{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T15:05:48Z","timestamp":1777129548158,"version":"3.51.4"},"reference-count":32,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T00:00:00Z","timestamp":1741910400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>\n            In this article we analyze low-cost accurate approximation of the function\n            <jats:inline-formula content-type=\"math\/tex\">\n              <jats:tex-math notation=\"LaTeX\" version=\"MathJax\">\\(1\/x\\)<\/jats:tex-math>\n            <\/jats:inline-formula>\n            using Chebyshev polynomials of the first kind and minimizing number of elementary operations in computer codes (in particular, by using the so-called magic constants). It is shown that Newton-Raphson iterative method is not optimal and a new approach is proposed. We prove that optimal Chebyshev polynomials can be factorized in terms of Chebyshev polynomials of lower order which leads to new optimal iteration schemes. We also construct a family of new algorithms by dividing the considered interval into sub-intervals where different magic constants and multiplicative factors are used (in order to increase the accuracy). Theoretical considerations and proofs are completed with numerical tests on three types of microcontroller processors.\n          <\/jats:p>","DOI":"10.1145\/3708472","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T03:13:36Z","timestamp":1734491616000},"page":"1-38","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Approximation of the 1\/\n            <i>x<\/i>\n            Function using Chebyshev Polynomials and Magic Constants"],"prefix":"10.1145","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2147-7222","authenticated-orcid":false,"given":"Cezary J.","family":"Walczyk","sequence":"first","affiliation":[{"name":"Faculty of Physics, University of Bialystok, Bialystok, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4131-309X","authenticated-orcid":false,"given":"Leonid V.","family":"Moroz","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical and Industrial Engineering, Institute of Production Systems Organization, Warsaw University of Technology, Warszawa, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2344-2576","authenticated-orcid":false,"given":"Volodymyr","family":"Samotyy","sequence":"additional","affiliation":[{"name":"Faculty of Electrical and Computer Engineering, Cracow University of Technology, Cracow, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1730-0950","authenticated-orcid":false,"given":"Jan L.","family":"Cie\u015bli\u0144ski","sequence":"additional","affiliation":[{"name":"University of Bialystok, Faculty of Physics, Bialystok, Poland"}]}],"member":"320","published-online":{"date-parts":[[2025,3,14]]},"reference":[{"key":"e_1_3_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396969"},{"key":"e_1_3_1_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.595279"},{"key":"e_1_3_1_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/ARITH.2007.17"},{"key":"e_1_3_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141885.1141890"},{"key":"e_1_3_1_6_1","first-page":"539","article-title":"Th\u00e9orie des m\u00e9canismes connus sous le nom de parall\u00e9logrammes","volume":"7","author":"Chebyshev Pafnuty L.","year":"1854","unstructured":"Pafnuty L. Chebyshev. 1854. Th\u00e9orie des m\u00e9canismes connus sous le nom de parall\u00e9logrammes. M\u00e9m. Acad. Imp. Sci. St-P\u00e9tersbourg 7 (1854), 539\u2013568.","journal-title":"M\u00e9m. Acad. Imp. Sci. St-P\u00e9tersbourg"},{"key":"e_1_3_1_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/ASAP.2010.5540952"},{"key":"e_1_3_1_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2005.54"},{"key":"e_1_3_1_9_1","doi-asserted-by":"publisher","DOI":"10.1049\/piee.1968.0142"},{"key":"e_1_3_1_10_1","doi-asserted-by":"publisher","DOI":"10.1109\/T-C.1970.223019"},{"key":"e_1_3_1_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/DELTA.2010.65"},{"key":"e_1_3_1_12_1","doi-asserted-by":"publisher","DOI":"10.1109\/DSD.2018.00020"},{"key":"e_1_3_1_13_1","doi-asserted-by":"publisher","DOI":"10.1109\/ISNE.2010.5669204"},{"key":"e_1_3_1_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2018.2852260"},{"key":"e_1_3_1_15_1","doi-asserted-by":"publisher","unstructured":"IEEE. 1985. IEEE standard for binary floating-point arithmetic (IEEE Std 754-1985). DOI:10.1109\/IEEESTD.1985.82928","DOI":"10.1109\/IEEESTD.1985.82928"},{"key":"e_1_3_1_16_1","doi-asserted-by":"publisher","unstructured":"IEEE. 2008. IEEE standard for floating-point arithmetic (IEEE Std 754-2008). DOI:10.1109\/IEEESTD.2008.4610935","DOI":"10.1109\/IEEESTD.2008.4610935"},{"key":"e_1_3_1_17_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2015.2441714"},{"key":"e_1_3_1_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.09.056"},{"key":"e_1_3_1_19_1","doi-asserted-by":"publisher","DOI":"10.1109\/DSD.2004.1333284"},{"key":"e_1_3_1_20_1","doi-asserted-by":"publisher","DOI":"10.1109\/ARITH.2011.31"},{"key":"e_1_3_1_21_1","first-page":"23","article-title":"A fast calculation of function Y=1\/X with the use of magic constant","volume":"821","author":"Moroz Leonid","year":"2015","unstructured":"Leonid Moroz and Andriy Hrynchyshyn. 2015. A fast calculation of function Y=1\/X with the use of magic constant. Bull. Lviv Polytech. Nat. Univ. (Automation, Measurement and Control Series) 821 (2015), 23\u201329.","journal-title":"Bull. Lviv Polytech. Nat. Univ. 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