Ln

Implementing the Natural Logarithm Function Using Taylor Series: A Theoretical and Practical Approach

Introduction

The natural logarithm (ln) is a fundamental mathematical function widely used in various scientific and engineering fields. Implementing ln in digital systems, especially for IEEE754 floating-point numbers, involves a blend of mathematical theory and practical digital design. This article delves into using the Taylor series for computing ln, explaining the theory behind it, and discussing its implementation in a digital system with Python simulation as a developmental aid.

Theoretical Background: Taylor Series for ln

The Taylor series is a powerful tool in mathematics for approximating functions. The natural logarithm, ln(x), can be expressed using a Taylor series expansion around a point close to the number for which the logarithm is being calculated. The series for ln(x) is:

​

This series converges for x in the range (0, 2]. Outside this range, direct computation using this series is not efficient or accurate due to slow convergence.

Python Simulation for Development

The Python function natural_log calculates ln(x) using the Taylor series. The series is truncated to the first 100 terms for practical reasons. The function make_less_than_two adjusts the input number to bring it within the range (0, 2] by repeatedly dividing it by 2 and keeping track of the number of iterations. The final ln value is adjusted accordingly using the logarithmic property ln(a^n) = n*ln(a), with a being 2 in this case.

Implementation in a Digital System

Implementing ln in a digital system like an FPGA or a custom processor involves several steps:

• Range Adjustment:

  • Similar to the Python function make_less_than_two, the input number is adjusted to bring it within the convergence range of the Taylor series.

  • This step can be implemented using division and counter modules.

• Taylor Series Computation:

  • The core of the ln function is the Taylor series computation.

  • This requires a loop structure, which can be realized in hardware using a state machine.

  • Each iteration of the series involves raising (x - 1) to a power (using a multiplication module), dividing by the iteration number (using a division module), and adding/subtracting from the cumulative sum (using sum and subtraction modules).

• Final Adjustment:

  • If the input number was adjusted in the first step, the final ln value is corrected by adding n*ln(2), where n is the number of divisions by 2 performed initially.

  • This requires a multiplication module to multiply n with ln(2), which can be precomputed or approximated.

The implementation of the natural logarithm function using the Taylor series in a digital system is a blend of mathematical ingenuity and digital design. By breaking down the computation into fundamental operations like multiplication, division, and addition/subtraction, and efficiently managing the range of inputs, we can compute ln(x) with reasonable accuracy and efficiency. This method's modularity also allows for reusing existing modules like sum, division, multiplication, and subtraction, demonstrating an elegant approach to implementing complex mathematical functions in hardware.

Introduction to the System Verilog module

In the pursuit of implementing complex mathematical functions in digital systems, the natural logarithm (ln) function presents unique challenges and opportunities. This article focuses on implementing the ln function in Verilog, tailored for IEEE754 floating-point numbers. This implementation demonstrates a practical application of mathematical theory and highlights the intricacies of handling floating-point arithmetic in digital electronics.

Design and Functionality of the ln Module

The Verilog module ln is constructed to compute the natural logarithm of a 32-bit floating-point number (inputA). It does so by utilizing the Taylor series approximation, a method effective within a specific range of input values. The module incorporates several sub-modules to handle various aspects of the computation:

• Adjustment for ln (adjustForLn Module):

  • To accommodate the convergence limitations of the Taylor series, which is effective only between 0 and 2, this submodule adjusts the input number to fall within this range. It divides the number by 2 repeatedly until it is less than 2, keeping track of how many divisions are made. This step is crucial for approximating the natural logarithm of larger numbers and ensuring that they fit within the interval where the Taylor series converges. 

• Taylor Series Computation:

  • The core of the ln computation is the iterative calculation of the Taylor series terms. Each term is computed as a power of (x - 1), divided by the term number, and alternately added or subtracted from the sum. This is facilitated by a series of generate loops.

• Final Adjustment:

  • After computing the ln value of the adjusted number, the module compensates for the initial adjustment. It multiplies the count of divisions by ln(2) and adds this value to the computed ln, obtaining the final ln value of the original input.

Conclusion: Bridging Theory and Practice

The development of the ln module in Verilog is a comprehensive example of how theoretical mathematical concepts can be transformed into practical digital solutions. With its intricate handling of floating-point arithmetic, this module illustrates the complexities of implementing advanced mathematical functions in hardware. It is a concrete application of digital design principles, showcasing Verilog's potential to realize sophisticated computational tasks.

Python simulation: