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Lang."],"published-print":{"date-parts":[[2022,1,16]]},"abstract":"<jats:p>\n            Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for\n            <jats:italic>n<\/jats:italic>\n            -bits, our key idea is to generate a polynomial approximation for a representation with\n            <jats:italic>n<\/jats:italic>\n            +2-bits using the\n            <jats:italic>round-to-odd<\/jats:italic>\n            mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with\n            <jats:italic>k<\/jats:italic>\n            -bits such that |\n            <jats:italic>E<\/jats:italic>\n            | +1 &lt;\n            <jats:italic>k<\/jats:italic>\n            \u2264\n            <jats:italic>n<\/jats:italic>\n            , where |\n            <jats:italic>E<\/jats:italic>\n            | is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with\n            <jats:italic>n<\/jats:italic>\n            +2-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries.\n          <\/jats:p>","DOI":"10.1145\/3498664","type":"journal-article","created":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T17:03:12Z","timestamp":1642006992000},"page":"1-28","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":17,"title":["One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modes"],"prefix":"10.1145","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7572-4017","authenticated-orcid":false,"given":"Jay P.","family":"Lim","sequence":"first","affiliation":[{"name":"Rutgers University, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5048-8548","authenticated-orcid":false,"given":"Santosh","family":"Nagarakatte","sequence":"additional","affiliation":[{"name":"Rutgers University, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2022,1,12]]},"reference":[{"key":"e_1_2_2_1_1","volume-title":"Ziegler","author":"Aigner Martin","year":"2009","unstructured":"Martin Aigner and Gnter M . 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