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.

Floating-point division is an expensive operation for processors in digital signal processing (DSP). The basis of the division operation is finding the reciprocal of the divisor. We present a reciprocal algorithm with four multiplications for single accuracy and six for double accuracy. The algorithm specifically includes bithack (known as magic constant) operations and floating-point addition, multiplication, and the fused multiply-add (fma) operation. The proposed approach improves two characteristics of the division process: accuracy and number of steps. This second characteristic has a direct impact on the division operation performance time and on the amount of equipment required for hardware implementation.

Detection of regularities in the parameters of the ateb­gabor method for biometric image filtration

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.

The exponential function $a^x$ is widespread in many fields of science. Its calculation is a complicated issue for Central Processing Units (CPUs) and Graphics Processing Units (GPUs), as well as for specialised Digital Signal Processing (DSP) processors, such as Intelligent Processor Units (IPUs), for the needs of neural networks. This article presents some simple and accurate exponential function calculation algorithms in half, single, and double precision that can be prototyped in Field-Programmable Gate Arrays (FPGAs). It should be noted that, for the approximation, the use of effective polynomials of the first degree was proposed in most cases. The characteristic feature of such algorithms is that they only contain fast ‘bithack’ operations (‘bit manipulation technique’) and Floating-Point (FP) addition, multiplication, and (if necessary) Fused Multiply-Add (FMA) operations. We published an article on algorithms for this class of function recently, but the focus was on the use of approximations of second-degree polynomials and higher, requiring two multiplications and two additions or more, which poses some complications in FPGA implementation. This article considers algorithms based on piecewise linear approximation, with one multiplication and one addition. Such algorithms of low complexity provide decent accuracy and speed, sufficient for practical applications such as accelerators for neural networks, power electronics, machine learning, computer vision, and intelligent robotic systems. These are FP-oriented algorithms; therefore, we briefly describe the characteristic parameters of such numbers.

A real time control system for balancing a ball on a platform with FPGA parallel implementation