Beating Posits at Their Own Game: Takum Arithmetic

Abstract: Recent evaluations have highlighted the tapered posit number format as a promising alternative to the uniform precision IEEE 754 floating-point numbers, which suffer from various deficiencies. Although the posit encoding scheme offers superior coding efficiency at values close to unity, its efficiency markedly diminishes with deviation from unity. This reduction in efficiency leads to suboptimal encodings and a consequent diminution in dynamic range, thereby rendering posits suboptimal for general-purpose computer arithmetic. This paper introduces and formally proves 'takum' as a novel general-purpose logarithmic tapered-precision number format, synthesising the advantages of posits in low-bit applications with high encoding efficiency for numbers distant from unity. Takums exhibit an asymptotically constant dynamic range in terms of bit string length, which is delineated in the paper to be suitable for a general-purpose number format. It is demonstrated that takums either match or surpass existing alternatives. Moreover, takums address several issues previously identified in posits while unveiling novel and beneficial arithmetic properties.