The Harmonized Parabolic Synthesis Methodology for Hardware Efficient Function Generation with Full Error Control

Abstract

The Harmonized Parabolic Synthesis methodology is a further development of the Parabolic Synthesis methodology for approximation of unary functions such as trigonometric functions, logarithms and the square root with moderate accuracy for ASIC implementation. These functions are extensively used in computer graphics, communication systems and many other application areas. For these high-speed applications, software solutions are not sufficient, and a hardware implementation is therefore needed. The Harmonized Parabolic Synthesis methodology has two outstanding advantages: it is parallel, thus reducing the execution time, and it is based on low complexity operations, thus being simple to implement in hardware. A difference compared to other approximation methodologies is that it is a multiplicative, and not additive, methodology. Compared to the Parabolic Synthesis methodologies it is possible to significantly enhance the performance in terms of reducing chip area, computation delay and power consumption. Furthermore, it increases the possibility to tailor the characteristics of the error, improving conditions for subsequent calculations. To evaluate the methodology, the fractional part of the logarithm is implemented and its performance is compared to the Parabolic Synthesis methodology. The comparison is made with 15-bit resolution. The design implemented using the proposed methodology performs 3× better than the Parabolic Synthesis implementation in terms of throughput, while consuming 90% less energy. The chip area is 70% smaller than for the Parabolic Synthesis methodology. In summary, the new technology further increases the advantages of Parabolic Synthesis.

Using Harmonized Parabolic Synthesis to Implement a Single-Precision Floating-Point Square Root Unit

“…In this section we will go through a brief description of the mathematics behind the proposed method [8], how to apply it to IEEE-754 single-precision floating-point numbers and the accuracy and error behavior of the method.…”

“…A significant advantage with the Harmonized Parabolic Synthesis methodology is that the distribution of the error of the approximation can be fully controlled [8]. To minimize the scattering effects on the distribution of the error it is important that it is evenly distributed around zero and with an advantageous shape of the distribution.…”

“…The proposed method relies on approximation that adopts Harmonized Parabolic Synthesis [7], [8]. It converges faster and provides faster computation and smaller chip area when compared to the other methods such as CORDIC [9], [10] and Newton Raphson [11], [12].…”

“…The accuracy increases with the number of intervals and for single-precision floating-point square root, 64 intervals are sufficient. This method can be used for computing unary operations such as trigonometric functions, logarithms, and square root as well as binary functions such as division [7], [8], [13].…”