Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
Identifieur interne : 001076 ( Istex/Corpus ); précédent : 001075; suivant : 001077Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
Auteurs : Vincent Lefèvre ; Damien Stehlé ; Paul ZimmermannSource :
- Lecture Notes in Computer Science [ 0302-9743 ]
Abstract
Abstract: We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10− 15 ulp, and we give the worst ones. In particular, the worst case for |x| ≥ 3 ×10− 11 is $\exp(9.407822313572878 \times 10^{-2}) = 1.098645682066338\,5\,0000000000000000\,278\ldots$ . This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.
Url:
DOI: 10.1007/978-3-540-85521-7_7
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In particular, the worst case for |x| ≥ 3 ×10− 11 is $\exp(9.407822313572878 \times 10^{-2}) = 1.098645682066338\,5\,0000000000000000\,278\ldots$ . This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.</div></front></TEI><istex><corpusName>springer-ebooks</corpusName><author><json:item><name>Vincent Lefèvre</name><affiliations><json:string>INRIA/ÉNS Lyon, Université de Lyon/LIP, 46 allée d’Italie, F-69364, Lyon Cedex 07, France</json:string><json:string>E-mail: Vincent.Lefevre@inria.fr</json:string></affiliations></json:item><json:item><name>Damien Stehlé</name><affiliations><json:string>CNRS/ÉNS Lyon, Université de Lyon/LIP/INRIA Arenaire, 46 allée d’Italie, F-69364, Lyon Cedex 07, France</json:string><json:string>E-mail: damien.stehle@gmail.com</json:string></affiliations></json:item><json:item><name>Paul Zimmermann</name><affiliations><json:string>LORIA/INRIA Lorraine, Bâtiment A, Technopôle de Nancy-Brabois, , 615 rue du jardin botanique, F-54602, Villers-lès-Nancy Cedex, France</json:string><json:string>E-mail: Paul.Zimmermann@loria.fr</json:string></affiliations></json:item></author><arkIstex>ark:/67375/HCB-7BVMKXZ6-N</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10− 15 ulp, and we give the worst ones. 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type="first">Madhu</forename><surname>Sudan</surname></persName></editor><editor><persName><forename type="first">Demetri</forename><surname>Terzopoulos</surname></persName></editor><editor><persName><forename type="first">Doug</forename><surname>Tygar</surname></persName></editor><editor><persName><forename type="first">Moshe</forename><forename type="first">Y.</forename><surname>Vardi</surname></persName></editor><editor><persName><forename type="first">Gerhard</forename><surname>Weikum</surname></persName></editor><idno type="pISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="seriesID">558</idno></series></biblStruct></sourceDesc></fileDesc><profileDesc><abstract xml:lang="en"><head>Abstract</head><p>We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10<hi rend="superscript">− 15</hi> ulp, and we give the worst ones. In particular, the worst case for |<hi rend="italic">x</hi>| ≥ 3 ×10<hi rend="superscript">− 11</hi> is <formula xml:id="IEq1" notation="TEX"><media mimeType="image" url=""></media>$\exp(9.407822313572878 \times 10^{-2}) = 1.098645682066338\,5\,0000000000000000\,278\ldots$
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This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.</p></abstract><textClass ana="subject"><keywords scheme="book-subject-collection"><list><label>SUCO11645</label><item><term>Computer Science</term></item></list></keywords></textClass><textClass ana="subject"><keywords scheme="book-subject"><list><label>I</label><item><term type="Primary">Computer Science</term></item><label>I12026</label><item><term type="Secondary" subtype="priority-1">Arithmetic and Logic Structures</term></item><label>I16021</label><item><term type="Secondary" subtype="priority-2">Algorithm Analysis and Problem Complexity</term></item><label>I17028</label><item><term type="Secondary" subtype="priority-3">Discrete Mathematics in Computer Science</term></item><label>I17052</label><item><term type="Secondary" subtype="priority-4">Symbolic and Algebraic 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ID="Chap7" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingStyle="Unnumbered" TocLevels="0"><ChapterID>7</ChapterID><ChapterDOI>10.1007/978-3-540-85521-7_7</ChapterDOI><ChapterSequenceNumber>7</ChapterSequenceNumber><ChapterTitle Language="En">Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format</ChapterTitle><ChapterFirstPage>114</ChapterFirstPage><ChapterLastPage>126</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2008</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>978-3-540-85521-7</BookID><BookTitle>Reliable Implementation of Real Number Algorithms: Theory and Practice</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Vincent</GivenName><FamilyName>Lefèvre</FamilyName></AuthorName><Contact><Email>Vincent.Lefevre@inria.fr</Email></Contact></Author><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>Damien</GivenName><FamilyName>Stehlé</FamilyName></AuthorName><Contact><Email>damien.stehle@gmail.com</Email></Contact></Author><Author AffiliationIDS="Aff3"><AuthorName DisplayOrder="Western"><GivenName>Paul</GivenName><FamilyName>Zimmermann</FamilyName></AuthorName><Contact><Email>Paul.Zimmermann@loria.fr</Email></Contact></Author><Affiliation ID="Aff1"><OrgDivision>INRIA/ÉNS Lyon</OrgDivision><OrgName>Université de Lyon/LIP</OrgName><OrgAddress><Street>46 allée d’Italie</Street><Postcode>F-69364</Postcode><City>Lyon Cedex 07</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>CNRS/ÉNS Lyon</OrgDivision><OrgName>Université de Lyon/LIP/INRIA Arenaire</OrgName><OrgAddress><Street>46 allée d’Italie</Street><Postcode>F-69364</Postcode><City>Lyon Cedex 07</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>LORIA/INRIA Lorraine, Bâtiment A, Technopôle de Nancy-Brabois</OrgDivision><OrgName> </OrgName><OrgAddress><Street>615 rue du jardin botanique</Street><Postcode>F-54602</Postcode><City>Villers-lès-Nancy Cedex</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10<Superscript>− 15</Superscript> ulp, and we give the worst ones. In particular, the worst case for |<Emphasis Type="Italic">x</Emphasis>| ≥ 3 ×10<Superscript>− 11</Superscript> is <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject FileRef="978-3-540-85521-7_7_Chapter_TeX2GIFIEq1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\exp(9.407822313572878 \times 10^{-2}) = 1.098645682066338\,5\,0000000000000000\,278\ldots$</EquationSource></InlineEquation>. This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format</title></titleInfo><name type="personal"><namePart type="given">Vincent</namePart><namePart type="family">Lefèvre</namePart><affiliation>INRIA/ÉNS Lyon, Université de Lyon/LIP, 46 allée d’Italie, F-69364, Lyon Cedex 07, France</affiliation><affiliation>E-mail: Vincent.Lefevre@inria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Damien</namePart><namePart type="family">Stehlé</namePart><affiliation>CNRS/ÉNS Lyon, Université de Lyon/LIP/INRIA Arenaire, 46 allée d’Italie, F-69364, Lyon Cedex 07, France</affiliation><affiliation>E-mail: damien.stehle@gmail.com</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Paul</namePart><namePart type="family">Zimmermann</namePart><affiliation>LORIA/INRIA Lorraine, Bâtiment A, Technopôle de Nancy-Brabois, , 615 rue du jardin botanique, F-54602, Villers-lès-Nancy Cedex, France</affiliation><affiliation>E-mail: Paul.Zimmermann@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="conference" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-BFHXPBJJ-3">conference</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2008</dateIssued><copyrightDate encoding="w3cdtf">2008</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10− 15 ulp, and we give the worst ones. In particular, the worst case for |x| ≥ 3 ×10− 11 is $\exp(9.407822313572878 \times 10^{-2}) = 1.098645682066338\,5\,0000000000000000\,278\ldots$ . This work can be extended to other elementary functions in the decimal64 format and allows the design of reasonably fast routines that will evaluate these functions with correct rounding, at least in some domains.</abstract><relatedItem type="host"><titleInfo><title>Reliable Implementation of Real Number Algorithms: Theory and Practice</title><subTitle>International Seminar Dagstuhl Castle, Germany, January 8-13, 2006 Revised Papers</subTitle></titleInfo><name type="personal"><namePart type="given">Peter</namePart><namePart type="family">Hertling</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Christoph</namePart><namePart type="given">M.</namePart><namePart type="family">Hoffmann</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Wolfram</namePart><namePart type="family">Luther</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Nathalie</namePart><namePart type="family">Revol</namePart><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2008</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11645">Computer Science</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="I">Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I12026">Arithmetic and Logic Structures</topic><topic authority="SpringerSubjectCodes" authorityURI="I16021">Algorithm Analysis and Problem Complexity</topic><topic authority="SpringerSubjectCodes" authorityURI="I17028">Discrete Mathematics in Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I17052">Symbolic and Algebraic Manipulation</topic><topic authority="SpringerSubjectCodes" authorityURI="I1701X">Numeric Computing</topic></subject><identifier type="DOI">10.1007/978-3-540-85521-7</identifier><identifier type="ISBN">978-3-540-85520-0</identifier><identifier type="eISBN">978-3-540-85521-7</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">182288</identifier><identifier type="BookID">978-3-540-85521-7</identifier><identifier type="BookChapterCount">12</identifier><identifier type="BookVolumeNumber">5045</identifier><identifier type="BookSequenceNumber">5045</identifier><part><date>2008</date><detail type="volume"><number>5045</number><caption>vol.</caption></detail><extent 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