arith22

09:45 to 10:45 SESSION 1: Keynote Talk

Chair: Sylvie Boldo

Calculating in Floating Sexagesimal Place Value Notation, 4000 years ago (slides)
Christine Proust, Laboratoire SPHERE (CNRS & Université Paris Diderot), France

Abstract: By the end of the third millennium BCE in Mesopotamia an innovation of major significance for the history of mathematics occurred: the sexagesimal place value notation. A sophisticated mathematical culture was subsequently developed by masters attached to the scribal schools that flourished in Iraq, Iran and Syria during the first centuries of the second millennium BCE. The best known aspect of this mathematical culture is the art of solving quadratic problems. The numerical algorithms exploiting the properties of base 60 and the floating notation are less known. This talk presents some of these algorithms, especially those based on factorization methods.

10:45 to 11:45 SESSION 2: Arithmetic Units 1

Chair: Neil Burgess

Low-Cost Duplicate Multiplication (slides)
Michael Sullivan and Earl Swartzlander

Minimizing Energy by Achieving Optimal Sparseness in Parallel Adders (slides)
Mustafa Aktan, Dursun Baran, and Vojin Oklobdzija

12:00 to 13:30 Lunch

13:30 to 14:30 SESSION 3: Arithmetic Units 2

Chair: Vojin Oklobdzija

An Efficient Softcore Multiplier Architecture for Xilinx FPGAs (slides)
Martin Kumm, Shahid Abbas, and Peter Zipf

Design and Implementation of an Embedded FPGA Floating Point DSP Block
Martin Langhammer and Bogdan Pasca

14:45 to 15:45 SESSION 4: Elementary and Special Functions 1

Chair: Peter Tang

Hardware implementations of fixed-point Atan2 (slides)
Matei Istoan and Florent de Dinechin

A robust general-purpose method for faithfully rounded floating-point function approximation in FPGAs (slides)
David Thomas

16:00 to 17:00 Ceremony

Presentation of the Medal of École Normale Supérieure de Lyon to Milos Ercegovac by the President of ENS Lyon (slides)

17:00 to 18:30 SPECIAL SESSION on the State of the Art of FP Units

Chair: Alberto Nannarelli

Intel(r) AVX-512 Instructions and Their Use in the Implementation of Math Functions (slides)
Marius Cornea, INTEL, USA

Floating-point Arithmetic in AMD Processors (slides)
Michael Schulte, AMD, USA

The IBM z13 SIMD Accelerators for Integer, String, and Floating-Point (slides)
Eric Schwarz, IBM, USA

ARM FPUs: Low Latency is Low Energy (slides)
David Lutz, ARM, USA

Tuesday, June 23rd, 2015

08:00 to 09:30 SESSION 5: Elementary and Special Functions 2

Chair: Naofumi Takagi

Precise and fast computation of elliptic integrals and functions (slides)
Toshio Fukushima

Code generators for mathematical functions (slides)
Nicolas Brunie, Florent de Dinechin, Olga Kupriianova, and Christoph Lauter

Semi-Automatic Floating-Point Implementation of Special Functions (slides)
Christoph Lauter and Marc Mezzarobba

09:30 to 10:00 Coffee Break

10:00 to 11:00 SESSION 6: Keynote Talk

Chair: David Hough

The End of Numerical Error (slides)
John Gustafson

Abstract: It is time to overthrow a century of methods based on floating point arithmetic. Current technical computing is based on the acceptance of rounding error using numerical representations that were invented in 1914, and acceptance of sampling error using algorithms designed for a time when transistors were very expensive. By sticking to an antiquated storage format (now codified as an IEEE standard) well into the exascale area, we are wasting power, energy, storage, bandwidth, and programmer effort. The pursuit of exascale floating point is ridiculous, since we do not need to be making 10¹⁸ sloppy rounding errors per second; we need instead to get provable, valid results for the first time, by turning the speed of parallel computers into higher quality answers instead of more junk per second.

