Genetic Improvement of Data for Maths Functions

2021 ◽  
Vol 1 (2) ◽  
pp. 1-30
Author(s):  
William B. Langdon ◽  
Oliver Krauss
Keyword(s):  
Open Source ◽  
Genetic Improvement ◽  
Root Function ◽  
Cube Root ◽  
Double Precision ◽  
Square Root ◽  
Smart Dust ◽  
Code Changes ◽  
Newton Raphson ◽  
Binary Logarithm

We use continuous optimisation and manual code changes to evolve up to 1024 Newton-Raphson numerical values embedded in an open source GNU C library glibc square root sqrt to implement a double precision cube root routine cbrt, binary logarithm log2 and reciprocal square root function for C in seconds. The GI inverted square root x -1/2 is far more accurate than Quake’s InvSqrt, Quare root. GI shows potential for automatically creating mobile or low resource mote smart dust bespoke custom mathematical libraries with new functionality.

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.

Correlation of Electrical, Structural, and Optical Properties of Erbium In Silicon

1993 ◽  
Vol 301 ◽  
Author(s):  
J. L. Benton ◽  
D. J. Eaglesham ◽  
M. Almonte ◽  
P. H. Citrin ◽  
M. A. Marcus ◽  
...  
Keyword(s):  
Root Function ◽  
Free Exciton ◽  
Excitation Power ◽  
Optically Active ◽  
Photoluminescence Intensity ◽  
Square Root ◽  
Structural And Optical Properties ◽  
X Ray ◽  
X Ray Absorption ◽  
Donor Activity

ABSTRACTAn understanding of the electrical, structural, and optical properites of Er in Si is necessary to evaluate this system as an opto-electronic material. Extended x-ray absorption fine structure, EXAFS, measurements of Er-implanted Si show that the optically active impurity complex is Er surrounded by an O cage of 6 atoms. The Er photoluminescence intensity is a square root function of excitation power, while the free exciton intensity increases linearly. The square root dependence of the 1.54μm-intensity is independent of measurement temperature and independent of co-implanted species. Ion-implantation of Er in Si introduces donor activity, but spreading resistance carrier concentration profiles indicate that these donors do not effect the optical activity of the Er.

Characterizing the Complexity of Code Changes in Open Source Software

2019 ◽  
pp. 1-14 ◽  
Author(s):  
Adarsh Anand ◽  
S. Bharmoria ◽  
Mangey Ram
Keyword(s):  
Open Source ◽  
Open Source Software ◽  
Code Changes

Design and FPGA implementation of ternary hardware IP core for square root function

2017 ◽  
Author(s):  
Siwar Ben Haj Hassine ◽  
Mehdi Jemai ◽  
Bouraoui Ouni
Keyword(s):  
Root Function ◽  
Fpga Implementation ◽  
Square Root ◽  
Ip Core

scholarly journals Platform-Independent Specification and Verification of the Standard Mathematical Square Root Function

2019 ◽  
Vol 53 (7) ◽  
pp. 595-616
Author(s):  
N. V. Shilov ◽  
D. A. Kondratyev ◽  
I. S. Anureev ◽  
E. V. Bodin ◽  
A. V. Promsky
Keyword(s):  
Root Function ◽  
Square Root ◽  
Specification And Verification

Fast algorithm for the three-dimensional Poisson equation in infinite domains

2020 ◽  
Author(s):  
Chunxiong Zheng ◽  
Xiang Ma
Keyword(s):  
Poisson Equation ◽  
Fast Algorithm ◽  
Root Function ◽  
Computational Cost ◽  
Three Dimensional ◽  
Chebyshev Approximation ◽  
Laplacian Operator ◽  
Square Root ◽  
Infinite Domain ◽  
Infinite Domains

Abstract This paper is concerned with a fast finite element method for the three-dimensional Poisson equation in infinite domains. Both the exterior problem and the strip-tail problem are considered. Exact Dirichlet-to-Neumann (DtN)-type artificial boundary conditions (ABCs) are derived to reduce the original infinite-domain problems to suitable truncated-domain problems. Based on the best relative Chebyshev approximation for the square-root function, a fast algorithm is developed to approximate exact ABCs. One remarkable advantage is that one need not compute the full eigensystem associated with the surface Laplacian operator on artificial boundaries. In addition, compared with the modal expansion method and the method based on Pad$\acute{\textrm{e}}$ approximation for the square-root function, the computational cost of the DtN mapping is further reduced. An error analysis is performed and numerical examples are presented to demonstrate the efficiency of the proposed method.

scholarly journals Scalable Fixed Point QRD Core Using Dynamic Partial Reconfiguration

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Gayathri R. Prabhu ◽  
Bibin Johnson ◽  
J. Sheeba Rani
Keyword(s):  
Square Root ◽  
Partial Reconfiguration ◽  
Adaptive Equalizer ◽  
Dynamic Partial Reconfiguration ◽  
Givens Rotation ◽  
Look Up Table ◽  
Array Architecture ◽  
Boundary Cell ◽  
Newton Raphson ◽  
Raphson Method

A Givens rotation based scalable QRD core which utilizes an efficient pipelined and unfolded 2D multiply and accumulate (MAC) based systolic array architecture with dynamic partial reconfiguration (DPR) capability is proposed. The square root and inverse square root operations in the Givens rotation algorithm are handled using a modified look-up table (LUT) based Newton-Raphson method, thereby reducing the area by 71% and latency by 50% while operating at a frequency 49% higher than the existing boundary cell architectures. The proposed architecture is implemented on Xilinx Virtex-6 FPGA for any real matrices of sizem×n, where4≤n≤8andm≥nby dynamically inserting or removing the partial modules. The evaluation results demonstrate a significant reduction in latency, area, and power as compared to other existing architectures. The functionality of the proposed core is evaluated for a variable length adaptive equalizer.

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