Decoding-Free Two-Input Arithmetic for Low-Precision Real Numbers

In this work, we present a novel method for directly computing functions of two real numbers using logic circuits without decoding; the real numbers are mapped to a particularly-chosen set of integer numbers. We theoretically prove that this mapping always exists and that we can implement any kind of binary operation between real numbers regardless of the encoding format. While the real numbers in the set can be arbitrary (rational, irrational, transcendental), we find practical applications to low-precision posit™ number arithmetic. We finally provide examples for decoding-free 4-bit Posit arithmetic operations, showing a reduction in gate count up to a factor of  7.6x (and never below 4.4x) compared to a standard two-dimensional tabulation.