We introduce the 'unum' (universal number), a superset of IEEE Floating Point, that contains extra metadata fields that actually save storage, yet give more accurate answers that do not round, overflow, or underflow. The potential they offer for improved programmer productivity is enormous. They also provide, for the first time, the hope of a numerical standard that guarantees bitwise identical results across different computer architectures. Unum format is the basis for the 'ubox' method, which redefines what is meant by "high performance" by measuring performance in terms of the knowledge obtained about the answer and not the operations performed per second. Examples are given for practical application to structural analysis, radiation transfer, the n-body problem, linear and nonlinear systems of equations, and Laplace's equation. This is a fresh approach to scientific computing that allows proper, rigorous representation of real number sets for the first time.

11:00 to 12:00 SESSION 7: Medium and Multiple Precision 1

Chair: Eric Schwarz

Faster FFTs in medium precision
Joris van der Hoeven and Grégoire Lecerf

Efficient implementation of elementary functions in the medium-precision range
Fredrik Johansson (slides)

12:00 to 13:30 Lunch

13:30 to 14:30 SESSION 8: Medium and Multiple Precision 2

Chair: Michael Schulte

Efficient divide-and-conquer multiprecision integer division (slides)
William Hart

Reliable evaluation of the Worst-Case Peak Gain matrix in multiple precision (slides)
Anastasia Volkova, Thibault Hilaire, and Christoph Lauter

14:30 to 15:00 Coffee Break

15:00 to 16:00 SESSION 9: Keynote Talk

Chair: Milos Ercegovac

Numerical challenges in long term integrations of the Solar system
Jacques Laskar, CNRS, Observatoire de Paris, France

Abstract: Long time integrations of the planetary motion in the Solar System has been a challenging work in the past decades. The progress have followed the improvements of computer technology, but also the improvements in the integration algorithms. This quest has led to the development of high order dedicated symplectic integrators that have a stable behavior over long time scales. As important in the increase of the computing performances is the use of parallel algorithms that have divided the computing times by an order of magnitude. A specific aspect of these long term computation is also a careful monitoring of the accumulation of the roundoff error in the numerical algorithms, where all bias should be avoided. It should also be noted that for these computations, not only compensated summation is required, but also 80 bits extended precision floating point arithmetics.

Integrating the equation of motion is only a part of the work. One needs also to determine precise initial conditions in order to ensure that the long time integration represent actually the motion of the real Solar System.

Once these steps are fulfilled, the main limitation in the obtention of a precise solution of the planetary motion will be given by the chaotic nature of the Solar system that will strictly limit the possibility of precise prediction for the motion of the planets to about 60 Myr.

16:00 to 17:00 SESSION 10: Residue Number Systems

Chair: David Matula

Contributions to the Design of Residue Number System Architectures (slides)
Benoît Gérard, Jean-Gabriel Kammerer, and Nabil Merkiche

RNS Arithmetic Approach in Lattice-based Cryptography - Accelerating the "Rounding-off" Core Procedure (slides)
Jean-Claude Bajard, Julien Eynard, Nabil Merkiche, and Thomas Plantard

17:45 to 22:00 Visit of the "Musée des Confluences" and Conference banquet

Wednesday, June 24th, 2015

08:00 to 10:00 SESSION 11: Modular and Finite-Field Arithmetic

Chair: Peter Kornerup

Modulo-(2ⁿ-2^q-1) Parallel Prefix Addition via Excess-Modulo Encoding of Residues (slides)
Seyed Hamed Fatemi Langroudi and Ghassem Jaberipur

New Bit-Level Serial GF(2^m) Multiplication Using Polynomial Basis (slides)
Hayssam El-Razouk and Arash Reyhani-Masoleh

Modular multiplication and division algorithms based on continued fraction expansion (slides)
Mourad Gouicem

Efficient Modular Exponentiation Based on Multiple Multiplications by a Common Operand (slides)
Christophe Negre, Thomas Plantard, and Jean-Marc Robert