LIP Laboratory, CNRS, ENSL, INRIA, UCBL · LIP Laboratory, CNRS, ENSL, INRIA, UCBL SymB INRIA...

Focus & Challenges Contributions Future

Computer Arithmetic

Arenaire project-team

LIP Laboratory, CNRS, ENSL, INRIA, UCBL

SymB INRIA Evaluation Seminar – Paris, November 14

Arenaire project-team Computer Arithmetic


The team, facts

Created in 1998, former head J.-M. Muller (1998-2004)

2002 change 2006Permanent professors and researchers 7 -2+3 8Permanent engineer 0 +0.4 0.4Temporary engineers, postdocs 1 -3+4 2PhD students 4 -5+9 8

5 PhD awarded: 2 researchers, 1 engineer1 temporary researcher, 1 industry

144 publications 2002-2006 (50 books/journals)



Outline

1 Scientific focus and challenges

2 ContributionsHardware arithmeticSoftware operators and toolsArithmetic and Formal ProofsAlgorithms and Arithmetics

3 Objectives 2007-2010



Scientific FocusImproving Knowledge in Computer Arithmetic

Arithmetic operators

Number representation+,−, ,×,÷,

√, ·/cst, . . .

Integers, real/complex numbersIntervals, exact computation

Elementary/special functions

Specific: DSP, cryptography, . . .

Improving quality

PerformanceAccuracy, reliability

portabilityPower consumption

Code size

Implementationtargets

Hardware: ASIC, FPGA (Verilog, VHDL, ...)

Low-level: dedicated and embedded chipsSoftware libraries (C++, Coq, Maple, ...)



Improving knowledge in Computer ArithmeticMain challenges

Increasing machine power/problem dimensionsComplex software

Various target architectures/constraintsSophisticated operator algorithms

◦ Guaranteeing the reliability of software-intensive systems

◦ Adequacy software/hardware (ex: embedded systems)



Scientific methodologyAlgorithmics, specification, automatic tools

Algorithmic, specification

Arithmetic algorithms and algorithms for arithmetic

Specification, normalization actions, properties of operators

Trade-off reliability/performance

Impact of the arithmetic choices on solving methods

Automatic tools

Code analysis and error diagnosing tools

Automatic tools : algorithm/code synthesis, automatic proof

Code optimization for performance



Algorithmic, specification, automatic toolsEx: Correctly rounded elementary function library

Better numericsLHC@home

36000 heterogeneous machines

Fine−grainarithmetic

Arithmeticand algorithms

Formal proofNumber theory

approximation

Tools for

polynomial TMD Gappa tool Validated infnorm

CRLibm

Coq proofs

Performance Proven correct rounding

Perfect double−precision elementary functions

Normalization

IEEE−754 R



Algorithmic, specification, automatic toolsOur strengths

From algebra and numerical analysis to code development

From hardware to software

From fixed-point numbers to intervals and exact computation


Focus & Challenges Contributions Future Hardware Operators Proofs Algorithms & Arithmetics

Outline






Contribution 1Hardware arithmetic

Automatic tools for the generation of arithmetic operators

Fixed-point elementary functionsMultipartite tablesSMSOSmallApproxHOTBM

Floating-point operators and elementary functionsFPLibrary

Algebraic functions: ÷,√

, 1/√

, n√

, . . .

DivGen

CryptographyModular operatorsApplications in encryption and signature



Contribution 1Hardware arithmetic

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

x

f(x)

0

0

1

0 0.25 0.5 0.75 1

x

f(x)

f(X)wO

..

.

wO + g

wO + g

wO + g

wO + g

XwI

T0( )A0

A

B

A0

A1

A2

A3

B1

Bn

B2

Tn( , )

T2( , )

T1( , )

An Bn

A2 B2

A1 B1

O

X

R

O

X

R

O

X

R

Ak

B′

k

b1

.

.

.

...

...

k

Kk( )

Kk( )

Kk( ) ·

O

X

R

O

X

R

O

X

R

O

X

R

Sk,1

Sk,2

B′

k

Ak,2

Ak,1

Ak,1

Ak,2

Sk,mkS ′

k,mkAk,mk

Ak,mk

Target architecture

Parameters

Constraints

Hardware operator

Input function

Function approximationError analysis

FPGA implementation

VHDL generation

Architectural exploration

Evaluation algorithm



Contribution 2Software operators and tools

Leading position on the correct rounding of elementaryfunctions in double-precision floating-point arithmetic

Meplib library, a key point for function evaluation: newmethods for generating best polynomial approximationsunder constraints such as size, accuracy, or evaluation costconstraints

Various algorithmic progress: use of fused multiply-add, errorrepresentation, range reduction, RN-coding, etc.



Contribution 2Software operators and tools

FLIP: A floating-point library for integer processors

Basic arithmetic operators±,×,÷,

√, fma, 1/x , 1/

√, ...

Adaptation of algorithmsto the target architecture

⇓

Speed-ups from 1.25 to 4without sacrificing accuracy

ST200 embedded processorsVLIW / No hardware floating-point

Multimedia applications(satellite boxes, TV cables, DVD, ...)

