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

Walczyk, Cezary J.; Moroz, Leonid; Cieśliński, Jan L.

doi:10.3390/e23010086

Cited by 13 publications

(14 citation statements)

References 25 publications

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“…Here, we briefly outline the main concepts of the basic FISR method used in the paper, which are defined in [28,29]. Suppose that we have a positive normalized FP number…”

Section: Brief Theory Of the Fisr Methodsmentioning

confidence: 99%

“…As proved in [28,29], in order to find the behavior of the relative error when calculating 0 y over the whole range of normalized FP numbers, it is sufficient to describe it in the range…”

Section: Brief Theory Of the Fisr Methodsmentioning

confidence: 99%

“…The accuracy of the RcpSqrt2 algorithm after the first iteration is 10.15 bits. Another similar algorithm from [29] has even better accuracy (20.97 bits) but requires one extra multiplication compared to Algorithms 1 and 2. return y; 10: } Our other recent modifications of the FISR algorithm are discussed in [26] and [32]. In both papers, besides the NR method, a modified second-order Householder method is used to improve the accuracy of the algorithms-in the former case, in the last iteration of the algorithm, and in the latter case, in the first iteration.…”

Section: State-of-the-art Fisr Algorithmsmentioning

confidence: 99%

“…Among the better-known methods of calculating the square root and reciprocal square root [1,2,15,20], the FISR method [20,[25][26][27][28][29] has recently gained increasing popularity in SW [8,20,27,[29][30][31][32] and HW [3,[33][34][35][36][37] applications. The algorithm was proposed for the first time in [24] but gained wider popularity through its use in the computer game Quake III Arena [27].…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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.

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“…Here, we briefly outline the main concepts of the basic FISR method used in the paper, which are defined in [28,29]. Suppose that we have a positive normalized FP number…”

Section: Brief Theory Of the Fisr Methodsmentioning

confidence: 99%

“…As proved in [28,29], in order to find the behavior of the relative error when calculating 0 y over the whole range of normalized FP numbers, it is sufficient to describe it in the range…”

Section: Brief Theory Of the Fisr Methodsmentioning

confidence: 99%

Section: State-of-the-art Fisr Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2021

Self Cite

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show abstract

“…To further improve the accuracy of the Algorithm 1, we can apply the additive correction of the parameter k 2 for each iteration, as proposed in [10,20,21]. In other words, the first iteration should be performed according to the modified Newton-Raphson formulas…”

Section: The Theory Of the Fast Inverse Cube Root Methods And Proposed Algorithmsmentioning

confidence: 99%

Fast Calculation of Cube and Inverse Cube Roots Using a Magic Constant and Its Implementation on Microcontrollers

et al. 2021

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We develop a bit manipulation technique for single precision floating point numbers which leads to new algorithms for fast computation of the cube root and inverse cube root. It uses the modified iterative Newton–Raphson method (the first order of convergence) and Householder method (the second order of convergence) to increase the accuracy of the results. The proposed algorithms demonstrate high efficiency and reduce error several times in the first iteration in comparison with known algorithms. After two iterations 22.84 correct bits were obtained for single precision. Experimental tests showed that our novel algorithm is faster and more accurate than library functions for microcontrollers.

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Fast and accurate approximation algorithms for computing floating point square root

Kokosiński,

Gepner,

Moroz

et al. 2024

Numer Algor

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The square root is one of the most used functions in many different engineering and scientific applications. We propose new methods for calculating the square root function that are based on the Newton–Raphson method with Heron iteration. A modification of Heron’s formula combined with an improved selection of the magic constants enables a significant reduction of the maximum relative error (MRE). Simple modifications to the Newton–Raphson formula and the magic number method enable implementation on platforms with limited hardware resources, such as microcontrollers and FPGAs, with variable accuracy. Implementations of new approximation algorithms in the C programming language were carefully tested and evaluated against their software and hardware counterparts on the most popular platforms, e.g., CPUs from Intel, AMD and ARM, GPU from Nvidia and IPU from Graphcore. The proposed numerical algorithms are shown to be superior in terms of computational time, the number of clock cycles, accuracy, MRE, and root mean square deviation.

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

Cited by 13 publications

References 25 publications

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

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

Fast Calculation of Cube and Inverse Cube Roots Using a Magic Constant and Its Implementation on Microcontrollers

Fast and accurate approximation algorithms for computing floating point square root

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