{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:14:16Z","timestamp":1750306456588,"version":"3.41.0"},"reference-count":10,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2016,5,10]],"date-time":"2016-05-10T00:00:00Z","timestamp":1462838400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100001665","name":"French National Research Agency","doi-asserted-by":"crossref","award":["ANR-13-INSE-0007 (MetaLibm project)"],"award-info":[{"award-number":["ANR-13-INSE-0007 (MetaLibm project)"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2016,6,15]]},"abstract":"<jats:p>\n            Assuming floating-point arithmetic with a fused multiply-add operation and rounding to nearest, the Cornea-Harrison-Tang method aims to evaluate expressions of the form\n            <jats:italic>ab<\/jats:italic>\n            +\n            <jats:italic>cd<\/jats:italic>\n            with high relative accuracy. In this article, we provide a rounding error analysis of this method, which unlike previous studies is not restricted to binary floating-point arithmetic but holds for any radix \u03b2. We show first that an asymptotically optimal bound on the relative error of this method is 2\u03b2\n            <jats:italic>u<\/jats:italic>\n            + 2\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            \/\u03b2 - 2\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            = 2\n            <jats:italic>u<\/jats:italic>\n            + 2\/\u03b2\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            +\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>3<\/jats:sup>\n            ), where\n            <jats:italic>u<\/jats:italic>\n            = 1\/2\u03b2\n            <jats:sup>\n              1-\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sup>\n            is the unit roundoff in radix \u03b2 and precision\n            <jats:italic>p<\/jats:italic>\n            . Then we show that the possibility of removing the\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            ) term from this bound is governed by the radix parity and the tie-breaking strategy used for rounding: if \u03b2 is odd or rounding is\n            <jats:italic>to nearest even<\/jats:italic>\n            , then the simpler bound 2\n            <jats:italic>u<\/jats:italic>\n            is obtained, while if \u03b2 is even and rounding is\n            <jats:italic>to nearest away<\/jats:italic>\n            , then there exist floating-point inputs\n            <jats:italic>a<\/jats:italic>\n            ,\n            <jats:italic>b<\/jats:italic>\n            ,\n            <jats:italic>c<\/jats:italic>\n            ,\n            <jats:italic>d<\/jats:italic>\n            that lead to a relative error larger than 2\n            <jats:italic>u<\/jats:italic>\n            + 2\/\u03b2\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            \u2014 4\n            <jats:italic>u<\/jats:italic>\n            <jats:sup>3<\/jats:sup>\n            . All these results hold provided underflows and overflows do not occur and under some mild assumptions on\n            <jats:italic>p<\/jats:italic>\n            satisfied by IEEE 754-2008 formats.\n          <\/jats:p>","DOI":"10.1145\/2824252","type":"journal-article","created":{"date-parts":[[2016,5,11]],"date-time":"2016-05-11T12:11:38Z","timestamp":1462968698000},"page":"1-20","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["A Radix-Independent Error Analysis of the Cornea-Harrison-Tang Method"],"prefix":"10.1145","volume":"42","author":[{"given":"Claude-Pierre","family":"Jeannerod","sequence":"first","affiliation":[{"name":"Inria, LIP (CNRS, ENSL, Inria, UCBL), Universit\u00e9 de Lyon, Lyon, France"}]}],"member":"320","published-online":{"date-parts":[[2016,5,10]]},"reference":[{"volume-title":"Scientific Computing on Itanium\u00ae-based Systems","author":"Cornea Marius","key":"e_1_2_1_1_1","unstructured":"Marius Cornea , John Harrison , and Ping Tak Peter Tang . 