{"id":"https://openalex.org/W2292537122","doi":"https://doi.org/10.1090/mcom/3234","title":"On relative errors of floating-point operations: Optimal bounds and applications","display_name":"On relative errors of floating-point operations: Optimal bounds and applications","publication_year":2016,"publication_date":"2016-11-11","ids":{"openalex":"https://openalex.org/W2292537122","doi":"https://doi.org/10.1090/mcom/3234","mag":"2292537122"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3234","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3234","pdf_url":"https://www.ams.org/mcom/2018-87-310/S0025-5718-2017-03234-8/S0025-5718-2017-03234-8.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2018-87-310/S0025-5718-2017-03234-8/S0025-5718-2017-03234-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110628690","display_name":"Claude-Pierre Jeannerod","orcid":null},"institutions":[{"id":"https://openalex.org/I100532134","display_name":"Universit\u00e9 Claude Bernard Lyon 1","ror":"https://ror.org/029brtt94","country_code":"FR","type":"education","lineage":["https://openalex.org/I100532134","https://openalex.org/I203339264"]},{"id":"https://openalex.org/I113428412","display_name":"\u00c9cole Normale Sup\u00e9rieure de Lyon","ror":"https://ror.org/04zmssz18","country_code":"FR","type":"education","lineage":["https://openalex.org/I113428412","https://openalex.org/I203339264"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210144566","display_name":"Laboratoire de l'Informatique du Parall\u00e9lisme","ror":"https://ror.org/04msnz457","country_code":"FR","type":"facility","lineage":["https://openalex.org/I100532134","https://openalex.org/I113428412","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I203339264","https://openalex.org/I203339264","https://openalex.org/I4210144566"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Claude-Pierre Jeannerod","raw_affiliation_strings":["Inria and Universit\u00c3\u00a9 de Lyon, laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL), 46 all\u00c3\u00a9e d\u00e2\u0080\u0099Italie 69364 Lyon cedex 07, France","ARIC - Arithmetic and Computing (46 All\u00e9e d'Italie \r\n69364 Lyon France - France)","LIP - Laboratoire de l'Informatique du Parall\u00e9lisme (46 All\u00e9e d'Italie 69364 LYON CEDEX 07 - France)"],"affiliations":[{"raw_affiliation_string":"Inria and Universit\u00c3\u00a9 de Lyon, laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL), 46 all\u00c3\u00a9e d\u00e2\u0080\u0099Italie 69364 Lyon cedex 07, France","institution_ids":["https://openalex.org/I100532134","https://openalex.org/I113428412","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"ARIC - Arithmetic and Computing (46 All\u00e9e d'Italie \r\n69364 Lyon France - France)","institution_ids":[]},{"raw_affiliation_string":"LIP - Laboratoire de l'Informatique du Parall\u00e9lisme (46 All\u00e9e d'Italie 69364 LYON CEDEX 07 - France)","institution_ids":["https://openalex.org/I4210144566","https://openalex.org/I113428412"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081963598","display_name":"Siegfried M. Rump","orcid":"https://orcid.org/0000-0002-4779-4800"},"institutions":[{"id":"https://openalex.org/I150744194","display_name":"Waseda University","ror":"https://ror.org/00ntfnx83","country_code":"JP","type":"education","lineage":["https://openalex.org/I150744194"]},{"id":"https://openalex.org/I159176309","display_name":"Universit\u00e4t Hamburg","ror":"https://ror.org/00g30e956","country_code":"DE","type":"education","lineage":["https://openalex.org/I159176309"]},{"id":"https://openalex.org/I884043246","display_name":"Hamburg University of Technology","ror":"https://ror.org/04bs1pb34","country_code":"DE","type":"education","lineage":["https://openalex.org/I884043246"]}],"countries":["DE","JP"],"is_corresponding":false,"raw_author_name":"Siegfried M. Rump","raw_affiliation_strings":["Hamburg University of Technology, Schwarzenbergstra\u00c3\u009fe 95, Hamburg 21071, Germany \u00e2\u0080\u0094 and \u00e2\u0080\u0094 Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan","IRC - Institute for Reliable Computing [Hamburg] (Technische Universit\u00e4t Hamburg-Harburg Institut f\u00fcr Zuverl\u00e4ssiges Rechnen Schwarzenbergstrasse 95 21073 Hamburg - Germany)"],"affiliations":[{"raw_affiliation_string":"Hamburg University of Technology, Schwarzenbergstra\u00c3\u009fe 95, Hamburg 21071, Germany \u00e2\u0080\u0094 and \u00e2\u0080\u0094 Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan","institution_ids":["https://openalex.org/I150744194","https://openalex.org/I159176309"]},{"raw_affiliation_string":"IRC - Institute for Reliable Computing [Hamburg] (Technische Universit\u00e4t Hamburg-Harburg Institut f\u00fcr Zuverl\u00e4ssiges Rechnen Schwarzenbergstrasse 95 21073 Hamburg - Germany)","institution_ids":["https://openalex.org/I884043246"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5110628690"],"corresponding_institution_ids":["https://openalex.org/I100532134","https://openalex.org/I113428412","https://openalex.org/I1294671590","https://openalex.org/I4210144566"],"apc_list":null,"apc_paid":null,"fwci":6.5143,"has_fulltext":true,"cited_by_count":44,"citation_normalized_percentile":{"value":0.966564,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"87","issue":"310","first_page":"803","last_page":"819"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11034","display_name":"Digital Filter Design and Implementation","score":0.9958999752998352,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9884999990463257,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/rounding","display_name":"Rounding","score":0.7894296646118164},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7544590830802917},{"id":"https://openalex.org/keywords/round-off-error","display_name":"Round-off error","score":0.6167629361152649},{"id":"https://openalex.org/keywords/base","display_name":"Base (topology)","score":0.6026899814605713},{"id":"https://openalex.org/keywords/arithmetic-underflow","display_name":"Arithmetic underflow","score":0.5958965420722961},{"id":"https://openalex.org/keywords/floating-point","display_name":"Floating point","score":0.5251036882400513},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.504752516746521},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4878668785095215},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4666559398174286},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.4551513195037842},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.444990336894989},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4287380874156952},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3719541132450104},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.32183852791786194},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1605447232723236},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15572509169578552}],"concepts":[{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.7894296646118164},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7544590830802917},{"id":"https://openalex.org/C61005703","wikidata":"https://www.wikidata.org/wiki/Q2145211","display_name":"Round-off error","level":2,"score":0.6167629361152649},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.6026899814605713},{"id":"https://openalex.org/C3265923","wikidata":"https://www.wikidata.org/wiki/Q669129","display_name":"Arithmetic underflow","level":2,"score":0.5958965420722961},{"id":"https://openalex.org/C84211073","wikidata":"https://www.wikidata.org/wiki/Q117879","display_name":"Floating point","level":2,"score":0.5251036882400513},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.504752516746521},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4878668785095215},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4666559398174286},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.4551513195037842},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.444990336894989},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4287380874156952},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3719541132450104},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.32183852791786194},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1605447232723236},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15572509169578552},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/mcom/3234","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3234","pdf_url":"https://www.ams.org/mcom/2018-87-310/S0025-5718-2017-03234-8/S0025-5718-2017-03234-8.