent width and mantissa width

bull FPLibrary was the first hardware library to include floating-point elementary functions (exp log and trig)

bull It has been used in tens of universities and companies worldwide The trigonometric operators have been included in a standardset of benchmarks defined by the Altera corporation and the university of Berkeley

bull The development of FPLibrary stopped in 2008 and all new development effort went to FloPoCo (see below) which providesoperators using the same floating-point format


bull State stable

FloPoCo a generator of hardware arithmetic cores

bull Type generator of hardware components

bull FloPoCo is the successor of FPLibrary It is a generator of non-standard arithmetic operators for FPGAs with a focus onfloating-point It supports research on application-specific operators specifically designed for reconfigurable computing

bull FloPoCo inputs a list of operator specifications (20 operators currently) and outputs synthesizable VHDL

bull FloPoCo offers more floating-point operators than vendor tools Its operators have comparable or better performance and aremore portable and more flexible


bull State stable

FLIP (Floating-point Library for Integer Processors)

bull Type software library

bull FLIP is a C library for the efficient software support of binary32 floating-point arithmetic on processors without floating-pointhardware units Although portable it is highly optimized for the ST200 family of processor cores

bull FLIP implements correctly-rounded addition subtraction multiplication division and square root It supports subnormal num-bers as well as the four rounding direction attributes required by the IEEE 754-2008 standard

bull FLIP is included in the standard development environment for the ST200


bull State stable


Gappa a Tool for Certifying Numerical Programs

bull Type software

bull Gappa is a tool intended to help verifying and formally proving properties on numerical programs dealing with floating-point orfixed-point arithmetic

bull Given a logical property involving interval enclosures of mathematical expressions Gappa tries to verify this property andgenerates a formal proof of its validity This formal proof can be machine-checked by an independent tool like the Coq proof-checker

bull Gappa is being used in several projects including CRlibm FLIP Sollya and CGal

bull Status free software GPL CeCILL

bull State stable

Participation to GNU MPFR

bull Type library

bull GNU MPFR is an efficient multiple-precision floating-point library with well-defined semantics (copying the good ideas fromthe IEEE-754 standard) in particular correct rounding

bull GNU MPFR provides about 80 mathematical functions in addition to utility functions (assignments conversions ) Specialdata (Not a Number infinities signed zeros) are handled like in the IEEE-754 standard

bull Since the end of 2006 MPFR has become a joint project between the Areacutenaire and CACAO (Loria Nancy) project-teams Cacaohas the leading contribution MPFR is now used and even required by GCC to evaluate constant expressions at compile timeIt is also widely used in Areacutenaire in particular by software like MPFI Gappa and Sollya and in the search for hard-to-roundcases


bull State stable

bull Depot APP IDDN FR 001 120020 00 R P 2000 000 10800 on 15 March 2000

Lattice package in MAGMA

bull Type computational algebra system

bull Magma is a computer algebra system designed to solve problems in algebra number theory geometry and combinatorics

bull MAGMA provides hundreds of mathematical functions in Group Theory Number Theory Module Theory Lattices Commuta-tive Algebra Representation Theory Invariant Theory Lie Algebra Algebraic Geometry Arithmetic Geometry Cryptographyetc

bull Damien Stehleacute has actively participated in the update of the lattice functions since 2006 In particular he wrote the new LLLfunctions and a series of functions related to the shortest lattice vector problem

bull Status non-free cost recovery (non-commercial proprietary) rights owned by the University of Sydney

bull State stable V215

MPFI Multiple Precision Floating-Point Interval Arithmetic

bull Type library

bull MPFI is a library written in C for interval arithmetic using arbitrary precision (arithmetic and algebraic operations elementaryfunctions operations on sets)

bull Used by scientists worldwide (France Belgium Germany UK USA India Colombia etc)


bull State stable

fpLLL lattice reduction using floating-point arithmetic

bull Type library

bull FPLLL is a library which allows to perform several fundamental tasks on euclidean lattices most notably compute a LLL-reduced basis and compute a shortest non-zero vector in a lattice

bull FPLLL is progressively becoming the reference for lattice reduction Last year it has been included in SAGE and adapted to beincluded in PARI GP A variant of it is also contained in the MAGMA computational algebra system (see above)


bull State stable


Metalibm a code generator for elementary functions

bull Type toolbox

bull The Metalibm project provides a tool for the automatic implementation of mathematical (libm) functions

bull A function f is automatically transformed into Gappa-certified C code implementing an approximation polynomial in a givendomain with given accuracy

bull Metalibm has been used to produce large parts of CRlibm


bull State alpha

382 Contribution to Research Infrastructures

F de Dinechin is Europractice representative for ENS-Lyon and manages the servers simulation software (ModelSim) and FPGAsynthesis software (Xilinx ISE and Altera Quartus suites) used by the Areacutenaire and Compsys teams

39 Industrialization patents and technology transfer

391 Patents (French European and WorldWide)

None

392 Creation of Startups

None

393 Consulting Activities

None

310 Educational Activities

3101 Supervision of Educational Programs

bull F de Dinechin has been the coordinator of international exchange programs for the department of computer science of ENSLyon since 2002

bull N Revol supervised the class for PhD students entitled Applications de lrsquoinformatique agrave la recherche et au deacuteveloppementtechnologique (30 hours per year) 2001-2007

bull N Brisebarre belonged to the committee that prepared the next computer science Master program of ENS Lyon in 2009

3102 Teaching

Name Course title (short) Level Institution No of Number ofhours academic years

N Brisebarre Analyse L1 Univ St-Eacutetienne 80 2N Brisebarre Initiation MapleMatlab M1 Univ St-Eacutetienne 20 2N Brisebarre Analyse complexe L3 Univ St-Eacutetienne 36 2N Brisebarre Arithmeacutetique des corps finis des courbes M2 ENS Lyon 15 1

applications agrave la cryptologieN Brisebarre Algegravebre algorithmique M2 UCB-Lyon 1 30 3

F de Dinechin Architecture Reacuteseau et Systegraveme L3 ENS Lyon 45 4F de Dinechin Programmation orienteacutee objet et geacutenie logiciel L3 ENS Lyon 45 1F de Dinechin Informatique pour les non-informaticiens L3 ENS Lyon 20 2F de Dinechin Compilation M1 ENS Lyon 45 2F de Dinechin Algorithmes Arithmeacutetiques M1 ENS Lyon 45 2F de Dinechin Nombres opeacuterateurs arithmeacutetique et circuits M2 ENS Lyon 45 1C-P Jeannerod Algegravebre algorithmique M2 UCB-Lyon 1 15 2C-P Jeannerod Algorithmes pour lrsquoarithmeacutetique M1 ENS Lyon 45 2C-P Jeannerod Calcul algeacutebrique M2 ENS Lyon 45 1and G Villard

V Lefegravevre Arithmeacutetique des ordinateurs M2 UCB-Lyon 1 30 3


N Louvet Systegravemes drsquoexploitation L3 UCB-Lyon 1 20 1N Louvet Algorithmique et programmation L1 UCB-Lyon 1 23 1N Louvet Architectures mateacuterielles et logicielles L2 UCB-Lyon 1 26 1N Louvet Programmation en Java L2 UCB-Lyon 1 15 1N Louvet Algorithmique numeacuterique M2 UCB-Lyon 1 46 1

J-M Muller Arithmeacutetique virgule flottante M2 ENS Lyon 45 2J-M Muller Fonctions eacuteleacutementaires M2 ENS Lyon 45 2

N Revol Validation in scientific computing M2 ENS Lyon 45 1D Stehleacute Algorithmique des reacuteseaux euclidiens et applications M2 ENS Lyon 45 1D Stehleacute Algorithmique des reacuteseaux euclidiens et applications M2 UCB-Lyon 1 46 1

3103 Other teaching activities

bull N Brisebarre has been the advisor for two M2 research internships (S Chevillard in 2006 and M Joldes in 2008)

bull N Brisebarre F de Dinechin C-P Jeannerod N Revol and D Stehleacute have been examiners for the EacuteNS admissions (2006 -2009)

bull F de Dinechin has been the advisor for the M2 research internship of B Pasca for 3 engineer final training periods and for 2L3 research training periods He has participated each year to at least one research internship committee (L3 M1 or M2) for thedepartment of computer science at EacuteNS-Lyon He has given a lecture at the Archi 09 winter school

bull C-P Jeannerod was the advisor for two M2 research internships (G Revy in 2006 and C Mouilleron in 2008) He was also theadvisor of three engineers (Eacute Bechetoille in 2005-2006 N Jourdan in 2007-2008 C Fagbohoun in 2009) He participated totwo internship committees at EacuteNS Lyon (M1 in 2006 and M2 in 2009)

bull D Stehleacute has been the advisor for the L3 research internships of Fodil Ali Rached (EacuteNS Lyon) and David Cadeacute (EacuteNS Paris)

bull D Stehleacute has been the advisor of the M2 research internship of Xavier Pujol (EacuteNS Lyon)

bull N Louvet and J-M Muller have been the advisors of the M2 research internship of A Panhaleux (EacuteNS Lyon) in 2008

bull V Lefegravevre gave practical sessions (first steps with MPFR) at the CEA-EDF-INRIA School Certified Numerical Computation atLORIA (October 2007)

bull V Lefegravevre gave a presentation of MPFR at the CNCrsquo2 summer school (Certified Numerical Computation) at LORIA (June2009)

311 Self-Assessment

Strong points The Areacutenaire team is unique in the world in its focus and breadth Areacutenaire has a recognized expertise in hardwarearithmetic software arithmetic cores arithmetic algorithms floating-point design and use validation and formal proof of arith-metic cores number-theoretic tools useful to computer arithmetic and linear algebra Most other computer arithmetic researchgroups in France or abroad focus only on a few of these fieldsMore importantly this expertise is shared to a large extent by most Areacutenaire members For instance performance and costare not the only metrics accuracy is also always considered Interval arithmetic is not only a subject of research it is alsopervasively used to build safer operators Asymptotic complexity is studied but also the constants in the complexity due totechnology constraints such as operator pipelines or cache sizes Validation is also a transversal concernAreacutenaire is an attractive team both for PhD students and for candidates to open positions There are still many areas where theteam could grow while keeping its unity towards VLSI towards cryptography towards signal processing towards languagestowards formal proofs among others

