Modified Fast Inverse Square Root and Square Root Approximation Algorithms: The Method of Switching Magic Constants

Many low-cost platforms that support floating-point arithmetic, such as microcontrollers and field-programmable gate arrays, do not include fast hardware or software methods for calculating the square root and/or reciprocal square root. Typically, such functions are implemented using direct lookup t...

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Published in:Computation Vol. 9; no. 2; p. 21
Main Authors: Moroz, Leonid V., Samotyy, Volodymyr V., Horyachyy, Oleh Y.
Format: Journal Article
Language:English
Published: MDPI AG 01.02.2021
Subjects:
algorithm design and analysis
elementary function approximation
fast inverse square root algorithm
fused multiply–add
IEEE 754 standard
Newton–Raphson method
ISSN:2079-3197, 2079-3197
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Abstract Many low-cost platforms that support floating-point arithmetic, such as microcontrollers and field-programmable gate arrays, do not include fast hardware or software methods for calculating the square root and/or reciprocal square root. Typically, such functions are implemented using direct lookup tables or polynomial approximations, with a subsequent application of the Newton–Raphson method. Other, more complex solutions include high-radix digit-recurrence and bipartite or multipartite table-based methods. In contrast, this article proposes a simple modification of the fast inverse square root method that has high accuracy and relatively low latency. Algorithms are given in C/C++ for single- and double-precision numbers in the IEEE 754 format for both square root and reciprocal square root functions. These are based on the switching of magic constants in the initial approximation, depending on the input interval of the normalized floating-point numbers, in order to minimize the maximum relative error on each subinterval after the first iteration—giving 13 correct bits of the result. Our experimental results show that the proposed algorithms provide a fairly good trade-off between accuracy and latency after two iterations for numbers of type float, and after three iterations for numbers of type double when using fused multiply–add instructions—giving almost complete accuracy.
AbstractList Many low-cost platforms that support floating-point arithmetic, such as microcontrollers and field-programmable gate arrays, do not include fast hardware or software methods for calculating the square root and/or reciprocal square root. Typically, such functions are implemented using direct lookup tables or polynomial approximations, with a subsequent application of the Newton–Raphson method. Other, more complex solutions include high-radix digit-recurrence and bipartite or multipartite table-based methods. In contrast, this article proposes a simple modification of the fast inverse square root method that has high accuracy and relatively low latency. Algorithms are given in C/C++ for single- and double-precision numbers in the IEEE 754 format for both square root and reciprocal square root functions. These are based on the switching of magic constants in the initial approximation, depending on the input interval of the normalized floating-point numbers, in order to minimize the maximum relative error on each subinterval after the first iteration—giving 13 correct bits of the result. Our experimental results show that the proposed algorithms provide a fairly good trade-off between accuracy and latency after two iterations for numbers of type float, and after three iterations for numbers of type double when using fused multiply–add instructions—giving almost complete accuracy.
Author Moroz, Leonid V.
Samotyy, Volodymyr V.
Horyachyy, Oleh Y.
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Cites_doi 10.1007/978-3-319-76526-6
10.1109/ISCE.2015.7177775
10.1109/ICAEE.2014.6838433
10.3390/e23010086
10.1109/ICPP.2017.52
10.47839/ijc.18.4.1616
10.1016/j.amc.2017.08.025
10.1109/VLSID.2017.60
10.1109/TC.2005.54
10.1109/TBCAS.2014.2376956
10.1109/MSP.2005.1406500
10.1109/DEVIC.2017.8073975
10.1023/A:1009984523264
10.1109/IDAACS.2019.8924302
10.1109/TC.2019.2947899
10.1201/b17296
10.1109/12.795125
10.1109/JPROC.2020.2991885
10.1109/IDAACS-SWS.2018.8525803
10.1016/j.sysarc.2017.06.005
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References Muller (ref_15) 2020; 108
Moroz (ref_28) 2018; 316
ref_14
ref_13
ref_35
ref_12
ref_11
Blinn (ref_24) 1997; 17
ref_33
ref_10
ref_32
ref_31
ref_30
Bruguera (ref_18) 2020; 69
ref_19
ref_17
ref_39
ref_16
ref_38
ref_37
Horyachyy (ref_26) 2019; 18
Lemaitre (ref_8) 2017; 79
Allie (ref_1) 2005; 22
Muller (ref_21) 1999; 5
Tisserand (ref_23) 2005; 54
ref_25
Schulte (ref_22) 1999; 48
ref_20
Hsieh (ref_36) 2015; 10
ref_40
ref_3
ref_2
ref_29
ref_27
ref_9
ref_5
Janhunen (ref_34) 2013; 78
ref_4
ref_7
ref_6
References_xml – ident: ref_7
– ident: ref_9
– volume: 78
  start-page: 645
  year: 2013
  ident: ref_34
  article-title: Novel detector implementations for 3G LTE downlink and uplink
  publication-title: Analog. Integr. Circuits Signal Process.