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-00934443v4","is_oa":true,"landing_page_url":"https://inria.hal.science/hal-00934443","pdf_url":"https://inria.hal.science/hal-00934443v4/file/JeannerodRump2016.pdf","source":{"id":"https://openalex.org/S4406922466","display_name":"SPIRE - Sciences Po Institutional REpository","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics of Computation, 2018, 87, pp.803-819. ⟨10.1090/mcom/3234⟩","raw_type":"Journal articles"}],"best_oa_location":{"id":"doi:10.1090/mcom/3234","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3234","pdf_url":"https://www.ams.org/mcom/2018-87-310/S0025-5718-2017-03234-8/S0025-5718-2017-03234-8.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2292537122.pdf","grobid_xml":"https://content.openalex.org/works/W2292537122.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W49576723","https://openalex.org/W581104972","https://openalex.org/W1968165559","https://openalex.org/W1969155620","https://openalex.org/W1984470719","https://openalex.org/W2002064155","https://openalex.org/W2012407419","https://openalex.org/W2020804487","https://openalex.org/W2023462640","https://openalex.org/W2032572332","https://openalex.org/W2036470874","https://openalex.org/W2043596201","https://openalex.org/W2047319360","https://openalex.org/W2075488055","https://openalex.org/W2296909587","https://openalex.org/W2329095262","https://openalex.org/W2752061190","https://openalex.org/W3022909171"],"related_works":["https://openalex.org/W3039927303","https://openalex.org/W2383134623","https://openalex.org/W1989068301","https://openalex.org/W4388911419","https://openalex.org/W2061091230","https://openalex.org/W4240706710","https://openalex.org/W4242955624","https://openalex.org/W4231731735","https://openalex.org/W196954033","https://openalex.org/W3185976014"],"abstract_inverted_index":{"Rounding":[0],"error":[1,177,194],"analyses":[2],"of":[3,14,19,62,70,85,168,179,191],"numerical":[4,181],"algorithms":[5],"are":[6,99],"most":[7],"often":[8],"carried":[9],"out":[10],"via":[11],"repeated":[12],"applications":[13],"the":[15,36,50,57,60,68,77,82,116,119,175,192],"so-called":[16],"standard":[17],"models":[18,34],"floating-point":[20,91,134],"arithmetic.":[21],"Given":[22],"a":[23],"round-to-nearest":[24],"function":[25],"$\\mathrm":[26],"{fl}$":[27],"and":[28,31,43,59,75,97,104,118,163],"barring":[29],"underflow":[30],"overflow,":[32],"such":[33,197],"bound":[35],"relative":[37],"errors":[38],"$E_1(t)":[39],"=":[40,45,123,129],"|t-\\mathrm":[41,46],"{fl}(t)|/|t|$":[42],"$E_2(t)":[44],"{fl}(t)|/|\\mathrm":[47],"{fl}(t)|$":[48],"by":[49,102],"unit":[51],"roundoff":[52],"$u$.":[53],"This":[54,166],"paper":[55],"investigates":[56],"possibility":[58],"usefulness":[61],"refining":[63],"these":[64,202],"bounds,":[65],"both":[66,157],"in":[67,76,145,159,183],"case":[69,78],"an":[71,86],"arbitrary":[72],"real":[73,110],"$t$":[74,80,108],"where":[79],"is":[81,109,171],"exact":[83],"result":[84],"arithmetic":[87],"operation":[88],"on":[89,115,201],"some":[90],"numbers.":[92,135],"We":[93,136],"show":[94],"that":[95,138],"$E_1(t)$":[96],"$E_2(t)$":[98],"optimally":[100],"bounded":[101],"$u/(1+u)$":[103],"$u$,":[105],"respectively,":[106],"when":[107,121],"or,":[111],"under":[112],"mild":[113],"assumptions":[114],"base":[117,146,160],"precision,":[120],"$t":[122,128],"x":[124],"\\pm":[125],"y$":[126],"or":[127],"xy$":[130],"with":[131],"$x,y$":[132],"two":[133],"prove":[137],"while":[139],"this":[140],"remains":[141],"true":[142],"for":[143,156,196],"division":[144,158],"$\\beta":[147,161],">":[148],"2$,":[149],"smaller,":[150],"attainable":[151],"bounds":[152,170,195,203],"can":[153],"be":[154],"derived":[155],"=2$":[162],"square":[164],"root.":[165],"set":[167],"optimal":[169],"then":[172],"applied":[173],"to":[174],"rounding":[176],"analysis":[178],"various":[180],"algorithms:":[182],"all":[184],"cases,":[185],"we":[186],"obtain":[187],"significantly":[188],"shorter":[189],"proofs":[190],"best-known":[193],"algorithms,":[198],"and/or":[199],"improvements":[200],"themselves.":[204]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3}],"updated_date":"2026-03-31T07:56:22.981413","created_date":"2025-10-10T00:00:00"}