Hardware starting approximation method and its application to the square root operation

Quadratically converging algorithms for high-order arithmetic operations typically are accelerated by a starting approximation. The higher the precision of the starting approximation, the less number of iterations required for convergence. Traditional methods have used look-up tables or polynomial approximations, or a combination of the two called piecewise linear approximations. This paper provides a revision and major extension to our study (1993) proposing a nontraditional method for reusing the hardware of a multiplier. An approximation is described in the form of partial product array (PPA) composed of Boolean elements. The Boolean elements are chosen such that their sum is a high-precision approximation to a high-order arithmetic operation such as square root, reciprocal, division, logarithm, exponential, and trigonometric functions. This paper derives a PPA that produces in the worst case a 16-bit approximation to the square root operation. The implementation of the PPA utilizes an existing 53 bit multiplier design requiring approximately 1,000 dedicated logic gates of function, additional repowering circuits, and has a latency of one multiplication.