Weak points The size and breadth of the team also means that it is more and more difficult to have an up-to-date idea of our colleaguesrsquolatest research interests Relatedly it is becoming increasingly difficult to carry team-wide federating projects such as ourHandbook of floating-point arithmeticThe team puts very strong emphasis on software development but receives comparatively little institutional support for it Whileit is fairly easy to get a few months of engineer we lack the stability that a full-time engineer would bring to secure softwareprojects typically initiated during a PhDArithmetic research should be driven by applications and Areacutenaire has to keep looking for new interdisciplinary or industrialcooperations Some directions such as signal processing or reconfigurable computing are well identified

Opportunities The publication of the Handbook of floating-point arithmetic which targets a wide audience should attract contactsand hopefully collaborations with people well outside our communityThe interval standardization effort is both an opportunity to exploit the breadth of Areacutenaire and an opportunity to strengthenour common culture The expertise acquired within Areacutenaire is needed in other standardization efforts in particular languagestandardsIn a wider sense the team is open to research opportunities brought by new technology targets (such as GPGPU reconfigurablecomputers massively multicore architectures nanoscale technologies or quantum computing) or new application domainsA great opportunity is the project refoundation in 2010 due to the 12-year age limit of INRIA project-teams


Risks Being the largest team in France in traditional computer arithmetic presents some risks for instance we have to ensure that ourformer PhD students keep finding interesting positions Being open to the wider community is the keyOur past successes in widening the scope of Areacutenaire by recruiting experts from different communities should not prevent usfrom exploiting external expertise too through new collaborations with leading teams in related fieldsAreacutenaire researchers suffer the current nationwide increase of research management tasks (writing proposals evaluating propos-als writing reports evaluating reports etc) This additional pressure leaves less time for research itself degrades our workingconditions and might degrade the teamrsquos atmosphere

312 Perspectives

Vision and new goals of the team

As long as computers compute they will need arithmetic at their very core Moreover as long as technology evolves the arithmeticunits will have to adapt to its evolution Areacutenaire researchers can be assured they always will have problems to solveHowever we want to create evolution not to follow it This is often a long-term goal It took twelve years to obtain the standardizationof correct rounding of elementary functions It took six years before formal proof tools could routinely be used in Areacutenaire whendesigning arithmetic cores and it may take as long before they are routinely used by our colleagues Pairing-based cryptography ormultiple-precision interval arithmetic are other examples that need long-term laboringIf we take as a starting point the two historical research directions of the team (building better operators and making best use of theexisting operators) three major changes in the research activities of the team have been initiated in the previous years and are expectedto evolve further

1 The notion of arithmetic operator has gotten coarser and coarser What we will call ldquoarithmetic operatorsrdquo is no longer limitedto hardware operators for addition multiplication or division It already includes elementary functions and numerical kernelssuch as linear algebra operators and FFT Such multivariate problems bring in new challenges

2 The two historical directions initially carried by different researchers have turned out to require the same expertise For instancebuilding a floating-point elementary function requires a fine knowledge of the existing floating-point operators

3 Operator validation using tools from interval arithmetic to formal theorem provers has become a routine part of the arithmeticoperator design

These observations have motivated the shift from designing operators to designing operator generators programs that output a low-level description of the operators taking care of their optimization and validation In other words our research is shifting from buildingarithmetic expertise to automating this expertiseAutomation of the arithmetic expertise requires new mathematical tools from the number theory and computer algebra communitiesThis explains the new recruits to the team over the period They are encouraged to develop their own personal expertisemdashwhich isnot strictly computer arithmeticmdashbut also to bring new arithmetic research directions most notably finite field arithmetic lattice-basedoptimization and cryptographyAutomation in turn opens a new frontier the design of arbitrary user-specified operators We have designed tools to obtain the bestpossible implementations of functions defined by standards In the close future we want to use them to implement arbitrary functionsWe now face a new problem how will the user specify the function to be implemented This question can be solved in two ways Oneapproach is to get closer to the compilation community so that the design of arithmetic operators becomes a part of the compilationprocess Another is to get closer to the computer algebra community that deals with symbolic-numeric computations In most currentprogramming languages an expression like sqrt(xx+yy) is defined as the composition of two library multiplications one libraryaddition and one library square root each hopefully implemented (in hardware or software) with well-defined standard-specifiedproperties However the programmer probably had in mind an atomic function

px2 + y2 which can be implemented much more

efficiently than the current composition and also possesses useful mathematical properties which are lost during composition Besidesin most current processors the current implementation will compute 53 bits of the result although the programmer might be contentedwith only 12 bits (that is 3 decimal digits) if he had the possibility to express itA consequence of such application-specific arithmetic core generation (compared to rather stable functions from general libraries) isthat each will have much fewer users Therefore we cannot count on a massive amount of users beta-testing and reporting bugsas general library designers do We must automatically validate the generated code In this respect the last decade concentrated onformalizing fixed-precision computation on scalar functions to a point where its validation could be usefully automated We now havetwo directions to explore machine-checkable proofs of multiple-precision algorithms and for multi-valued functions such as linearalgebra operators The ultimate goal is to automatically generate formal proofs of arbitrary numerical code but progress towards thisgoal has to be incremental A promising direction is to use techniques based on floating-point error-free transformations which on oneside rely on standard floating-point therefore could build up on existing knowledge and on the other side have been shown to addressboth arbitrary precision and multi-valued functions

Structuration of scientific perspectives

Designing an optimized complex operator involves many steps An approximation may be used for example a polynomial to approxi-mate a function Perspectives related to approximation are reviewed in Section 3122 Then there are usually several possible ways toevaluate the computation For a polynomial there are many evaluation schemes with different performancecostaccuracy trade-offsbut the same applies to the evaluation of general arithmetic expressions and to matrix operations Perspectives related to evaluation


are detailed in Section 3123 Perspectives related to certifying approximation and evaluation will be detailed in Section 3124 Sec-tion 3121 will refine the issues related to the automation of operator design The remaining sections deal with perspectives thatmay look more applicative but actually nurture our research in computer arithmetic pairing-based cryptology in Section 3126 andeuclidean lattices in Section 3125

3121 Towards the automatic design of operators and functions

What we have successfully automated so far (mostly in the previous 4 years) is program generation for statically-defined elementaryfunctions of one real variable These techniques will certainly need refining as we try to apply them to more functions in particularto special functions (such as erf) and multivariate functions (such as euclidean norms) To apply these techniques to arbitrary codeat compile time further leads to the following challenges The first one is to identify a relevant compound function in a programalong with the useful information that will allow to implement it efficiently range of the input values needed output accuracy andorrounding mode etc This is a static analysis problem It requires interaction with compilation people working on classical compilersbut also in the emerging field of high-level programming tools for FPGAs We will bring in expertise in interval arithmetic A secondchallenge then is to analyze such a function automatically which typically implies the following tasks

bull Compute maximal-width subranges on which the output is trivial that is requires no computation at all (zero infinity etc)This can be extremely tedious and error-prone if done by hand An important side effect of this step is to generate test vectorsfor corner cases

bull Identify computer arithmetic properties answering to questions like ldquoIs further range reduction possiblerdquo ldquoAre there floating-point midpointsrdquo ldquoCan overflow occurrdquo etc Today this is essentially handwritten by experts so the challenge is to automateit and to implement dictionaries containing such knowledge

bull Investigate automating range reduction Generic reduction schemes can be used (for example interval splitting into sub-intervals) but involve many parameters and we have to model the associated costperformanceaccuracy tradeoffs Morefunction-specific range reduction steps can be an outcome of the previous analysis steps

This general automation process will be progressively set up and refined by working on concrete implementations in software orin hardware of a significant set of operators hopefully driven by applications and through industrial collaborations all the C99elementary functions other functions such as the ICDF function used in Gaussian random number generators variations around theeuclidean norm such as x

px2 + y2 complex arithmetic interval operators FFTs and other signal processing transforms We also

plan to study operators similar to the now classical FMA like DP2 which computes a times b + c times d For example this instruction isavailable in the latest x86 instruction set extensions where it involves three intermediate roundings Implementing it with one roundingonly would improve the complex product and other operationsIt is worth mentioning that although similar design methods will apply for software and hardware implementations time scales aredifferent ldquocompile-timerdquo allows for a few seconds for software and for a few hours when designing hardware Other challenges morespecifically related to software and hardware targets are given below

Software operator design

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

bull Embedded processors without a floating-point unit (FPU) for which we need to implement the basic operators coarser ele-mentary functions and compile-time arbitrary functions On such targets memory may be limited and power consumption isimportant

bull General-purpose processors that do possess an FPU for the basic operations The challenge is then to use this FPU to its best toimplement coarser operators and compile-time functions The main metrics here are performance and to a lesser extent codesize

A challenge is to model instruction-level parallelism on such diverse targets and to manage to exploit it automatically

Hardware operator design

Automating hardware operator design is a must when targeting FPGAs or reconfigurable circuits which are increasingly being usedto accelerate scientific or financial calculations The challenge here is to design operators specifically for these applications not onlyshould their low-level architecture match the peculiar metrics of FPGAs but also the high-level architecture and even the operatorspecifications should be as application-specific as possible and probably completely different to what we are used to design into VLSIcircuits Such operators can only be produced by generators and this is the motivation of the FloPoCo generator framework still inearly development In addition to designing the operators themselves current challenges include accurately modelling the FPGAspower estimation and interaction with high-level (also known as C-to-hardware) compilersHowever we will also keep working on more traditional arithmetic for processors in particular to enhance FPUs Here automation ismostly useful for design exploration and testing Besides the already mentioned DP2 operator one operator we wish to study is theAdd3 operator that returns the correctly rounded sum of its three arguments It would improve the performance of general code andalso boost ldquotriple-doublerdquo arithmetic Since Add3 has the same instruction-set footprint as the FMA the challenge is to reuse as muchof the FMA datapath as possible In addition to the expertise already being developed in Areacutenaire around numerical validation we alsowant to work on more hardware-specific aspects of validation such as fault coverage or built-in self-test Links should be built with thehardware verification community


