标题： IEEE 754、Posit 和 Takum 算术中的整数表示 显示英文标题

标题： Integer Representations in IEEE 754, Posit, and Takum Arithmetics

摘要： 定点和takum机器数格式被提出作为IEEE 754浮点标准的替代方案。 由于浮点数经常用于表示整数值，某些应用明确依赖此功能，因此有必要评估这些新格式在这一功能上的表现与它们试图取代的标准相比如何。 虽然已知posits的实证结果，但这一方面尚未被正式研究。 本文提供了posits和takums整数表示能力的严格推导和证明，考察了表示给定整数所需的位数以及用指定位数可以表示的最大连续整数。 研究结果表明，尽管posits在这方面通常不如IEEE 754浮点数有效，但takums在表示能力上总体上优于IEEE 754和posits。 显示英文摘要

摘要： The posit and takum machine number formats have been proposed as alternatives to the IEEE 754 floating-point standard. As floating-point numbers are frequently employed to represent integral values, with certain applications explicitly relying on this capability, it is pertinent to evaluate how effectively these new formats fulfil this function compared to the standard they seek to replace. While empirical results are known for posits, this aspect has yet to be formally investigated. This paper provides rigorous derivations and proofs of the integral representation capabilities of posits and takums, examining both the exact number of bits required to represent a given integer and the largest consecutive integer that can be represented with a specified number of bits. The findings indicate that, while posits are generally less effective than IEEE 754 floating-point numbers in this regard, takums demonstrate overall superior representational strength compared to both IEEE 754 and posits.