– ident: ref_5
– ident: ref_17
  doi: 10.1007/978-3-319-76526-6
– ident: ref_35
  doi: 10.1109/ISCE.2015.7177775
– ident: ref_33
  doi: 10.1109/ICAEE.2014.6838433
– ident: ref_11
– ident: ref_29
  doi: 10.3390/e23010086
– ident: ref_30
  doi: 10.1109/ICPP.2017.52
– ident: ref_16
– volume: 18
  start-page: 461
  year: 2019
  ident: ref_26
  article-title: Simple effective fast inverse square root algorithm with two magic constants
  publication-title: Int. J. Comput.
  doi: 10.47839/ijc.18.4.1616
– ident: ref_39
– ident: ref_40
– volume: 17
  start-page: 80
  year: 1997
  ident: ref_24
  article-title: Floating-point tricks
  publication-title: IEEE Eng. Med. Boil. Mag.
– volume: 316
  start-page: 245
  year: 2018
  ident: ref_28
  article-title: Fast calculation of inverse square root with the use of magic constant–analytical approach
  publication-title: Appl. Math. Comput.
  doi: 10.1016/j.amc.2017.08.025
– ident: ref_37
  doi: 10.1109/VLSID.2017.60
– ident: ref_14
– volume: 54
  start-page: 319
  year: 2005
  ident: ref_23
  article-title: Multipartite table methods
  publication-title: IEEE Trans. Comput.
  doi: 10.1109/TC.2005.54
– volume: 10
  start-page: 219
  year: 2015
  ident: ref_36
  article-title: A UWB Radar Signal Processing Platform for Real-Time Human Respiratory Feature Extraction Based on Four-Segment Linear Waveform Model
  publication-title: IEEE Trans. Biomed. Circuits Syst.
  doi: 10.1109/TBCAS.2014.2376956
– volume: 22
  start-page: 93
  year: 2005
  ident: ref_1
  article-title: A root of less evil digital signal processing
  publication-title: IEEE Signal Process. Mag.
  doi: 10.1109/MSP.2005.1406500
– ident: ref_3
  doi: 10.1109/DEVIC.2017.8073975
– volume: 5
  start-page: 279
  year: 1999
  ident: ref_21
  article-title: A Few Results on Table-Based Methods
  publication-title: Dev. Reliab. Comput.
  doi: 10.1023/A:1009984523264
– ident: ref_32
  doi: 10.1109/IDAACS.2019.8924302
– ident: ref_6
– volume: 69
  start-page: 274
  year: 2020
  ident: ref_18
  article-title: Low Latency Floating-Point Division and Square Root Unit
  publication-title: IEEE Trans. Comput.
  doi: 10.1109/TC.2019.2947899
– ident: ref_25
– ident: ref_4
– ident: ref_31
– ident: ref_27
– ident: ref_2
– ident: ref_20
  doi: 10.1201/b17296
– ident: ref_12
– ident: ref_10
– volume: 48
  start-page: 842
  year: 1999
  ident: ref_22
  article-title: Approximating elementary functions with symmetric bipartite tables
  publication-title: IEEE Trans. Comput.
  doi: 10.1109/12.795125
– volume: 108
  start-page: 2136
  year: 2020
  ident: ref_15
  article-title: Elementary Functions and Approximate Computing
  publication-title: Proc. IEEE
  doi: 10.1109/JPROC.2020.2991885
– ident: ref_38
  doi: 10.1109/IDAACS-SWS.2018.8525803
– ident: ref_13
– ident: ref_19
– volume: 79
  start-page: 1
  year: 2017
  ident: ref_8
  article-title: Cholesky factorization on SIMD multi-core architectures
  publication-title: J. Syst. Arch.
  doi: 10.1016/j.sysarc.2017.06.005
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SubjectTerms algorithm design and analysis
elementary function approximation
fast inverse square root algorithm
fused multiply–add
IEEE 754 standard
Newton–Raphson method
Title Modified Fast Inverse Square Root and Square Root Approximation Algorithms: The Method of Switching Magic Constants
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