3122 New mathematical tools for function approximation

As explained in section 333 the classical methods for computing approximations to functions cannot be used without loss in termsof accuracy or efficiency Our algorithms for computing best or ldquonearly bestrdquo approximations under constraints (implemented in theSollya toolbox) will be generalized in two directions we aim at solving the several variable polynomial and the rational cases and adaptour techniques to other bases than the classical (xj) oneAnother important target is the computation of worst cases for the correct rounding of (elementary and special) functions

From one variable to several variables

We plan to adapt our algorithms (and their implementations) for computing polynomials with machine-number coefficients that verywell approximate a given function to the several variable case We first need to implement a generalization of the Remez algorithm (inthe real coefficient case) We might need to design some linear programming based tools to do so We will also address the case ofrational approximation in one variable We will develop an implementation in Sollya of the computation of these best approximationsin the real caseA second general step will consist in the suitable adaptation of the LLL-based approach we developed in the one variable case Theapplications we have in mind are the design of function evaluation tools in several variable and making practical a hardware-orientedprocess called E-method for evaluating elementary functions using polynomial and rational function approximations

From monomials to other function basis

Currently our algorithms and their implementations address the case in one variable of a basis of the form (xi) where i belongs to afinite subset of N with a cardinality bounded by say 100We aim at dealing with other bases in particular the classical basis made of Chebyshev polynomials that should lead to a betternumerical analysis without losing any efficiency during evaluation and the trigonometric polynomials case which could allow for newapplications in the design of numerical filters

Challenges in the search for correct rounding worst cases

With our current algorithms the cost of finding the worst case accuracy needed for correct rounding grows exponentially with thewidth of the floating-point format getting the worst cases for the most common functions in double precision (64-bit width) as wedid was already quite a challenge which required very carefully tuned implementation and months of calculation on a small networkof computersWe wish to tackle larger formats (x87-extended and quadruple precisions) decimal formats and more functions (including the mostfrequently used special functions) This will require work both on the algorithms themselves and on their implementation In particularwe wish to improve the work on numerically irregular functions such as the trigonometric functions (sine cosine tangent) on hugeargumentsAlso the programs that compute worst cases are very complex which might cast some doubt on the correctness of the returned resultsWe wish our next programs to be formally proven (at least for the critical parts)

3123 Linear algebra and polynomial evaluation

Linear algebra and polynomial evaluation are key tools for the design synthesis and validation of fast and accurate arithmeticsConversely arithmetic features can have a strong impact on the cost and numerical quality of computations with matrices vectors andpolynomials Thus deepening our understanding of such interactions is one of our major long-term goals To achieve this the expertiseacquired in the design of basic arithmetic operators will surely be quite valuable But the certification methods that we are planning todevelop simultaneously (see Section 3124) should be even more crucial

Certified linear algebra

We will investigate the theoretical and practical costs of computing certified and effective error bounds on standard linear algebraoperations like factorizations inversion linear equation solving and the computation of eigenvalues A first direction is symboliccomputing of a priori error bounds by manipulating truncated series Another direction deals with improving the efficiency and qualityof self-validating methods for computing error bounds at run time Concerning structured linear algebra we have two additional goalsIn exact arithmetic (algebraic complexity model) we want to finish our investigation of complexity reductions from one structure toanother In floating-point and fixed-point arithmetics we plan to study how the structure of the matrix can be used to speed up thecomputation of condition numbers and of certified and effective error bounds

Certified code generation

We plan to improve our work on code generation for polynomials and to extend it to general arithmetic expressions as well as tooperations typical of level 1 BLAS like sums and dot products Due to the intrinsic multivariate nature of such problems the numberof evaluation schemes is huge and a challenge here is to compute and certify in a reasonable amount of time evaluation programsthat satisfy both efficiency and accuracy constraints To achieve this goal we will in particular study the design of specific programtransformation techniques driven by the tight and guaranteed error bounds that the certification tools of Section 3124 will produce


3124 Methods for Certified Computing

Throughout every research direction and result of the team the certification of the quality of the computed result is required More pre-cisely the goal is to verify using computer arithmetic whether certain mathematical assumptions on the result hold The verification ofthese assumptions can usually be carried out using symbolic computations but when dealing with numerical problems the computationof rigorous error bounds or enclosures rather than their exact values is often sufficient to conclude In this case interval arithmetic orfloating-point computations with a careful control of the generated rounding errors can often be used to improve the performances

Certified bounds on roundoff errors

Many error analysis techniques are well known for obtaining a priori bounds on the global roundoff error generated by a floating-point program Computation of such error bounds can already be done automatically with Gappa in the case of straight-line programsassuming the precision of every arithmetic operations is fixed and known in advance One of our next challenge will be to handle a priorierror bounds in algorithms where the computing precision varies with each operation for programs involving loops of variable lengthThis involves symbolic computations of expressions for these roundoff errors and simplifications based on numerical computations ofthese expressions in order to get simple but not overly simplified upper boundsOn the other hand techniques are also available to compute rigorous a posteriori error bounds interval arithmetic being probably thebest established technique with this respect But certified a posteriori error bounds can also be computed using the specifications of theIEEE-754 floating-point arithmetic in particular using the directed rounding modes and sometimes faster implementations than withinterval arithmetic can be obtained As a consequence it would be interesting to develop code generators to automate the design ofcertified algorithms based on these techniquesWhen several a posteriori error bounds for a given algorithm are available studying the tradeoff been the cost and the sharpness of thecomputed error bound is also a topic we plan to investigate

Certified enclosures

As mentioned above interval methods can be used to control roundoff errors in numerical computations but can also be used moregenerally for computing a rigorous enclosure for the range of a function on a given domain Indeed computing enclosures or rangescorresponds to the case where input data vary in a set One can then deduce properties such as the occurrence of overflows or the signof the result The approach of choice is interval arithmetic An ongoing work is the standardization of this arithmetic an importanteffort is made towards the standardization of interval arithmetic since early 2008 and it will be maintained upon the completion of astandard planned at earliest in 2012A first issue we will address is the development of an efficient library of interval arithmetic operations for general purpose computationsThe goal is to reduce the overhead incurred by interval arithmetic compared to floating-point arithmetic One main source for thisoverhead is the number of changes of the rounding mode A second issue is the design and implementation of general algorithms thatcompute enclosures of the solutions of linear and nonlinear systems Another direction of research is the use of arbitrary precisionwhen the prescribed accuracy is not reached For efficiency purposes the difficulty is to increase scarcely the precision only wheneverything else fails

Higher order techniques

Automatic differentiation is a tool that is more and more used to get either sensitivity measures or more generally Taylor modelsObtaining condition numbers is a well known technique to measure the sensitivity of the solution to a problem to perturbations of itsinput data Combined with backward error analysis techniques they can also be used to obtain first order error bounds on the computedsolution to a given problem Future research will focus on algorithms to compute or to estimate condition numbers using automaticdifferentiation in particular on efficient implementations and on complexity issuesUsual or naive interval arithmetic evaluates an expression as it is written To diminish the so-called ldquodependency problemrdquo Taylormodels arithmetic is employed When it is implemented using floating-point arithmetic it now faces the problem of the limitedprecision A challenge is to intertwine Taylor models arithmetic with double-double (or triple-double or higher) arithmeticthe latter being preferred over arbitrary precision floating-point arithmetic for efficiency reasons This arithmetic will be a very usefultool for the computation of the supremum norm of a function in one variable

Formal proof

The methods mentioned so far certify the quality of the result up to a bug in their implementation To get a higher degree of confi-dence the certification can be checked by a theorem prover such as Coq The main difficulties are twofold First the chosen arithmeticor methods must be implemented and proven within the theorem prover In particular neither the symbolic-numeric computation oferror bounds in the case or variable precision nor Taylor models arithmetic based on floating-point arithmetic exist yet Second andmost important a theorem prover calculates extremely slowly This implies that it cannot perform the same operations as the originalalgorithm the computational path must be reworked in order to factorize and simplify parts of it and to reduce its number of operationsThis approach has been initiated in Gappa it should be enlarged to a wider variety of methods of certification


Representation issues

A general challenge is to represent and store not only programming information such as the numerical type and value of a variablebut also information known by the programmer such as the mathematically corresponding value or the fact that a given floating-pointoperation is actually exact without rounding error In other words the question is how to represent the semantics of a piece of code Wehave begun the elaboration of a language for the representation of numerical meta-information LEMA Langage pour les ExpressionsMatheacutematiques Annoteacutees enabling the programmer to precisely specify the semantic of numerical programsBuilding a database of properties that are frequently used either mathematical general properties or mathematical properties concerningfloating-point arithmetic will constitute a step towards the automation of code synthesis and certification This is another issue we willinvestigate

3125 Euclidean lattice reduction

Lattice reduction has proved useful for several different problems arising in computer arithmetic including the computation of the worstcases for the rounding of mathematical functions in fixed precisions and the approximation of functions by floating-point polynomialsLattice reduction has many other applications in computer science and computational mathematics making it worthwhile to devisefaster general-purpose lattice reduction algorithms

Using floating-point arithmetic within lattice reduction algorithms

We wish to finish the work started with the L2 algorithm and already largely simplified by the H-LLL algorithm We want to provide amore complete answer to the question of how to best use floating-point arithmetic within LLL-type algorithms and carry the developedtechniques over to stronger reduction algorithms (eg BKZ) and also to faster reduction LLL-type algorithms (eg Segment-LLL)

Studying Schnorrrsquos hierarchy of reductions

There exists a natural hierarchy reductions and reduction algorithms (due to Schnorr) between fast but weak reductions (LLL) andstronger but slower reductions(HKZ) A related question is to compute a shortest non-zero lattice vector as efficiently as possible(theoretically and in practice with massive parallelism and hardware implementation with and without heuristics) We will also try tosimplify and improve upon the large diversity of different trade-offs between efficiency and output quality of reduction algorithms Todevise better algorithms we will investigate the elaborate techniques used in the proofs of security of lattice-based cryptosystems suchas discrete Gaussian distributions transference theorems and quantum lattice algorithms

Faster LLL-type algorithms

From a theoretical point of view we will try to use floating-point arithmetic along with blocking techniques to speed up LLL reductionand to devise a quasi-linear time LLL-reduction algorithm (for fixed dimensions) From a practical point of view we will use mod-els developed by the LaRedA ANR project such as the sand-pile model to optimize the codes and combine the different developedalgorithms in a clever way to rigorously achieve the result in the fastest possible way A good test for these improvements would bethe automatic verification of all bounds obtained with Coppersmithrsquos technique for attacking different variants of the RSA cryptosys-tem (which would require huge amounts of large LLL reductions) The latter is likely to provide interesting results on lattice-basedtechniques for solving the Table Makerrsquos Dilemma including for functions of several variables

