Handbook of Floating Point Arithmetic.pdf

Jean-Michel Muller

Nicolas Brisebarre

Florent de Dinechin

Claude-Pierre Jeannerod

Vincent Lefevre

Guillaume Melquiond

Nathalie Revol

Damien Stehle

Serge Torres

Handbook of

Floating-Point

Arithmetic

Birkhauser

Boston Basel Berlin

Jean-Michel Muller

Nicolas Brisebarre

Florent de Dinechin

CNRS, Laboratoire LIP

ENSL, Laboratoire LIP

Ecole Normale

Superieure de Lyon

46, allee d’Italie

69364 Lyon Cedex 07

France

jean-michel.muller@

nicolas.brisebarre@

orent.de.dinechin@

ens-lyon.fr

Claude-Pierre Jeannerod

Vincent Lefevre

Guillaume Melquiond

INRIA, Laboratoire LIP

INRIA Saclay – Ile-de-

Ecole Normale

France

Superieure de Lyon

Parc Orsay Universite

46, allee d’Italie

4, rue Jacques Monod

69364 Lyon Cedex 07

91893 Orsay Cedex

France

claude-pierre.jeannerod@

ens-lyon.fr

vincent@vinc17.net

guillaume.melquiond@

Nathalie Revol

Damien Stehle

Serge Torres

INRIA, Laboratoire LIP

CNRS, Macquarie University,

ENSL, Laboratoire LIP

Ecole Normale

and University of Sydney

Ecole Normale

Superieure de Lyon

School of Mathematics

Superieure de Lyon

46, allee d’Italie

and Statistics

46, allee d’Italie

69364 Lyon Cedex 07

University of Sydney

69364 Lyon Cedex 07

France

Sydney NSW 2006

France

nathalie.revol@ens-lyon.fr

Australia

serge.torres@ens-lyon.fr

damien.stehle@gmail.com

ISBN 978-0-8176-4704-9

e-ISBN 978-0-8176-4705-6

DOI 10.1007/978-0-8176-4705-6

Library of Congress Control Number: 2009939668

Mathematics Subject Classication (2000): 65Y99, 68N30

ACM Subject Classication: G.1.0, G.4

Birkhauser Boston, a part of Springer Science+Business Media, LLC 2010

the publisher (Birkhauser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY

10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with

any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar

methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identied

as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

Birkhauser Boston is part of Springer Science+Business Media (www.birkhauser.com)

inria.fr

Contents

Preface

ListofFigures xvii

ListofTables

xxi

IIntroduction,BasicDenitions,andStandards 1

1Introduction 3

1.1SomeHistory............................ 3

1.2DesirableProperties......................... 6

1.3SomeStrangeBehaviors...................... 7

1.3.1Somefamousbugs..................... 7

1.3.2Difcultproblems ..................... 8

2DenitionsandBasicNotions 13

2.1Floating-PointNumbers...................... 13

2.2Rounding............................... 20

2.2.1Roundingmodes...................... 20

2.2.2Usefulproperties...................... 22

2.2.3Relativeerrorduetorounding.............. 23

2.3Exceptions.............................. 25

2.4LostorPreservedPropertiesoftheArithmeticontheReal

Numbers............................... 27

2.5NoteontheChoiceoftheRadix.................. 29

2.5.1Representationerrors.................... 29

2.5.2Acaseforradix10..................... 30

2.6ToolsforManipulatingFloating-PointErrors.......... 32

2.6.1Theulpfunction...................... 32

2.6.2Errorsinulpsandrelativeerrors............. 37

2.6.3Anexample:iteratedproducts.............. 37

2.6.4Unitroundoff........................ 39

2.7NoteonRadixConversion..................... 40

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