Control

8BR(1bit)

IPUException

Control

16x32

Mult

16x32

MultDPU

Prefetch

buffer

4 set

D$

32KB

Store

Load

Unit

64

GPR

(32b)

File

Reg

32KB

I$

Branch Br RegFile

ALU ALU ALU ALU

Pre−D

ecod

e

Cluster0 Cluster

Unit

Reg’s

Impact on technology:Integration of FLIP into the industrial version of the ST200 compiler (2005)

Ongoing extension to other targets like Texas Instruments DSPs



Contribution 3Arithmetic and formal proofs

@@

@@R

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Basic arithmetic blocks

/∗ Mul12 : e r r o r−f r e e m u l t i p l i c a t i o n ∗/double c = 134217729.;up = u * c; vp = v * c;u1 = (u - up) + up;v1 = (v - vp) + vp;u2 = u - u1; v2 = v - v1;p = u * v;e = u1 * v1 - p + u1 * v2 + u2 * v1 +

u2 * v2;

/∗ Add12 : e r r o r−f r e e a d d i t i o n ∗/s = u + v;t = s - u;e = v - t;

Hand-written proof in Coq

Complete function

/∗ Compute app rox imat i on to r = 1/ s q r t (m) ∗/r0 = SQRTPOLYC0 + m * (SQRTPOLYC1 + m * (

SQRTPOLYC2 + m * (SQRTPOLYC3 + m *SQRTPOLYC4)));

/∗ I t e r a t e two t imes on doub l e ∗/r1 = 0.5 * r0 * (3 - m * (r0 * r0));r2 = 0.5 * r1 * (3 - m * (r1 * r1));

/∗ I t e r a t e two t imes on double−doub l e ∗/Mul12(r2Sqh , r2Sql , r2 , r2);Add12(r2PHr2h , r2PHr2l , r2 , 0.5 * r2);Mul12(mMr2h , mMr2l , m, r2);Mul22(mMr2Ch , mMr2Cl , mMr2h , mMr2l , r2Sqh ,

r2Sql);

MHmMr2Ch = -0.5 * mMr2Ch;MHmMr2Cl = -0.5 * mMr2Cl;Add22(r3h , r3l , r2PHr2h , r2PHr2l , MHmMr2Ch ,

MHmMr2Cl);Mul22(r3Sqh , r3Sql , r3h , r3l , r3h , r3l);Mul22(mMr3Sqh , mMr3Sql , m,0, r3Sqh , r3Sql);Mul22(r4h , r4l , r3h , r3l , 1, -0.5* mMr3Sql);

/∗ Mu l t i p l y r e c i p r o c a l squa r e r oo t by m ∗/Mul22(srtmh , srtml , m, 0, r4h , r4l);

Automatic proof by Gappa



Contribution 3Floating-point arithmetic and proofs

A Coq library of basic properties and algorithms in floating-pointarithmetic

Motivation:

Improve confidence in proofs

Accurate yet generic hypotheses

A comprehensive basis of formal blocks to build upon

Examples:

- Argument reduction, Horner’s scheme

- Knuth’s error-free addition, use of the FMA operator

- Exotic rounding modes, etc.



Contribution 3Certification of numerical algorithms

Verifying correctness of complex numerical algorithms is long,tedious, and error-prone, when done by hand

Objective: automate this process

Given some source code, bound values and rounding errors:Gappa

Bound truncation and approximation errors introduced bynumerical algorithms

Applications:

- FLIP library: high confidence in the embedded code

- CRlibm library: faster design and tighter bounds

- CGAL library: robust geometric predicates



Contribution 4Comparing and combining arithmetics

Algorithmic Complexity

algebraic model

bit model

division−free

polynomials

integers

for linear algebra

Guaranteed Enclosures

expansions withinterval remainders

Taylor modelarithmetic

floating−pointarithmetic roundoff errors

Software developmentLinBox: finite field linear algebra via floats, faster than BLASMPFI: arbitrary precision interval arithmetic



Contribution 4Algorithmic complexity in linear algebra

Divideand

conquer

Hint: try to at most double the data size (or the precision) whenthe dimension is divided by two

Application: essentially optimal computation of the inverse of ann × n polynomial matrix of degree d in O˜(n3d) operations(output size = n3d)

2000-2006 progress: reductions to matrix multiplication over thepolynomials or in bit complexity



Contribution 4Algorithmic complexity in linear algebra

Divide-doubleand

conquer

Hint: try to at most double the data size (or the precision) whenthe dimension is divided by two

Application: essentially optimal computation of the inverse of ann × n polynomial matrix of degree d in O˜(n3d) operations(output size = n3d)

2000-2006 progress: reductions to matrix multiplication over thepolynomials or in bit complexity



Industrial transfer

STMicroelectronics

-STMicroelectonics and Region Rhone-Alpes grant-STMicroelectronis/Arenaire contract-Joint implication in the Pole de competitivite mondialMinalogic (function approximation at compile-time)

Intel (correctly rounded functions)

Maple (linear algebra, matrix polynomials)

NIA-NASA (automatic proof)



Outline






Challenge

New automated development approach for handling softwarecomplexity and hardware/software intrication in computerarithmetic

- From specification to automatic proving- Diagnose and circumvent numerical errors- Expression or critical code evaluation- Higher level of programmation, algorithm synthesis- Code synthesis

Problem

Algorithm

Mathematical approach

Model

Code

Chip/Computer



Three main directions

Arithmetics and algorithms. New algorithms for coarseroperators. Standardization actions. Function approximation,linear algebra, lattice basis reduction, global optimization.

Quality issues: specification and validation. Specification:mathematics, accuracy, performance, property to prove, etc.Algorithmic complexity of certified computation.

Synthesis of the implementation. Analysis of thespecification. Approximation/evaluation and code generation.Proof design and check. Prototyping future compilers.



Synthesis/compilation for computer arithmetic


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