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Softw."],"published-print":{"date-parts":[[2013,7]]},"abstract":"<jats:p>\n            A very simple test, introduced by Ziv, allows one to determine if an approximation to the value\n            <jats:italic>f(x)<\/jats:italic>\n            of an elementary function at a given point\n            <jats:italic>x<\/jats:italic>\n            suffices to return the floating-point number nearest\n            <jats:italic>f(x)<\/jats:italic>\n            . The same test may be used when implementing floating-point operations with input and output operands of different formats, using arithmetic operators tailored for manipulating operands of the same format. That test depends on a \u201cmagic constant\u201d\n            <jats:italic>e<\/jats:italic>\n            . We show how to choose that constant\n            <jats:italic>e<\/jats:italic>\n            to make the test reliable and efficient. Various cases are considered, depending on the availability of an fma instruction, and on the range of\n            <jats:italic>f(x)<\/jats:italic>\n            .\n          <\/jats:p>","DOI":"10.1145\/2491491.2491495","type":"journal-article","created":{"date-parts":[[2013,7,24]],"date-time":"2013-07-24T13:26:35Z","timestamp":1374672395000},"page":"1-19","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":7,"title":["On Ziv's rounding test"],"prefix":"10.1145","volume":"39","author":[{"given":"Florent","family":"de Dinechin","sequence":"first","affiliation":[{"name":"\u00c9cole Normale Sup\u00e9rieure de Lyon, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christoph","family":"Lauter","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Pierre et Marie Curie Paris VI, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Michel","family":"Muller","sequence":"additional","affiliation":[{"name":"CNRS, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Serge","family":"Torres","sequence":"additional","affiliation":[{"name":"\u00c9cole Normale Sup\u00e9rieure de Lyon, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2013,7,23]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2007.70819"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/ARITH.2005.46"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01397083"},{"key":"e_1_2_1_4_1","first-page":"754","article-title":"IEEE Standard for Floating-Point Arithmetic","author":"IEEE.","year":"2008","unstructured":"IEEE. 2008 . IEEE Standard for Floating-Point Arithmetic . IEEE Standard 754 - 2008 . http:\/\/ieeexplore.ieee. org\/servlet\/opac&quest;punumber=4610933. IEEE. 2008. IEEE Standard for Floating-Point Arithmetic. IEEE Standard 754-2008. http:\/\/ieeexplore.ieee. org\/servlet\/opac&quest;punumber=4610933.","journal-title":"IEEE Standard"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2010.144"},{"key":"e_1_2_1_6_1","unstructured":"Kahan W. 1983. Minimizing q*m-n. http:\/\/http.cs.berkeley.edu\/&sim;wkahan\/(At the beginning of the file \u201cnearpi.c\u201d).  Kahan W. 1983. Minimizing q*m-n. http:\/\/http.cs.berkeley.edu\/&sim;wkahan\/(At the beginning of the file \u201cnearpi.c\u201d)."},{"key":"e_1_2_1_7_1","unstructured":"Kahan W. 2004. A logarithm too clever by half. http:\/\/http.cs.berkeley.edu\/&sim;wkahan\/LOG10HAF.TXT.  Kahan W. 2004. A logarithm too clever by half. http:\/\/http.cs.berkeley.edu\/&sim;wkahan\/LOG10HAF.TXT."},{"key":"e_1_2_1_8_1","volume-title":"The Art of Computer Programming","author":"Knuth D.","unstructured":"Knuth , D. 1998. The Art of Computer Programming 3 rd Ed. Vol. 2 . Addison-Wesley , Reading, MA . Knuth, D. 1998. The Art of Computer Programming 3rd Ed. Vol. 2. Addison-Wesley, Reading, MA.","edition":"3"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/1353445.1353446"},{"key":"e_1_2_1_10_1","volume-title":"Elementary Functions, Algorithms and Implementation","author":"Muller J.-M.","unstructured":"Muller , J.-M. 2006. Elementary Functions, Algorithms and Implementation 2 nd Ed., Birkh\u00e4user , Boston, MA . Muller, J.-M. 2006. Elementary Functions, Algorithms and Implementation 2nd Ed., Birkh\u00e4user, Boston, MA.","edition":"2"},{"key":"e_1_2_1_11_1","doi-asserted-by":"crossref","unstructured":"Muller J.-M. Brisebarre N. de Dinechin F. Jeannerod C.-P. Lef\u00e8vre V. Melquiond G. Revol N. Stehl\u00e9 D. and Torres S. 2010. Handbook of Floating-Point Arithmetic. Birkh\u00e4user Boston MA.   Muller J.-M. Brisebarre N. de Dinechin F. Jeannerod C.-P. Lef\u00e8vre V. Melquiond G. Revol N. Stehl\u00e9 D. and Torres S. 2010. Handbook of Floating-Point Arithmetic. Birkh\u00e4user Boston MA.","DOI":"10.1007\/978-0-8176-4705-6"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19740540106"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1137\/030601818"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01404922"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/114697.116813"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2491491.2491495","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2491491.2491495","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T07:28:50Z","timestamp":1750231730000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2491491.2491495"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2013,7]]}},"alternative-id":["10.1145\/2491491.2491495"],"URL":"https:\/\/doi.org\/10.1145\/2491491.2491495","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,7]]},"assertion":[{"value":"2012-05-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2013-01-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2013-07-23","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}