{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T16:27:07Z","timestamp":1780936027303,"version":"3.54.1"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"310","license":[{"start":{"date-parts":[[2018,7,7]],"date-time":"2018-07-07T00:00:00Z","timestamp":1530921600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function\n \n \n \n \n f<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n \\mathrm {fl}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and barring underflow and overflow, such models bound the relative errors\n \n \n \n \n \n E<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n t<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n f<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n |<\/mml:mo>\n <\/mml:mrow>\n t<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n E_1(t) = |t-\\mathrm {fl}(t)|\/|t|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n E<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n t<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n f<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n f<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n E_2(t) = |t-\\mathrm {fl}(t)|\/|\\mathrm {fl}(t)|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by the unit roundoff\n \n \n \n u<\/mml:mi>\n u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and in the case where\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the exact result of an arithmetic operation on some floating-point numbers. We show that\n \n \n \n \n \n E<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n E_1(t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n E<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n E_2(t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are optimally bounded by\n \n \n \n \n u<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n u\/(1+u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n u<\/mml:mi>\n u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , respectively, when\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is real or, under mild assumptions on the base and the precision, when\n \n \n \n \n t<\/mml:mi>\n =<\/mml:mo>\n x<\/mml:mi>\n \n \u00b1\n \n <\/mml:mo>\n y<\/mml:mi>\n <\/mml:mrow>\n t = x \\pm y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n t<\/mml:mi>\n =<\/mml:mo>\n x<\/mml:mi>\n y<\/mml:mi>\n <\/mml:mrow>\n t = xy<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n <\/mml:mrow>\n x,y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n two floating-point numbers. We prove that while this remains true for division in base\n \n \n \n \n \n \u03b2\n \n <\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n \\beta > 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , smaller, attainable bounds can be derived for both division in base\n \n \n \n \n \n \u03b2\n \n <\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n \\beta =2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and\/or improvements on these bounds themselves.\n <\/p>","DOI":"10.1090\/mcom\/3234","type":"journal-article","created":{"date-parts":[[2016,11,11]],"date-time":"2016-11-11T07:53:18Z","timestamp":1478850798000},"page":"803-819","source":"Crossref","is-referenced-by-count":31,"title":["On relative errors of floating-point operations: Optimal bounds and applications"],"prefix":"10.1090","volume":"87","author":[{"given":"Claude-Pierre","family":"Jeannerod","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Siegfried M.","family":"Rump","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2017,7,7]]},"reference":[{"issue":"259","key":"1","doi-asserted-by":"publisher","first-page":"1469","DOI":"10.1090\/S0025-5718-07-01931-X","article-title":"Error bounds on complex floating-point multiplication","volume":"76","author":"Brent, Richard","year":"2007","journal-title":"Math. 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