A Modification of the Fast Inverse Square Root Algorithm

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

doi:10.3390/computation7030041

Cited by 12 publications

(21 citation statements)

References 31 publications

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“…The idea of increasing the accuracy by a modification of the Newton–Raphson formulas is motivated by the fact that

for any

(see [ 31 , 32 ]). Therefore, we can try to shift the graph of

upwards (making it more symmetric with respect to the horizontal axis).…”

Section: Modified Newton–raphson Formulasmentioning

confidence: 99%

“…Therefore, we can try to shift the graph of

upwards (making it more symmetric with respect to the horizontal axis). Then, the errors of the first correction are expected to decrease twice and the errors of the second correction are expected to decrease by about eight times (for more details, see [ 32 ]). Indeed, according to ( 8 ), reducing the first correction by a factor of 2 will reduce the second correction by a factor of 4.…”

Section: Modified Newton–raphson Formulasmentioning

confidence: 99%

“…Parameters

and

are determined by minimization of

. Then, the parameter

is determined by minimization of

(for details, see [ 32 ]). As a result, we get:

and

(see ( 10 )).…”

Section: Modified Newton–raphson Formulasmentioning

confidence: 99%

“…Recently, we presented a new approach to the fast inverse square root code InvSqrt , presenting a rigorous derivation of the well-known code [ 31 ]. Then, this approach was used to construct a more accurate modification (called InvSqrt1 ) of the fast inverse square root (see [ 32 ]). It will be developed and generalized in the next sections, where we will show how to increase the accuracy of the InvSqrt code without losing its advantages, including the low computational cost.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we focus on the Newton–Raphson corrections, which form the second part of the InvSqrt code. Following and developing ideas presented in our recent papers [ 31 , 32 ], we propose modifications of the Newton–Raphson formulas, which result in algorithms that have the same or similar computational cost as/to InvSqrt , but improve the accuracy of the original code, even by several times. The modifications consist in changing both the Newton–Raphson coefficients and the magic constant.…”

Section: Introductionmentioning

confidence: 99%

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

Walczyk

Moroz

Cieśliński

2021

Entropy

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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.

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“…The idea of increasing the accuracy by a modification of the Newton–Raphson formulas is motivated by the fact that

for any

(see [ 31 , 32 ]). Therefore, we can try to shift the graph of

upwards (making it more symmetric with respect to the horizontal axis).…”

Section: Modified Newton–raphson Formulasmentioning

confidence: 99%

“…Therefore, we can try to shift the graph of

Section: Modified Newton–raphson Formulasmentioning

confidence: 99%

“…Parameters

and

are determined by minimization of

. Then, the parameter

is determined by minimization of

(for details, see [ 32 ]). As a result, we get:

and

(see ( 10 )).…”

Section: Modified Newton–raphson Formulasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Walczyk

Moroz

Cieśliński

2021

Entropy

Self Cite

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Novel seed generation and quadrature-based square rooting algorithms

Altamimi

Youssef

2022

Sci Rep

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The square root operation is indispensable in a myriad of computational science and engineering applications. Various computational techniques have been devised to approximate its value. In particular, convergence methods employed in this regard are highly affected by the initial approximation of the seed value. Research shows that the provision of an initial approximation with higher accuracy yields fewer additional iterations to calculate the square root. In this article, we propose two novel algorithms. The first one presents a seed generation technique that depends on bit manipulation and whose output is to be used as an initial value in the calculation of square roots. The second one describes a quadrature-based square rooting method that utilizes a rectangle as the plane figure for squaring. We provide error estimation of the former using the vertical parabola equation and employ a suitable lookup table, for the latter, to store needed cosine values. The seed generation approach produces a significant reduction in the number of iterations of up to 84.42% for selected convergence methods. The main advantages of our proposed square rooting algorithm lie in its high accuracy and in its requirement of just a single iteration. Our proposed algorithm also provides for lower computational latency, measured in the number of clock cycles, compared to Newton–Raphson’s and Bakhshali’s square rooting methods.

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