Faithful interpolation in reciprocal tables

Describes a table compression method employing finite-precision linear interpolation in reciprocal tables. The interpolation method employs a compressed look-up table and a small-sized multiplier to yield an output reciprocal as a simple direct operation. The leading bits of the arbitrarily precise input are used to index the table, and a limited number of succeeding fractional bits are used to interpolate on the table employing a multiply/add operation. The low-order bits of the product are rounded off so that the output reciprocals are guaranteed to be correct to a unit in the last place and provide a round-to-nearest reciprocal for over 90% of arbitrarily precise input arguments. The interpolation method generates 2k-bit faithful reciprocals employing a k-bits-in, (2k+2)-bits-out table and a (k+3)×(k+3)-bit multiplier. A single-precision faithful reciprocal can be generated employing a table of size 13 kBytes and a 15×15-bit multiplier, compared to a table size of 46 MBytes for conventional reciprocal tables. The table and dedicated small multiplier efficiently characterize a functional reciprocator unit with, at most, a 2-cycle latency