An accurate, high speed implementation of division by reciprocal approximation

While unlimited accuracy is theoretically possible, it is very important to minimize the number of iteration steps to improve performance and/or to reduce hardware requirements. Consequently, there is an important accuracy/speed/cost tradeoff in reciprocal approximation implementations. A reciprocal approximation implementation is discussed, with special attention given to these tradeoffs. An interpolation method is used to ensure that an initial approximation, held in a ROM table, is as accurate as possible. A method for implementing the iteration steps is given. Special instructions are used so that maximum accuracy can be carried between iteration operations. For 64-b floating-point operands (53-b mantissa), a table lookup and only two iterations are required, and high accuracy is maintained. The rounded reciprocal rarely differs from a true round-to-nearest value based on an infinite precision result. When the results do differ (less than once every 1000 calculations), the difference in accuracy is shown to be less than 0.025 of a least significant bit (LSB).<>