The Hackenbush number system for compression of numerical data

Abstract

The Hackenbush number system is a method of representing numbers by expressing the integer part in unary and the fractional part in binary. The representation absorbs the sign bit and binary point into the string of binary digits which represents the number. Conversion between the Hackenbush representation and the more conventional fixed and floating point representations requires approximately the same small amount of computation as is required to convert between the fixed and floating representations. If the numbers to be represented are selected from a known zero-mean distribution which has bounded height and tails that fall off exponentially or faster, the m-bit Hackenbush representation is typically more accurate than any floating point representation. For example, if the numbers are selected from the normal Gaussian distribution, the mean-square error of the m-bit Hackenbush representation is less than the mean-square error of any (arbitrarily complex) (m − 1)-bit representation, and it is comparable to the mean square quantization error of the (m + n)-bit normalized floating point representation which has 1 bit of sign, n + 1 bits of exponent, and (m − 2) bits of characteristic, for all n.