2002. Scientific Computing on Itanium\u00ae-based Systems . Intel Press , Hillsboro, OR . xvii+406 pages. Marius Cornea, John Harrison, and Ping Tak Peter Tang. 2002. Scientific Computing on Itanium\u00ae-based Systems. Intel Press, Hillsboro, OR. xvii+406 pages."},{"volume-title":"Accuracy and Stability of Numerical Algorithms","author":"Higham Nicholas J.","key":"e_1_2_1_2_1","unstructured":"Nicholas J. Higham . 2002. Accuracy and Stability of Numerical Algorithms ( second ed.). SIAM , Philadelphia . xxx+680 pages. Nicholas J. Higham. 2002. Accuracy and Stability of Numerical Algorithms (second ed.). SIAM, Philadelphia. xxx+680 pages."},{"key":"e_1_2_1_3_1","first-page":"754","article-title":"IEEE Standard for Floating-Point Arithmetic","author":"IEEE Computer Society","year":"2008","unstructured":"IEEE Computer Society . 2008 . IEEE Standard for Floating-Point Arithmetic , IEEE Standard 754 - 2008 . IEEE Computer Society, New York, NY. http:\/\/ieeexplore.ieee.org\/servlet\/opac?punumber&equals;4610933 IEEE Computer Society. 2008. IEEE Standard for Floating-Point Arithmetic, IEEE Standard 754-2008. IEEE Computer Society, New York, NY. http:\/\/ieeexplore.ieee.org\/servlet\/opac?punumber&equals;4610933","journal-title":"IEEE Standard"},{"key":"e_1_2_1_4_1","unstructured":"Claude-Pierre Jeannerod Peter Kornerup Nicolas Louvet and Jean-Michel Muller. 2015. Error bounds on complex floating-point multiplication with an FMA. Math. Comp. Preprint available at https:\/\/hal.inria.fr\/ hal-00867040v4.  Claude-Pierre Jeannerod Peter Kornerup Nicolas Louvet and Jean-Michel Muller. 2015. Error bounds on complex floating-point multiplication with an FMA. Math. Comp. Preprint available at https:\/\/hal.inria.fr\/ hal-00867040v4."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2013-02679-8"},{"key":"e_1_2_1_6_1","volume-title":"Rump","author":"Jeannerod Claude-Pierre","year":"2014","unstructured":"Claude-Pierre Jeannerod and Siegfried M . Rump . 2014 . On relative errors of floating-point operations: optimal bounds and applications. (2014). Preprint available at https:\/\/hal.inria.fr\/hal-00934443. Claude-Pierre Jeannerod and Siegfried M. Rump. 2014. On relative errors of floating-point operations: optimal bounds and applications. (2014). Preprint available at https:\/\/hal.inria.fr\/hal-00934443."},{"volume-title":"The Art of Computer Programming","author":"Knuth Donald E.","key":"e_1_2_1_7_1","unstructured":"Donald E. Knuth . 1998. The Art of Computer Programming , Volume 2 , Seminumerical Algorithms (3rd ed.). Addison-Wesley , Reading, MA. xiii+762 pages. Donald E. Knuth. 1998. The Art of Computer Programming, Volume 2, Seminumerical Algorithms (3rd ed.). Addison-Wesley, Reading, MA. xiii+762 pages."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/2629615"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/050645671"},{"volume-title":"Floating-Point Computation","author":"Sterbenz Pat H.","key":"e_1_2_1_10_1","unstructured":"Pat H. Sterbenz . 1974. Floating-Point Computation . Prentice-Hall , Englewood Cliffs, NJ . xiv+316 pages. Pat H. Sterbenz. 1974. Floating-Point Computation. Prentice-Hall, Englewood Cliffs, NJ. xiv+316 pages."}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2824252","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2824252","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T05:43:22Z","timestamp":1750225402000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2824252"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,10]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,6,15]]}},"alternative-id":["10.1145\/2824252"],"URL":"https:\/\/doi.org\/10.1145\/2824252","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"type":"print","value":"0098-3500"},{"type":"electronic","value":"1557-7295"}],"subject":[],"published":{"date-parts":[[2016,5,10]]},"assertion":[{"value":"2014-04-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2015-09-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-05-10","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}