Lattice-based cryptography

One of our goals is to devise and make more practical lattice-based cryptography With code-based cryptosystems lattice-basedcryptosystems are the only public-key cryptosystems known today that resist would-be quantum computers They also provide veryprecise and strong security properties the security often provably reduces to worst-case instances of problems closely related to NP-hard problems One can also expect quasi-optimal complexities for encrypting and decrypting (O(1) bit operations per message bit)by using structured lattices However for the moment it seems hard to make these cryptosystems practical and we plan to tackle thatproblem by assessing precisely the practical security and by optimizing the implementation by using our arithmetical expertise

Other applications

Finally we plan to investigate the applications of lattice reduction in other fields such as MIMO decoding for wireless communicationsnumerical analysis and optimization

3126 Pairing-based cryptology

Wersquove just mentioned lattice-based cryptology Curve-based cryptology is the other public-key cryptological tool we plan to tackle inthe years to come More precisely we will mainly focus on the arithmetic and hardware implementation issues related to pairing-basedcryptographyWe plan to extend our previous works on the ηT pairing to the genus 2 case in order to compare it to the elliptic curve case Moreover wewish to study in detail the other existing pairings (Ate optimal pairings etc) we first need a better understanding of their algorithmicand we have to check whether the current architecture that we use must be adapted in order to take into account the course of the(moreover possibly new) operations occurring in the modified algorithms We also plan to study the case of finite fields with large


characteristic (its use is a recommendation of the NSA in particular) This will lead us to work with ordinary curves which will forceus to reconsider all the existing solutions

313 Publications and Productions

International and national peer reviewed journals [ACL]

[1] Bernhard Beckermann George Labahn and Gilles Villard Normal forms for general polynomial matrices Journal of SymbolicComputation 41(6)708ndash737 2006

[2] Jean-Luc Beuchat Nicolas Brisebarre Jeacutereacutemie Detrey Eiji Okamoto Masaaki Shirase and Tsuyoshi Takagi Algorithms andarithmetic operators for computing the ηt pairing in characteristic three IEEE Transactions on Computers 57(11)1454ndash1468November 2008

[3] Jean-Luc Beuchat Takanori Miyoshi Jean-Michel Muller and Eiji Okamoto Hornerrsquos rule-based multiplication over GF(p)and GF(pn) A survey International Journal of Electronics 95(7)669ndash685 2008

[4] Jean-Luc Beuchat and Jean-Michel Muller Automatic generation of modular multipliers for FPGA applications IEEE Trans-actions on Computers 57(12) December 2008

[5] Sylvie Boldo and Guillaume Melquiond Emulation of a FMA and correctly rounded sums Proved algorithms using roundingto odd IEEE Transactions on Computers 57(4)462ndash471 April 2008

[6] Alin Bostan Claude-Pierre Jeannerod and Eacuteric Schost Solving structured linear systems with large displacement rank Theo-retical Computer Science 407(13)155ndash181 November 2008

[7] Nicolas Boullis and Arnaud Tisserand Some optimizations of hardware multiplication by constant matrices IEEE Transactionson Computers 54(10)1271ndash1282 October 2005

[8] Nicolas Brisebarre David Defour Peter Kornerup Jean-Michel Muller and Nathalie Revol A new range reduction algorithmIEEE Transactions on Computers 54(3)331ndash339 2005

[9] Nicolas Brisebarre and Jean-Michel Muller Correct rounding of algebraic functions Theoretical Informatics and Applications4171ndash83 January 2007

[10] Nicolas Brisebarre and Jean-Michel Muller Correctly rounded multiplication by arbitrary precision constants IEEE Transac-tions on Computers 57(2)165ndash174 February 2008

[11] Nicolas Brisebarre Jean-Michel Muller and Arnaud Tisserand Computing machine-efficient polynomial approximations ACMTransactions on Mathematical Software 32(2)236ndash256 June 2006

[12] Nicolas Brisebarre Jean-Michel Muller Arnaud Tisserand and Serge Torres Hardware operators for function evaluation usingsparse-coefficient polynomials IEE Electronics Letters 42(25)1441ndash1442 December 2006

[13] Nicolas Brisebarre and Georges Philibert Effective lower and upper bounds for the Fourier coefficients of powers of the modularinvariant j J Ramanujan Math Soc 20(4)255ndash282 2005

[14] Herveacute Broumlnnimann Guillaume Melquiond and Sylvain Pion The design of the Boost interval arithmetic library TheoreticalComputer Science 351111ndash118 2006

[15] Octavian Cret Ionut Trestian Radu Tudoran Laura Darabant Lucia Vacariu and Florent de Dinechin Accelerating the com-putation of the physical parameters involved in transcranial magnetic stimulation using FPGA devices Romanian Journal ofInformation Science and Technology 10(4)361ndash379 2008

[16] Alain Darte Robert Schreiber and Gilles Villard Lattice-based memory allocation IEEE Transactions on Computers54(10)1242ndash1257 October 2005 Special Issue Tribute to B Ramakrishna (Bob) Rau

[17] Florent de Dinechin Christoph Quirin Lauter and Jean-Michel Muller Fast and correctly rounded logarithms in double-precision Theoretical Informatics and Applications 4185ndash102 2007

[18] Florent de Dinechin and Arnaud Tisserand Multipartite table methods IEEE Transactions on Computers 54(3)319ndash330 2005

[19] Florent de Dinechin and Gilles Villard High precision numerical accuracy in physics research Nuclear Inst and Methods inPhysics Research A 559(1)207ndash210 2006

[20] Jeacutereacutemie Detrey and Florent de Dinechin Outils pour une comparaison sans a priori entre arithmeacutetique logarithmique et arithmeacute-tique flottante Technique et science informatiques 24(6)625ndash643 2005

[21] Jeacutereacutemie Detrey and Florent de Dinechin Parameterized floating-point logarithm and exponential functions for FPGAs Micro-processors and Microsystems Special Issue on FPGA-based Reconfigurable Computing 31(8)537ndash545 December 2007

[22] Jeacutereacutemie Detrey and Florent de Dinechin A tool for unbiased comparison between logarithmic and floating-point arithmeticJournal of VLSI Signal Processing 49(1)161ndash175 2007

[23] Jeacutereacutemie Detrey and Florent de Dinechin Fonctions eacuteleacutementaires en virgule flottante pour les acceacuteleacuterateurs reconfigurablesTechnique et Science Informatiques 27(6)673ndash698 2008

[24] Milos Ercegovac and Jean-Michel Muller Complex square root with operand prescaling Journal of VLSI Signal Processing49(1)19ndash30 October 2007


[25] Milos Ercegovac and Jean-Michel Muller An efficient method for evaluating complex polynomials Journal of VLSI SignalProcessing Systems 2009 To appear

[26] Laurent Fousse Guillaume Hanrot Vincent Lefegravevre Patrick Peacutelissier and Paul Zimmermann MPFR A multiple-precisionbinary floating-point library with correct rounding ACM Trans Math Softw 33(2) 2007

[27] Ryan Glabb Laurent Imbert Graham Jullien Arnaud Tisserand and Nicolas Veyrat-Charvillon Multi-mode operator for SHA-2 hash functions Journal of Systems Architecture Special issue on ldquoEmbedded Cryptographic Hardwarerdquo 53(2-3)127ndash1382007

[28] Stef Graillat Jean-Luc Lamotte and Diep Nguyen Hong Extended precision with a rounding mode toward zero environmentapplication on the CELL processor International Journal of Reliability and Safety Special issue on Reliable EngineeringComputing 2009 To appear

[29] Stef Graillat Philippe Langlois and Nicolas Louvet Algorithms for accurate validated and fast polynomial evaluation JapanJournal of Industrial and Applied Mathematics 2009 To appear

[30] Claude-Pierre Jeannerod and Gilles Villard Essentially optimal computation of the inverse of generic polynomial matricesJournal of Complexity 21(1)72ndash86 2005

[31] Claude-Pierre Jeannerod and Gilles Villard Asymptotically fast polynomial matrix algorithms for multivariable systems Inter-national Journal of Control 79(11)1359ndash1367 2006

[32] E Kaltofen and G Villard On the complexity of computing determinants Computational Complexity 1391ndash130 2005

[33] Peter Kornerup Christoph Quirin Lauter Vincent Lefegravevre Nicolas Louvet and Jean-Michel Muller Computing correctlyrounded integer powers in floating-point arithmetic ACM Transactions on Mathematical Software 37(1) 2009 To appear

[34] Peter Kornerup and Jean-Michel Muller Choosing starting values for certain Newton-Raphson iterations Theoretical ComputerScience 351(1)101ndash110 February 2006

[35] Peter Kornerup and Jean-Michel Muller Leading guard digits in finite-precision redundant representations IEEE Transactionson Computers 55(5)541ndash548 May 2006

[36] Christoph Quirin Lauter and Vincent Lefegravevre An efficient rounding boundary test for pow(xy) in double precision IEEETransactions on Computers 58(2)197ndash207 February 2009

[37] Guillaume Melquiond and Sylvain Pion Formally certified floating-point filters for homogeneous geometric predicates Theo-retical Informatics and Applications 41(1) 2007

[38] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon Optimisation drsquoopeacuterateurs arithmeacutetiques mateacuteriels agrave basedrsquoapproximations polynomiales Technique et science informatiques 27(6)699ndash718 June 2008

[39] Ivan Morel Damien Stehleacute and Gilles Villard Analyse numeacuterique et reacuteduction des reacuteseaux Technique et Science Informatiques2009 To appear

[40] Phong Q Nguyen and Damien Stehleacute An LLL algorithm with quadratic complexity SIAM Journal on Computing 2009 Toappear

[41] Phong Q Nguyen and Damien Stehleacute Low-dimensional lattice basis reduction revisited ACM Transactions on Algorithms2009

[42] Jose-Alejandro Pineiro Stuart F Oberman Jean-Michel Muller and Javier D Bruguera High-speed function approximationusing a minimax quadratic interpolator IEEE Transactions on Computers 54(3)304ndash318 March 2005

