Reciprocal generator using piece-wise-linear segments of varying width with floating-point format

Abstract

A reciprocal generator is useful for perspective correction for 3D graphics. The input range is divided into many sections. A lookup table contains reciprocal outputs for only two of the sections, the smallest-inputs section and the largest-inputs section. Entries in the table for the smallest section contain a base and a scale factor to indicate the reciprocal value. One entry is provided for each possible input value in the smallest section. This provides high precision where the outputs have the largest values, reducing visible distortions caused by relatively small changes in the large output values. Each section is divided into intervals, with one table entry for each interval. For the largest section, each table entry has an initial reciprocal and a slope of a line approximating the reciprocal curve in that interval. Reciprocals for inputs within the interval are calculated by multiplying an offset into the interval by the slope, and then adding to the initial reciprocal for that interval. Each section is twice as wide as the previous section, but has the same number of intervals. A scale factor is generated by counting a number of leading zeros in the input. Then the input is shifted by the number of leading zeros and input of the table. Thus the inputs are scaled to points within the largest section, so intermediate sections do not have to be stored in the table.