Improving the Accuracy of the Fast Inverse Square Root by Modifying Newton–Raphson Corrections

Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-...

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Vydané v:	Entropy (Basel, Switzerland) Ročník 23; číslo 1; s. 86
Hlavní autori:	Walczyk, Cezary J., Moroz, Leonid V., Cieśliński, Jan L.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Switzerland MDPI AG 09.01.2021 MDPI
Predmet:	Accuracy Algorithms Approximation approximation of functions Computer graphics Field programmable gate arrays Floating point arithmetic inverse square root Libraries magic constant Mathematical analysis Mathematical functions Newton-Raphson method Storage capacity magic constant floating-point arithmetic approximation of functions inverse square root Newton–Raphson method
ISSN:	1099-4300, 1099-4300
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Abstract	Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton–Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs.
AbstractList	Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton–Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs. Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton-Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs.Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton-Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs.
Author	Moroz, Leonid V. Walczyk, Cezary J. Cieśliński, Jan L.
AuthorAffiliation	2 Department of Security Information and Technology, Lviv Polytechnic National University, st. Kn. Romana 1/3, 79000 Lviv, Ukraine; moroz_lv@polynet.lviv.ua 1 Wydział Fizyki, Uniwersytet w Białymstoku, ul. Ciołkowskiego 1L, 15-245 Białystok, Poland; c.walczyk@uwb.edu.pl
AuthorAffiliation_xml	– name: 1 Wydział Fizyki, Uniwersytet w Białymstoku, ul. Ciołkowskiego 1L, 15-245 Białystok, Poland; c.walczyk@uwb.edu.pl – name: 2 Department of Security Information and Technology, Lviv Polytechnic National University, st. Kn. Romana 1/3, 79000 Lviv, Ukraine; moroz_lv@polynet.lviv.ua
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SubjectTerms	Accuracy Algorithms Approximation approximation of functions Computer graphics Field programmable gate arrays Floating point arithmetic inverse square root Libraries magic constant Mathematical analysis Mathematical functions Newton-Raphson method Storage capacity
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Title	Improving the Accuracy of the Fast Inverse Square Root by Modifying Newton–Raphson Corrections
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