[43] N Revol K Makino and M Berz Taylor models and floating-point arithmetic proof that arithmetic operations are validatedin COSY Journal of Logic and Algebraic Programming 64135ndash154 2005

[44] N Revol and F Rouillier Motivations for an arbitrary precision interval arithmetic and the MPFI library Reliable Computing11(4)275ndash290 2005

Invited conferences seminars and tutorials [INV]

[45] Jean-Michel Muller Exact computations with an arithmetic known to be approximate Invited lectures (3 hours) at the conferenceNumeration Mathematics and Computer Science CIRM Marseille Mar 2009

[46] Jean-Michel Muller Exact computations with approximate arithmetic Invited conference at the Journees en lrsquohonneur deDonald Knuth for the Doctor Honoris Causa diploma given to D Knuth Bordeaux France 29-31 Oct 2007

[47] Jean-Michel Muller Some algorithmic improvements due to the availability of an FMA invited talk at the Dagstuhl seminar08021 numerical validation in current hardware architectures January 2008

[48] Nathalie Revol Automatic adaptation of the computing precision January 2008 Dagstuhl seminar 08021 on Numerical Vali-dation in Current Hardware Architectures

[49] Nathalie Revol Introduction to interval analysis and to some interval-based software systems and libraries In ECMI 2008 TheEuropean Consortium For Mathematics In Industry July 2008

[50] Damien Stehleacute Floating-point LLL Theoretical and practical aspects LLL+25 conference in honour of the 25th anniversaryof the LLL algorithm Caen June 2007


[51] Damien Stehleacute Numerical analysis and LLL reduction Oberwolfach bi-annual workshop on Explicit Methods in NumberTheory Germany July 2007

[52] Arnaud Tisserand Algorithms and number systems for hardware computer arithmetic In International Symposium on Symbolicand Algebraic Computation (ISSAC) Beijing China July 2005 Invited tutorial

[53] Gilles Villard Efficient algorithms in linear algebra May 2005 33rd Theoretical Computer Science Spring School Computa-tional Complexity Montagnac-les-truffes

[54] Gilles Villard Some recent progress in symbolic linear algebra and related questions (invited tutorial) In Proc InternationalSymposium on Symbolic and Algebraic Computation Waterloo Canada pages 391ndash392 ACM Press August 2007

International and national peer-reviewed conference proceedings [ACT]

[55] Ali Akhavi and Damien Stehleacute Speeding-up lattice reduction with random projections In Proc 8th Latin American TheoreticalInformatics (LATINrsquo08) volume 4957 of Lecture Notes in Computer Science pages 293ndash305 Springer 2008

[56] Jean-Claude Bajard Philippe Langlois Dominique Michelucci Geacuteraldine Morin and Nathalie Revol Towards guaranteedgeometric computations with approximate arithmetics (12 pages) In Advanced Signal Processing Algorithms Architecturesand Implementations XVIII part of the SPIE Optics amp Photonics 2008 Symposium volume 7074 August 2008

[57] Rachid Beguenane Jean-Luc Beuchat Jean-Michel Muller and Steacutephane Simard Modular multiplication of large integers onFPGA In 39th Asilomar Conference on Signals Systems amp Computers IEEE November 2005

[58] Jean-Luc Beuchat Nicolas Brisebarre Jeacutereacutemie Detrey and Eiji Okamoto Arithmetic operators for pairing-based cryptographyIn Cryptographic Hardware and Embedded Systems ndash CHES 2007 number 4727 in Lecture Notes in Computer Science pages239ndash255 Springer 2007 Best paper award

[59] Jean-Luc Beuchat Nicolas Brisebarre Jeacutereacutemie Detrey Eiji Okamoto and Francisco Rodriacuteguez-Henriacutequez A comparisonbetween hardware accelerators for the modified Tate pairing over F2m and F3m In Second International Conference on Pairing-based Cryptography (Pairingrsquo08) volume 5209 of Lecture Notes in Computer Science pages 297ndash315 Springer Verlag 2008

[60] Jean-Luc Beuchat Nicolas Brisebarre Masaaki Shirase Tsuyoshi Takagi and Eiji Okamoto A coprocessor for the final expo-nentiation of the ηT pairing in characteristic three In Proceedings of Waifi 2007 volume 4547 of Lecture Notes in ComputerScience pages 25ndash39 Springer 2007

[61] Jean-Luc Beuchat and Jean-Michel Muller Multiplication algorithms for radix-2 RN-codings and tworsquos complement numbersIn Proceedings of the 16th IEEE International Conference on Application-Specific Systems Architectures and Processors (ASAP2005) pages 303ndash308 IEEE 2005

[62] Jean-Luc Beuchat and Jean-Michel Muller RN-codes algorithmes drsquoaddition de multiplication et drsquoeacuteleacutevation au carreacute InSympArsquo2005 10e eacutedition du SYMPosium en Architectures nouvelles de machines pages 73ndash84 April 2005

[63] Sylvie Boldo and Guillaume Melquiond When double rounding is odd In Proceedings of the 17th IMACS World Congress onComputational and Applied Mathematics Paris France 2005

[64] Sylvie Boldo and Jean-Michel Muller Some functions computable with a fused-MAC In Proc 17th IEEE Symposium onComputer Arithmetic (ARITH-17) pages 52ndash58 Cape Cod USA 2005

[65] Alin Bostan Claude-Pierre Jeannerod and Eacuteric Schost Solving Toeplitz- and Vandermonde-like linear systems with largedisplacement rank In Proc International Symposium on Symbolic and Algebraic Computation (ISSAC 2007) Waterloo Canadapages 33ndash40 ACM Press August 2007

[66] Nicolas Brisebarre and Sylvain Chevillard Efficient polynomial Linfin-approximations In Proceedings of the 18th IEEE Sympo-sium on Computer Arithmetic (ARITH-18) pages 169ndash176 IEEE 2007

[67] Nicolas Brisebarre Sylvain Chevillard Milos Ercegovac Jean-Michel Muller and Serge Torres An efficient method for eval-uating polynomial and rational function approximations In Application-specific Systems Architectures and Processors (ASAP2008) pages 245ndash250 IEEE 2008

[68] Nicolas Brisebarre Florent de Dinechin and Jean-Michel Muller Integer and floating-point constant multipliers for FPGAs InApplication-specific Systems Architectures and Processors (ASAP 2008) pages 239ndash244 IEEE 2008

[69] Nicolas Brisebarre Florent de Dinechin and Jean-Michel Muller Multiplieurs et diviseurs constants en virgule flottante avecarrondi correct In RenParrsquo18 SympArsquo2008 CFSErsquo6 2008

[70] Nicolas Brisebarre and Guillaume Hanrot Floating-point L2 approximations to functions In Proceedings of the 18th IEEESymposium on Computer Arithmetic (ARITH-18) pages 177ndash184 IEEE 2007

[71] Nicolas Brisebarre and Jean-Michel Muller Correctly rounded multiplication by arbitrary precision constants In Proc 17thIEEE Symposium on Computer Arithmetic (ARITH-17) IEEE Computer Society Press June 2005

[72] Francisco Chaacuteves and Marc Daumas A library to Taylor models for PVS automatic proof checker In Proceedings of the NSFworkshop on reliable engineering computing pages 39ndash52 Savannah Georgia 2006

[73] Sylvain Chevillard Mioara Joldes and Christoph Lauter Certified and fast computation of supremum norms of approximationerrors In 19th IEEE Symposium on Computer Arithmetic (ARITH-19) Portland Oregon USA 2009


[74] Sylvain Chevillard and Christoph Quirin Lauter A certified infinite norm for the implementation of elementary functions InSeventh International Conference on Quality Software (QSIC 2007) pages 153ndash160 IEEE 2007

[75] Sylvain Chevillard and Nathalie Revol Computation of the error function erf in arbitrary precision with correct rounding InRNC 8 Proceedings 8th Conference on Real Numbers and Computers pages 27ndash36 Javier D Bruguera and Marc Daumas July2008

[76] Sylvain Collange Jeacutereacutemie Detrey and Florent de Dinechin Floating point or LNS choosing the right arithmetic on an appli-cation basis In 9th Euromicro Conference on Digital System Design Architectures Methods and Tools (DSDrsquo2006) pages197ndash203 Dubrovnik Croatia August 2006 IEEE

[77] Marc Daumas Guillaume Melquiond and Cesar Muntildeoz Guaranteed proofs using interval arithmetic In Proceedings of the17th IEEE Symposium on Computer Arithmetic pages 188ndash195 Cape Cod Massachusetts USA 2005

[78] Florent de Dinechin Alexey Ershov and Nicolas Gast Towards the post-ultimate libm In 17th Symposium on ComputerArithmetic pages 288ndash295 IEEE 2005

[79] Florent de Dinechin Mioara Joldes Bogdan Pasca and Guillaume Revy Racines carreacutees multiplicatives sur FPGA In 13e

SYMPosium en Architectures nouvelles de machines (SYMPA) Toulouse September 2009

[80] Florent de Dinechin Cristian Klein and Bogdan Pasca Generating high-performance custom floating-point pipelines InProceedings of the 19th International Conference on Field Programmable Logic and Applications IEEE August 2009

[81] Florent de Dinechin Christoph Quirin Lauter and Guillaume Melquiond Assisted verification of elementary functions usingGappa In Proceedings of the 2006 ACM Symposium on Applied Computing pages 1318ndash1322 2006

[82] Florent de Dinechin and Sergey Maidanov Software techniques for perfect elementary functions in floating-point intervalarithmetic In Real Numbers and Computers July 2006

[83] Florent de Dinechin Eric McIntosh and Franck Schmidt Massive tracking on heterogeneous platforms In 9th InternationalComputational Accelerator Physics Conference (ICAP) October 2006

[84] Florent de Dinechin and Bogdan Pasca Large multipliers with less DSP blocks In Proceedings of the 19th InternationalConference on Field Programmable Logic and Applications IEEE August 2009

[85] Florent de Dinechin Bogdan Pasca Octavian Cret and Radu Tudoran An FPGA-specific approach to floating-point accumula-tion and sum-of-products In Field-Programmable Technologies pages 33ndash40 IEEE 2008

