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.

翻译：近期评估凸显了渐变精度波西特数格式作为统一精度IEEE 754浮点数（存在诸多缺陷）的潜在替代方案。尽管波西特编码方案在数值接近1时具有优越的编码效率，但其效率随数值偏离1而显著降低。这种效率下降导致编码次优，进而缩小动态范围，使波西特不适合通用计算机算术。本文引入并严格证明"塔库姆"是一种新型通用对数渐变精度数格式，综合了波西特在低位应用中的优势与远离1的数值的高编码效率。塔库姆在比特串长度方面展现出渐近恒定的动态范围，本文阐明其适合作为通用数格式。研究表明，塔库姆在性能上可与现有替代方案媲美甚至超越之。此外，塔库姆解决了波西特此前发现的多个问题，同时揭示了新颖且有益的算术性质。