scholarly journals A Modification of the Fast Inverse Square Root Algorithm

Computation ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 41 ◽  
Author(s):  
Cezary J. Walczyk ◽  
Leonid V. Moroz ◽  
Jan L. Cieśliński
Keyword(s):  
Analytical Approach ◽  
Floating Point ◽  
Square Root ◽  
Approximate Evaluation ◽  
Single Precision ◽  
Seed Solution ◽  
Numerical Tests ◽  
Newton Raphson ◽  
Relative Errors ◽  
Magic Constant

We present a new algorithm for the approximate evaluation of the inverse square root for single-precision floating-point numbers. This is a modification of the famous fast inverse square root code. We use the same “magic constant” to compute the seed solution, but then, we apply Newton–Raphson corrections with modified coefficients. As compared to the original fast inverse square root code, the new algorithm is two-times more accurate in the case of one Newton–Raphson correction and almost seven-times more accurate in the case of two corrections. We discuss relative errors within our analytical approach and perform numerical tests of our algorithm for all numbers of the type float.

scholarly journals A Modification of the Fast Inverse Square Root Algorithm

2019 ◽  
Author(s):  
Cezary J. Walczyk ◽  
Leonid V. Moroz ◽  
Jan L. Cieśliński
Keyword(s):  
Computational Cost ◽  
Floating Point ◽  
Square Root ◽  
Single Precision ◽  
Fast Calculation ◽  
Newton Raphson ◽  
Floating Point Numbers ◽  
Improved Algorithm

We present an improved algorithm for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithm is much more accurate than the famous fast inverse square root algorithm and has a similar computational cost. The presented modification concern Newton-Raphson corrections and can be applied when the distribution of these corrections is not symmetric (for instance, in our case they are always negative).

scholarly journals Posit Arithmetic Hardware Implementations with The Minimum Cost Divider and SquareRoot

Electronics ◽  
2020 ◽  
Vol 9 (10) ◽  
pp. 1622
Author(s):  
Feibao Xiao ◽  
Feng Liang ◽  
Bin Wu ◽  
Junzhe Liang ◽  
Shuting Cheng ◽  
...  
Keyword(s):  
Hardware Implementation ◽  
Minimum Cost ◽  
Number System ◽  
Floating Point ◽  
Square Root ◽  
Subtraction Method ◽  
Embedded Devices ◽  
Hardware Implementations ◽  
Newton Raphson ◽  
Raphson Method

As a substitute for the IEEE 754-2008 floating-point standard, Posit, a new kind of number system for floating-point numbers, was put forward recently. Hitherto, some studies have proven that Posit is a better floating-point style than IEEE 754-2008 in some fields. However, most of these studies presented the advantages of Posit from the arithmetical aspect, but none of them suggested it had a better hardware implementation than that of IEEE 754-2008. In this paper, we propose several hardware implementations that contain the Posit adder/subtractor, multiplier, divider, and square root. Our goal is to achieve an arbitrary Posit format and exploit the minimum circuit area, which is required in embedded devices. To implement the minimum circuit area for the divider and square root, the alternating addition and subtraction method is used rather than the Newton–Raphson method. Compared with other works, the area of our divider is about 0.2×–0.7× (FPGA). Furthermore, this paper provides the synthesis results for each critical module with the Xilinx Virtex-7 FPGA VC709 platform.

scholarly journals SIMPLE EFFECTIVE FAST INVERSE SQUARE ROOT ALGORITHM WITH TWO MAGIC CONSTANTS

2019 ◽  
pp. 461-470
Author(s):  
Oleh Horyachyy ◽  
Leonid Moroz ◽  
Viktor Otenko
Keyword(s):  
Computer Game ◽  
Initial Approximation ◽  
Floating Point ◽  
Square Root ◽  
Original Algorithm ◽  
Gate Arrays ◽  
Field Programmable ◽  
Floating Point Number ◽  
Programmable Gate Arrays ◽  
Relative Errors

The purpose of this paper is to introduce a modification of Fast Inverse Square Root (FISR) approximation algorithm with reduced relative errors. The original algorithm uses a magic constant trick with input floating-point number to obtain a clever initial approximation and then utilizes the classical iterative Newton-Raphson formula. It was first used in the computer game Quake III Arena, causing widespread discussion among scientists and programmers, and now it can be frequently found in many scientific applications, although it has some drawbacks. The proposed algorithm has such parameters of the modified inverse square root algorithm that minimize the relative error and includes two magic constants in order to avoid one floating-point multiplication. In addition, we use the fused multiply-add function and iterative methods of higher order in the second iteration to improve the accuracy. Such algorithms do not require storage of large tables for initial approximation and can be effectively used on field-programmable gate arrays (FPGAs) and other platforms without hardware support for this function.

scholarly journals Fast calculation of inverse square root with the use of magic constant – analytical approach

2018 ◽  
Vol 316 ◽  
pp. 245-255 ◽  
Author(s):  
Leonid V. Moroz ◽  
Cezary J. Walczyk ◽  
Andriy Hrynchyshyn ◽  
Vijay Holimath ◽  
Jan L. Cieśliński
Keyword(s):  
Analytical Approach ◽  
Square Root ◽  
Fast Calculation ◽  
Magic Constant

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

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 86
Author(s):  
Cezary J. Walczyk ◽  
Leonid V. Moroz ◽  
Jan L. Cieśliński
Keyword(s):  
Storage Capacity ◽  
Root Function ◽  
Low Complexity ◽  
Double Precision ◽  
Square Root ◽  
Direct Computation ◽  
Single Precision ◽  
Fast Calculation ◽  
Computational Costs ◽  
Newton Raphson

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.

scholarly journals Approximate Reciprocal Square Root with Single - and Half-Precision Floats

2018 ◽  
Author(s):  
Matheus M. Susin ◽  
Lucas Wanner
Keyword(s):  
Power Consumption ◽  
Floating Point ◽  
Square Root ◽  
Single Precision ◽  
Approximation Techniques ◽  
Point Number ◽  
Floating Point Number ◽  
Floating Point Numbers

In this work, we compared the precision, speed, and power consumption of the reciprocal square root of a single-precision floating point number, using different approximation techniques. We also devised an equivalent approximation for half-precision floating point numbers, and evaluated its performance across the whole range of positive non-zero 16-bit floating point values.

scholarly journals A Low-Cost High Radix Floating-Point Square-Root Circuit

Electronics ◽  
2021 ◽  
Vol 10 (16) ◽  
pp. 1988
Author(s):  
Yuheng Yang ◽  
Qing Yuan ◽  
Jian Liu
Keyword(s):  
Power Consumption ◽  
Low Cost ◽  
Floating Point ◽  
Square Root ◽  
Single Precision ◽  
Dynamic Power ◽  
Low Area ◽  
High Radix

In this paper, we propose an efficient architecture of floating-point square-root circuit with low area cost, which is in accordance with the IEEE-754 standard. We extend the principle of the standard SRT algorithm so that the latency and area cost of the proposed circuit are linear with the radix. In addition, no extra computation cycles are required. With 65 nm technology, the area cost of the single-precision floating-point square-root circuit based on proposed architecture is only 6450.84 μm2, and the dynamic power consumption is only 0.764 mW at 300 MHz. The implementation results show that the proposed square-root circuit can reduce the area cost by 60%~90% compared with other designs in the literature.

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