[86] Jeacutereacutemie Detrey TutoRISC un RISC dans mon FPGA In 9es Journeacutees Peacutedagogiques du CNFM pages 153ndash158 Saint-MaloFrance November 2006

[87] Jeacutereacutemie Detrey and Florent de Dinechin A parameterizable floating-point logarithm operator for FPGAs In 39th AsilomarConference on Signals Systems amp Computers IEEE November 2005

[88] Jeacutereacutemie Detrey and Florent de Dinechin A parameterized floating-point exponential function for FPGAs In IEEE InternationalConference on Field-Programmable Technology (FPTrsquo05) IEEE December 2005

[89] Jeacutereacutemie Detrey and Florent de Dinechin Table-based polynomials for fast hardware function evaluation In 16th Intl Conferenceon Application-specific Systems Architectures and Processors (ASAP 2005) IEEE July 2005

[90] Jeacutereacutemie Detrey and Florent de Dinechin Opeacuterateurs trigonomeacutetriques en virgule flottante sur FPGA In RenParrsquo17 SympArsquo2006CFSErsquo5 et JCrsquo2006 pages 96ndash105 Perpignan France October 2006

[91] Jeacutereacutemie Detrey and Florent de Dinechin Floating-point trigonometric functions for FPGAs In International Conference onField-Programmable Logic and Applications pages 29ndash34 IEEE August 2007

[92] Jeacutereacutemie Detrey Florent de Dinechin and Xavier Pujol Return of the hardware floating-point elementary function In 18thSymposium on Computer Arithmetic (ARITH-18) pages 161ndash168 IEEE June 2007

[93] Pouya Dormiani Milos Ercegovac and Jean-Michel Muller Design and implementation of a radix-4 complex division unitwith prescaling In Proceedings of the 20th IEEE International Conference on Application-specific Systems Architectures andProcessors (ASAP 2009) Boston USA 2009 IEEE Computer Society

[94] Wayne Eberly Mark Giesbrecht Pascal Giorgi Arne Storjohann and Gilles Villard Solving sparse integer linear systems InProc International Symposium on Symbolic and Algebraic Computation Genova Italy pages 63ndash70 ACM Press July 2006

[95] Wayne Eberly Mark Giesbrecht Pascal Giorgi Arne Storjohann and Gilles Villard Faster inversion and other black box matrixcomputation using efficient block projections In Proc International Symposium on Symbolic and Algebraic ComputationWaterloo Canada pages 143ndash150 ACM Press August 2007

[96] Milos Ercegovac and Jean-Michel Muller Arithmetic processor for solving tri-diagonal systems of linear equations In proc40th Conference on Signals Systems and Computers Pacific Grove CA November 2006

[97] Milos Ercegovac and Jean-Michel Muller Complex multiply-add and other related operators In Proceedings of SPIE ConfAdvanced Signal Processing Algorithms Architectures and Implementation XVII August 2007

[98] Milos Ercegovac and Jean-Michel Muller A hardware-oriented method for evaluating complex polynomials In Proceedings of18th IEEE Conference on Application-Specific Systems Architectures and Processors (ASAP 2007) IEEE July 2007


[99] Milos D Ercegovac Jean-Michel Muller and Arnaud Tisserand Simple seed architectures for reciprocal and square rootreciprocal In Proc 39th Asilomar Conference on Signals Systems and Computers Pacific Grove California USA October2005

[100] Ryan Glabb Laurent Imbert Graham Jullien Arnaud Tisserand and Nicolas Veyrat-Charvillon Multi-mode operator for SHA-2 hash functions In Proc International Conference on Engineering of Reconfigurable Systems and Algorithms (ERSA) pages207ndash210 Las Vegas Nevada USA June 2006

[101] Stef Graillat Jean-Luc Lamotte and Hong Diep Nguyen Error-free transformation in rounding mode toward zero In DagstuhlSeminar on Numerical Validation in Current Hardware Architectures volume 5492 of Lecture Notes in Computer Science pages217ndash229 2009

[102] Guillaume Hanrot Vincent Lefegravevre Damien Stehleacute and Paul Zimmermann Worst cases of a periodic function for large argu-ments In Proceedings of the 18th IEEE Symposium on Computer Arithmetic (ARITH-18) pages 133ndash140 IEEE 2007

[103] Guillaume Hanrot and Damien Stehleacute Improved analysis of Kannanrsquos shortest lattice vector algorithm (extended abstract) InProceedings of Crypto 2007 volume 4622 of LNCS pages 170ndash186 Springer-Verlag 2007

[104] Claude-Pierre Jeannerod Herveacute Knochel Christophe Monat and Guillaume Revy Faster floating-point square root for integerprocessors In IEEE Symposium on Industrial Embedded Systems (SIESrsquo07) 2007

[105] Claude-Pierre Jeannerod Herveacute Knochel Christophe Monat Guillaume Revy and Gilles Villard A new binary floating-pointdivision algorithm and its software implementation on the ST231 processor In Proceedings of the 19th IEEE Symposium onComputer Arithmetic (ARITH-19) Portland OR June 2009

[106] R Baker Kearfott John D Pryce and Nathalie Revol Discussions on an interval arithmetic standard at Dagstuhl seminar 08021In Dagstuhl Seminar on Numerical Validation in Current Hardware Architectures volume 5492 of Lecture Notes in ComputerScience pages 1ndash6 2009

[107] Peter Kornerup Vincent Lefegravevre Nicolas Louvet and Jean-Michel Muller On the computation of correctly-rounded sums In19th IEEE Symposium on Computer Arithmetic (ARITH-19) Portland Oregon USA 2009

[108] Peter Kornerup Vincent Lefegravevre and Jean-Michel Muller Computing integer powers in floating-point arithmetic In Proceed-ings of 41th Conference on signals systems and computers IEEE November 2007

[109] Peter Kornerup and Jean-Michel Muller RN-coding of numbers definition and some properties In Proceedings of the 17thIMACS World Congress on Scientific Computation Applied Mathematics and Simulation Paris July 2005

[110] Philippe Langlois and Nicolas Louvet Compensated Horner algorithm in K times the working precision In RNC 8 Proceedings8th Conference on Real Numbers and Computers pages 157ndash166 Javier D Bruguera and Marc Daumas July 2008

[111] Christoph Quirin Lauter and Florent de Dinechin Optimising polynomials for floating-point implementation In RNC 8 Pro-ceedings 8th Conference on Real Numbers and Computers pages 7ndash16 July 2008

[112] Vincent Lefegravevre Damien Stehleacute and Paul Zimmermann Worst cases for the exponential function in the IEEE 754r decimal64format In Reliable Implementation of Real Number Algorithms Theory and Practice volume 5045 of Lecture Notes in Com-puter Science pages 114ndash126 Dagstuhl Germany 2008 Springer

[113] Guillaume Melquiond and Sylvain Pion Formal certification of arithmetic filters for geometric predicates In Proceedings ofthe 17th IMACS World Congress on Computational and Applied Mathematics Paris France 2005

[114] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon Divgen a divider unit generator In Proc Advanced SignalProcessing Algorithms Architectures and Implementations XV volume 5910 page 59100M San Diego California USAAugust 2005 SPIE

[115] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon Eacutetude statistique de lrsquoactiviteacute de la fonction de seacutelectiondans lrsquoalgorithme de E-meacutethode In 5es journeacutees drsquoeacutetudes Faible Tension Faible Consommation (FTFC) pages 61ndash65 ParisMay 2005

[116] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon Eacutevaluation de polynocircmes et de fractions rationnelles surFPGA avec des opeacuterateurs agrave additions et deacutecalages en grande base In 10e SYMPosium en Architectures nouvelles de machines(SYMPA) pages 85ndash96 Le Croisic April 2005

[117] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon Small FPGA polynomial approximations with 3-bit coeffi-cients and low-precision estimations of the powers of x In Proc 16th International Conference on Application-specific SystemsArchitectures and Processors (ASAP 2005) pages 334ndash339 Samos Greece July 2005 IEEE Best Paper Award

[118] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon New identities and transformations for hardware poweroperators In Proc Advanced Signal Processing Algorithms Architectures and Implementations XVI San Diego CaliforniaUSA August 2006 SPIE

[119] Romain Michard Arnaud Tisserand and Nicolas Veyrat-Charvillon Optimisation drsquoopeacuterateurs arithmeacutetiques mateacuteriels agrave basedrsquoapproximations polynomiales In 11e SYMPosium en Architectures nouvelles de machines (SYMPA) Perpignan October 2006

[120] Ivan Morel Damien Stehleacute and Gilles Villard H-LLL Using Householder inside LLL In Proc International Symposium onSymbolic and Algebraic Computation (ISSAC 2009) Seould Korea pages 271ndash278 ACM Press August 2009


[121] Jean-Michel Muller Generating functions at compile-time In Proc 40th Conference on Signals Systems and ComputersPacific Grove CA November 2006

[122] Jean-Michel Muller Arnaud Tisserand Benoicirct Dupont de Dinechin and Christophe Monat Division by constant for the ST100DSP microprocessor In Proc 17th Symposium on Computer Arithmetic (ARITH-17) pages 124ndash130 Cape Cod MA USAJune 2005 IEEE Computer Society

[123] Xavier Pujol and Damien Stehleacute Rigorous and efficient short lattice vectors enumeration In Proceedings of ASIACRYPTrsquo08volume 5350 of Lecture Notes in Computer Science pages 390ndash405 Springer 2008

[124] Arne Storjohann and Gilles Villard Computing the rank and a small nullspace basis of a polynomial matrix In Proc Interna-tional Symposium on Symbolic and Algebraic Computation Beijing China pages 309ndash316 ACM Press July 2005

[125] Ionut Trestian Octavian Cret Laura Cret Lucia Vacariu Radu Tudoran and Florent de Dinechin FPGA-based computationof the inductance of coils used for the magnetic stimulation of the nervous system In Biomedical Electronics and Devicesvolume 1 pages 151ndash155 2008

[126] Gilles Villard Certification of the QR factor R and of lattice basis reducedness In Proc International Symposium on Symbolicand Algebraic Computation Waterloo Canada pages 361ndash368 ACM Press August 2007

[127] Gilles Villard Differentiation of Kaltofenrsquos division-free determinant algorithm In MICArsquo2008 Milestones in ComputerAlgebra Stonehaven Bay Trinidad and Tobago May 2008

