Browse Prior Art Database

Look-up Table Approach in Taylor Series Approximation for increased accuracy and reduced hardware

IP.com Disclosure Number: IPCOM000234558D
Publication Date: 2014-Jan-17
Document File: 5 page(s) / 1M

Publishing Venue

The IP.com Prior Art Database

Abstract

Rather than using the bandwidth of the processor for complex arithmetic operations, more and more data processing is implemented in the hardware itself. These hardware accelerators usually implement complex mathematical equations including the power series functions. Implementation of power functions ( where X and Y are real numbers) in hardware is a tricky task as one needs to make judicious choice between area and accuracy of the design. The Taylor series Approximation is best suited for power function implementation in Hardware. However, Taylor series does not converge when base (X) is not close to 1. In order to get the higher accuracy, more number of terms in Taylor series needs to be used which in turn will increase size of hardware and reduce system throughput. The proposed algorithm in this paper describes a method to compute using Look-up Table (LUT) approach in Taylor series approximation by which targeted accuracy can be achieved with even lower number of terms which in turn will reduce hardware and increase the system performance.

This text was extracted from a Microsoft Word document.
At least one non-text object (such as an image or picture) has been suppressed.
This is the abbreviated version, containing approximately 40% of the total text.

Look-up Table Approach in Taylor Series Approximation for increased accuracy and reduced hardware

                             


Abstract

Rather than using the bandwidth of the processor for complex arithmetic operations, more and more data processing is implemented in the hardware itself. These hardware accelerators usually implement complex mathematical equations including the power series functions.  Implementation of power functions (  where X and Y are real numbers) in hardware is a tricky task as one needs to make judicious choice between area and accuracy of the design. The Taylor series Approximation is best suited for power function implementation in Hardware. However, Taylor series does not converge when base (X) is not close to 1. In order to get the higher accuracy, more number of terms in Taylor series needs to be used which in turn will increase size of hardware and reduce system throughput.

The proposed algorithm in this paper describes a method to compute  using Look-up Table (LUT) approach in Taylor series approximation by which targeted accuracy can be achieved with even lower number of terms which in turn will reduce hardware and increase the system performance.

Keywords

Look-up Table (LUT), Taylor Series.

Introduction

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. Depending upon the accuracy required, finite number of terms of Taylor series is used to approximate a function. Calculation of   (where X and Y are real numbers) using Taylor series approximation is explained here with an improved method to achieve higher accuracy and reduced hardware with less number of terms.

  Direct Taylor series expansion of   is given by :  


Y

The number of terms to be used in the above Taylor series expansion depends upon the targeted accuracy. If higher accuracy is needed, more number of terms needs to be used in series approximation.

      As the number of terms increases, size of multiplier and internal registers used to store the result increases which will increase the area and time in which the result is computed. 

     For example,   compare the cases when 8 terms and 9 terms are used in Taylor series approximation. Four additional bits will be needed to store the value of factorial 9 as compared to factorial 8. Bigger multiplier will be needed in case of 9 terms which will blow up the hardware area.

 Proposed Method

If range of base (X) is known, the calculation of  (where X and Y are real numbers) using Taylor series approximation  with less number of terms give correct results if X is close to 1 and Y is in the range -1 < Y < 1 .

The flow chart in Figure 1 explains the detailed procedure.

1.       Range Reduction  of Y :

If Y is not in the range of -1 to +1, error in the function approximation could be higher. Following steps can be used to reduce the error.

Separate the...