Simple Effective Fast Inverse Square Root Algorithm With Two Magic Constants

Horyachyy, Oleh; Moroz, Leonid; Otenko, Viktor

doi:10.47839/ijc.18.4.1616

Cited by 4 publications

(2 citation statements)

References 15 publications

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“…It can be adjusted to yield the desired accuracy by controlling the number of polynomial coefficients 25,48 . Besides, the polynomial approximation method can be used for estimating the seed value of the square root for iterative algorithms 49,50 . The closer the value to the exact root, the fewer iterations are needed.…”

Section: Polynomial Approximation Methodmentioning

confidence: 99%

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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Section: Polynomial Approximation Methodmentioning

confidence: 99%

Novel seed generation and quadrature-based square rooting algorithms

Altamimi

Youssef

2022

Sci Rep

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“…In the field of computer 3D graphics, and scientific computing, accurate representation of real numbers in computers often requires the use of floating-point arithmetic. One common task is the calculation of inverse square root, which is typically implemented through iterative algorithms with initial approximations obtained from lookup tables or magic constants [1]. As a result, it makes it computationally difficult as more divisions as well as direct square root operation is needed which makes more usage of hardware and hence more utilization cost.…”

Section: Introductionmentioning

confidence: 99%

Fast Inverse Square Root using FPGA

Chopde,

Bodas,

Deshmukh

et al. 2024

Advancements in Communication and Systems

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The Fast Inverse Square Root (FISR) algorithm, originally introduced in the Quake III source code, accomplishes the vector normalization task required in graphics application through basic multiplication and bit-shifting operations. The core of this algorithm relies on the use of approximation techniques to enhance an initial estimation, which is primarily based on a designated “magic” constant. The implemented Verilog code utilizes the Newton-Raphson iterations, modified booth’s multiplier, and the inverse square root, featuring a core “Inverse Square Root” module with 32-bit input and output. This paper makes use of two magic constants “0x5f3e34bc” and “0x5f3759df” aiming to improve the accuracy. It selects a magic constant based on the exponent bit. The approximation occurs through two Newton-Raphson iterations. A Booth Multiplier is used, that is using a Radix-4 encoding scheme to reduce partial product generation, making it faster.

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