Short communications [COM] and posters [AFF] in conferences and workshops

[128] Sylvie Boldo Marc Daumas William Kahan and Guillaume Melquiond Proof and certification for an accurate discriminantIn SCAN 2006 - 12th GAMM - IMACS International Symposium on Scientific Computing Computer Arithmetic and ValidatedNumerics Duisburg Germany 2006

[129] Alin Bostan Claude-Pierre Jeannerod and Eacuteric Schost Solving structured linear systems of large displacement rank Poster atISSAC 2006 July 2006

[130] Herveacute Broumlnnimann Guillaume Melquiond and Sylvain Pion Proposing interval arithmetic for the C++ standard In SCAN2006 - 12th GAMM - IMACS International Symposium on Scientific Computing Computer Arithmetic and Validated NumericsDuisburg Germany 2006

[131] Francisco Joseacute Chaacuteves Alonso Marc Daumas Cesar Muntildeoz and Nathalie Revol Automatic strategies to evaluate formulas onTaylor models and generate proofs in PVS In 6th International Congress on Industrial and Applied Mathematics (ICIAMrsquo07)July 2007

[132] Sylvain Chevillard and Christoph Quirin Lauter Certified infinite norm using interval arithmetic In SCAN 2006 - 12th GAMM- IMACS International Symposium on Scientific Computing Computer Arithmetic and Validated Numerics Duisburg Germany2006

[133] Florent de Dinechin Elementary functions for double-precision interval arithmetic In SCAN 2006 - 12th GAMM - IMACSInternational Symposium on Scientific Computing Computer Arithmetic and Validated Numerics Duisburg Germany 2006

[134] Florent de Dinechin Jeacutereacutemie Detrey Ionut Trestian Octavian Cret and Radu Tudoran When FPGAs are better at floating-pointthan microprocessors Poster at FPGArsquo2008 2008

[135] Claude-Pierre Jeannerod Nicolas Louvet Nathalie Revol and Gilles Villard Computing condition numbers with automaticdifferentiation In SCAN 2008 - 13th GAMM - IMACS International Symposium on Scientific Computing Computer Arithmeticand Validated Numerics El Paso Texas 2008

[136] Claude-Pierre Jeannerod Nicolas Louvet Nathalie Revol and Gilles Villard On the computation of some componentwisecondition numbers In SNSCrsquo08 - 4th International Conference on Symbolic and Numerical Scientific Computing HagenbergAustria 2008

[137] Claude-Pierre Jeannerod Christophe Mouilleron and Gilles Villard Extending Cardinalrsquos algorithm to a broader class ofstructured matrices Poster at ISSAC 2009 July 2009

[138] Claude-Pierre Jeannerod Saurabh Kumar Raina and Arnaud Tisserand High-radix floating-point division algorithms for em-bedded VLIW integer processors (extended abstract) In Proc 17th World Congress on Scientific Computation Applied Mathe-matics and Simulation IMACS Paris France July 2005

[139] Philippe Langlois and Nicolas Louvet Accurate solution of triangular linear system In SCAN 2008 - 13th GAMM - IMACSInternational Symposium on Scientific Computing Computer Arithmetic and Validated Numerics El Paso Texas 2008

[140] Christoph Quirin Lauter A survey of multiple precision computation using floating-point arithmetic In Taylor Models 2006Florida 2006

[141] Ivan Morel From an LLL-reduced basis to another Poster at ISSAC 2008 July 2008

[142] Hong-Diep Nguyen and Nathalie Revol Solving and certifying the solution of a linear system In SCAN 2008 - 13th GAMM -IMACS International Symposium on Scientific Computing Computer Arithmetic and Validated Numerics El Paso Texas 2008

[143] Nathalie Revol Bounding roundoff errors in Taylor models arithmetic In 17th IMACS Conf on Scientific Computation AppliedMath and Simulation Paris France July 2005


[144] Nathalie Revol Design choices and their limits for an interval arithmetic library The example of MPFI In SCAN 2006 - 12thGAMM - IMACS International Symposium on Scientific Computing Computer Arithmetic and Validated Numerics DuisburgGermany 2006

[145] Nathalie Revol Implementing Taylor models arithmetic with floating-point arithmetic In 6th International Congress on Indus-trial and Applied Mathematics (ICIAMrsquo07) July 2007

[146] Nathalie Revol Automatic adaptation of the computing precision In V Taylor Models Workshop May 2008

[147] Nathalie Revol Survey of proposals for the standardization of interval arithmetic In SWIM 08 Small Workshop on IntervalMethods June 2008

Scientific books and book chapters [OS]

[148] Florent de Dinechin Milos Ercegovac Jean-Michel Muller and Nathalie Revol Encyclopedia of Computer Science and Engi-neering chapter Digital Arithmetic Wiley 2008

[149] Jean-Michel Muller Elementary Functions Algorithms and Implementation Birkhaumluser Boston 2nd Edition 2006

[150] Jean-Michel Muller Nicolas Brisebarre Florent de Dinechin Claude-Pierre Jeannerod Vincent Lefegravevre Guillaume MelquiondNathalie Revol Damien Stehleacute and Serge Torres Handbook of Floating-Point Arithmetic Birkhauser Boston 2009 ACMG10 G12 G4 B20 B24 F21 To appear

[151] Damien Stehleacute LLL+25 25th Anniversary of the LLL Algorithm Conference chapter Floating-point LLL theoretical andpractical aspects Springer 2009 To appear

Scientific popularization [OV]

[152] Vincent Lefegravevre and Jean-Michel Muller Erreurs en arithmeacutetique des ordinateurs Images des matheacutematiques June 2009httpimagesmathcnrsfrErreurs-en-arithmetique-deshtml

[153] Jean-Michel Muller Ordinateurs en quecircte drsquoarithmeacutetique In Histoire des Nombres Taillendier 2007

Book or Proceedings editing [DO]

[154] Marc Daumas and Nathalie Revol editors Special issue on Real Numbers and Computers volume 351 of Theoretical ComputerScience 2006

[155] Peter Hertling Christoph M Hoffmann Wolfram Luther and Nathalie Revol editors Special issue on Reliable Implementationof Real Number Algorithms Theory and Practice volume 5045 of Lecture Notes in Computer Science 2008

[156] Peter Kornerup Paolo Montuschi Jean-Michel Muller and Eric Schwarz editors Special Section on Computer Arithmeticvolume 58 of IEEE Transactions on Computers February 2009

[157] Peter Kornerup and Jean-Michel Muller editors Proceedings of the 18th IEEE Symposium on Computer Arithmetic IEEEConference Publishing Services June 2007

Other Publications [AP]

[158] Sylvie Boldo and Guillaume Melquiond Emulation of a FMA and correctly-rounded sums proved algorithms using roundingto odd Technical report INRIA 2006 Available at httphalinriafrinria-00080427

[159] Sylvie Boldo and Cesar Muntildeoz A formalization of floating-point numbers in PVS Technical report National Institute forAerospace 2005 A much shorter version of this paper has been published in Proceedings of the 21st Annual ACM Symposiumon Applied Computing (2006)

[160] Sylvain Chevillard Polynocircmes de meilleure approximation agrave coefficients flottants Master thesis Laboratoire de lrsquoInformatiquedu Paralleacutelisme - ENS Lyon 2006

[161] Sylvain Chevillard The functions erf and erfc computed with arbitrary precision Technical Report RR2009-04 Laboratoire delrsquoInformatique du Paralleacutelisme (LIP) 46 alleacutee drsquoItalie 69 364 Lyon Cedex 07 January 2009

[162] Florent de Dinechin Jeacutereacutemie Detrey Ionut Trestian Octavian Cret and Radu Tudoran When FPGAs are better at floating-pointthan microprocessors Technical Report ensl-00174627 Eacutecole Normale Supeacuterieure de Lyon 2007

[163] Florent de Dinechin Christoph Quirin Lauter and Guillaume Melquiond Certifying floating-point implementations usingGappa Technical Report ensl-00200830 arXiv08010523 Eacutecole Normale Supeacuterieure de Lyon 2007

[164] Guillaume Hanrot and Damien Stehleacute Worst-case Hermite-Korkine-Zolotarev reduced lattice bases Technical report MathsArxiv 2008

[165] Claude-Pierre Jeannerod LSP matrix decomposition revisited Research report RR2006-28 Laboratoire de lrsquoInformatique duParalleacutelisme ENS Lyon September 2006

[166] Claude-Pierre Jeannerod Herveacute Knochel Christophe Monat and Guillaume Revy Computing floating-point square roots viabivariate polynomial evaluation Technical Report 2008-38 ensl-00335792 LIP Eacutecole Normale Supeacuterieure de Lyon 2008


[167] Claude-Pierre Jeannerod Nicolas Louvet Jean-Michel Muller and Adrien Panhaleux Midpoints and exact points of somealgebraic functions in floating-point arithmetic Technical Report ensl-00409366 LIP Eacutecole Normale Supeacuterieure de Lyon2009

[168] Claude-Pierre Jeannerod and Guillaume Revy Optimizing correctly-rounded reciprocal square roots for embedded VLIW coresTechnical Report 2009-21 ensl-00391185 LIP Eacutecole Normale Supeacuterieure de Lyon 2009

[169] Mioara Joldes Computation of various infinite norms in one or several variables designed for the development of mathematicallibraries Master thesis Laboratoire de lrsquoInformatique du Paralleacutelisme - ENS Lyon 2008

[170] Christoph Quirin Lauter Basic building blocks for a triple-double intermediate format Technical Report RR2005-38 LIPSeptember 2005

[171] Christoph Quirin Lauter Exact and mid-point rounding cases of power(xy) Technical Report 2006-46 Laboratoire delrsquoInformatique du Paralleacutelisme 2006

[172] Bogdan Pasca Customizing floating-point operators for linear algebra acceleration on FPGAs Master thesis Laboratoire delrsquoInformatique du Paralleacutelisme - ENS Lyon 2008

[173] Cleacutement Pernet Aude Rondepierre and Gilles Villard Computing the Kalman form Research report ccsd-00009558 arXivcsSC0510014 IMAG Grenoble October 2005

[174] Guillaume Revy Analyse et implantation drsquoalgorithmes rapides pour lrsquoeacutevaluation polynomiale sur les nombres flottants Masterthesis Laboratoire de lrsquoInformatique du Paralleacutelisme - ENS Lyon 2006

[175] Gilles Villard Certification of the QR Factor R and of Lattice Basis Reducedness (extended version of [126]) Research reporthal-00127059 arXiv0701183 CNRS Universiteacute de Lyon INRIA 2007

[176] Gilles Villard Kaltofenrsquos division-free determinant algorithm differentiated for matrix adjoint computation (journal submission)Research report ensl-00335918 arXiv08105647 CNRS Universiteacute de Lyon INRIA 2008

Doctoral Dissertations and Habilitation Theses [TH]

[177] Francisco Joseacute Chaacuteves Alonso Utilisation et certification de lrsquoarithmeacutetique drsquointervalles dans un assistant de preuves PhDthesis Eacutecole Normale Supeacuterieure de Lyon France September 2007

[178] Sylvain Chevillard Eacutevaluation efficace de fonctions numeacuteriques - Outils et exemples PhD thesis Eacutecole Normale Supeacuterieure deLyon France July 2009

[179] Florent de Dinechin Mateacuteriel et logiciel pour lrsquoeacutevaluation de fonctions numeacuteriques Preacutecision performance et validationHabilitation agrave diriger des recherches Universiteacute Claude Bernard - Lyon 1 June 2007

[180] Jeacutereacutemie Detrey Arithmeacutetiques reacuteelles sur FPGA virgule fixe virgule flottante et systegraveme logarithmique PhD thesis EacutecoleNormale Supeacuterieure de Lyon France January 2007

[181] Christoph Quirin Lauter Arrondi correct de fonctions matheacutematiques - fonctions univarieacutees et bivarieacutees certification et automa-tisation PhD thesis Eacutecole Normale Supeacuterieure de Lyon France October 2008

[182] Guillaume Melquiond De lrsquoarithmeacutetique drsquointervalles agrave la certification de programmes PhD thesis Eacutecole Normale Supeacuterieurede Lyon France November 2006

[183] Romain Michard Opeacuterateurs arithmeacutetiques mateacuteriels optimiseacutes PhD thesis Eacutecole Normale Supeacuterieure de Lyon France June2008

[184] Louvet Nicolas Algorithmes compenseacutes en arithmeacutetique flottante preacutecision validation performances PhD thesis Universiteacutede Perpignan France November 2007

[185] Saurabh Kumar Raina FLIP a floating-point library for integer processors PhD thesis Eacutecole Normale Supeacuterieure de LyonFrance September 2006

[186] Guillaume Revy Mathematical functions in floating-point arithmetic for integer processors - Algorithms implementationpolynomial evaluation and code generation PhD thesis Eacutecole Normale Supeacuterieure de Lyon France 2009

[187] Nicolas Veyrat-Charvillon Opeacuterateurs arithmeacutetiques mateacuteriels pour des applications speacutecifiques PhD thesis Eacutecole NormaleSupeacuterieure de Lyon France June 2007

2005 2006 2007 2008 2009 TotalACL - International and national peer-reviewed journal 11 8 8 9 8 44INV - Invited conferences 2 0 4 3 1 10ACT - International and national peer-reviewed conference proceedings 20 13 16 14 10 73COM AFF - Short communications and posters in international and nationalconferences and workshops 2 7 2 8 1 20

OS - Scientific books and book chapters 0 1 0 1 2 4OV - Scientific popularization 0 0 1 0 1 2


DO - Book or Proceedings editing 0 1 1 1 1 4AP - Other Publications 3 5 3 5 3 19TH - Doctoral Dissertations and Habilitation Theses 0 2 5 2 2 11

Total 38 37 40 43 29 187

Other Productions Software [AP]

[188] David Cadeacute Xavier Pujol and Damien Stehleacute fplll 2005ndash2009httppersoens-lyonfrdamienstehle

[189] Sylvain Chevillard Mioara Joldes Nicolas Jourdan and Christoph Lauter Sollya a toolbox for manipulating numerical func-tions 2006ndash2009httpsollyagforgeinriafr

[190] Sylvain Collange Jeacutereacutemie Detrey Florent de Dinechin Bogdan Pasca Cristian Klein and Xavier Pujol FloPoCo a generatorof arithmetic cores for FPGAs 2008ndash2009httpwwwens-lyonfrLIPArenaireWareFloPoCo

[191] Sylvain Collange Jeacutereacutemie Detrey Florent de Dinechin and Xavier Pujol FPLibrary a library of floating-point operators forFPGAs 2002ndash2008httpwwwens-lyonfrLIPArenaireWareFPLibrary

[192] Catherine Daramy-Loirat David Defour Florent de Dinechin Matthieu Gallet Nicolas Gast Christoph Lauter and Jean-MichelMuller CRlibm a library of correctly rounded elementary functions in double-precision 2004ndash2009httplipforgeens-lyonfrwwwcrlibm

[193] Laurent Fousse Guillaume Hanrot Vincent Lefegravevre Patrick Peacutelissier Philippe Theacuteveny and Paul Zimmermann GNU MPFRMarch 2000httpwwwmpfrorg

[194] The LinBox Group Linbox 1998ndash2009httpwwwlinalgorg

[195] Claude-Pierre Jeannerod Nicolas Jourdan Saurabh Kumar Raina Guillaume Revy and Arnaud Tisserand FLIP a floating-pointlibrary for integer processors 2003ndash2009httpflipgforgeinriafr

[196] Christoph Lauter Metalibm a code generator for elementary functions 2008httplipforgeens-lyonfrwwwmetalibm

[197] Guillaume Melquiond Gappa a tool for certifying numerical programs 2005ndash2009httplipforgeens-lyonfrwwwgappa



TORRES Serge Engineer (40) MESR 01092001VAacuteZQUEZ AacuteLVAREZ Aacutelvaro Post-doc INRIA 01102009

Doctoral Students

Name First name InstitutionDate of first registrationas doctoral student

JOLDES Mioara Reacutegion Rhocircne-Alpes 01092008JOURDAN LU Jingyan CIFRE 01032009MARTIN-DOREL Eacuterik MESR 01092009MOREL Ivan EacuteNS Lyon 01092007MOUILLERON Christophe EacuteNS Lyon 01092008NGUYEN Hong Diep INRIA CORDI 01102007PANHALEUX Adrien EacuteNS Lyon 01092008PASCA Bogdan MESR 01092008PUJOL Xavier EacuteNS Lyon 01092009REVY Guillaume MESR 01102006

Past team members

Past Members Oct 2005-Oct 2009

Name First name PositionParentInstitution Arrival Departure Current position

DAUMAS Marc Research Scientist CNRS 01101998 30112005 Professor Perpignan University

STEHLEacute Damien Research Scientist CNRS 01102006 15072008CNRS scientist seconded tothe Universities of Sydney andMacquarie until 15072010

Past Doctoral students

Name First nameFirstregistration

Departure InstitutionDoctoralschool

SupervisorCurrentposition

CHAacuteVESALONSO

FranciscoJoseacute

01112004 31102007 EacuteNS Lyon (MarieCurie grant)

MathIFM DaumasN Revol

Prof U LosAndes BogotaColombia

CHEVILLARD Sylvain 01092006 01092009 EacuteNS Lyon MathIFN BrisebarreJ-M Muller

Post-DocINRIA Lorraine

DETREY Jeacutereacutemie 01092003 15012007 EacuteNS Lyon MathIFF de DinechinJ-M Muller

CR2 INRIA

LAUTER Christoph 01092005 17102008 EacuteNS Lyon MathIFF de DinechinJ-M Muller

Intel PortlandOr

MELQUIOND Guillaume 01092003 21112006 EacuteNS Lyon MathIF M Daumas CR2 INRIA

MICHARD Romain 01092004 25062008 EacuteNS LyonINRIA MathIFJ-M MullerA Tisserand

PostDoc INSALyon

RAINASaurabhKumar

01102003 30092006 EacuteNS Lyon (ReacutegionRhocircne-Alpes grant)

MathIFC-P JeannerodJ-M MullerA Tisserand

Ass Pr at ITSEng College(Greater NoidaIndia)

VEYRAT-CHARVILLON

Nicolas 01092004 28062007 EacuteNS Lyon MathIFJ-M MullerA Tisserand

PostDoc at UC Louvain

Past post-doctoral researchers engineers and visitors (greater than or equal to 3 months)

Name First name Function Arrival Departure Home Institution(if appropriate)

Current position

BECHETOILLE Eacutedouard Engineer 01102005 30112006 INRIACNRS Research Engineer

(IPNL)

BEUCHAT Jean-Luc Post-doc 01112001 30112005Swiss National ScienceFoundation

Associate Professor atTsukuba University (Japan)

GIESBRECHT Mark Visitor 01122005 31062006University of WaterlooCanada

Ass Professor (WaterlooUniversity)

JOURDAN Nicolas Engineer 23072007 29022008 INRIAIn charge of ind relations

and techno transfer at IN-RIA Rhocircne-Alpes




nion)











































331 Number Systems





























































People Awarded












Editorial Boards

































National expertise


























































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187
















nion)











































331 Number Systems





























































People Awarded












Editorial Boards

































National expertise


























































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187



































331 Number Systems





























































People Awarded












Editorial Boards

































National expertise


























































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187































































People Awarded












Editorial Boards

































National expertise


























































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187
























Editorial Boards

































National expertise


























































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187




















National expertise


























































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187


















































Library

Program

developed using

links against

Development tools


Userminusspace

tools and libraries

SollyaGappa

FLIP

MPFI

CRlibm FloPoCo

FPLLLMPFR




























bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187
















bull Type library







bull Type toolbox





bull State stable









bull State stable







bull State stable







bull State stable



bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187















bull Type software





bull State stable


bull Type library





bull State stable










bull Type library




bull State stable


bull Type library




bull State stable



bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187















bull Type toolbox





bull State alpha





None


None


None






3102 Teaching




















311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187



























311 Self-Assessment






312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187















312 Perspectives




















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187






















































Formal proof















Other applications























































































































































































































Total 38 37 40 43 29 187


























Other applications























































































































































































































Total 38 37 40 43 29 187
































































































































































































































Total 38 37 40 43 29 187













3. arénaire - computer arithmetic · initially, the team was mainly focused on hardware-oriented...

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