Logarithmic number systems (LNS) are used to represent real numbers in many applications using a constant base raised to a fixed-point exponent making its distribution exponential. This greatly simplifies hardware multiply, divide, and square root. LNS with base-2 is most common, but in this article, we show that for low-precision LNS the choice of base has a significant impact. We make four main contributions. First, LNS is not closed under addition and subtraction, so the result is approximate. We show that choosing a suitable base can manipulate the distribution to reduce the average error. Second, we show that low-precision LNS addition and subtraction can be implemented efficiently in logic rather than commonly used ROM lookup tables, the complexity of which can be reduced by an appropriate choice of base. A similar effect is shown where the result of arithmetic has greater precision than the input. Third, where input data from external sources is not expected to be in LNS, we can reduce the conversion error by selecting a LNS base to match the expected distribution of the input. Thus, there is no one base that gives the global optimum, and base selection is a trade-off between different factors. Fourth, we show that circuits realized in LNS require lower area and power consumption for short word lengths.

中文翻译：

---

低精度对数系统

对数系统 (LNS) 用于在许多应用程序中表示实数，使用一个常数基数提升到一个定点指数，使其分布呈指数级。这极大地简化了硬件乘法、除法和平方根。以 2 为基数的 LNS 最为常见，但在本文中，我们展示了对于低精度 LNS，基数的选择具有显着影响。我们做出了四个主要贡献。首先，LNS在加减法下不是闭合的，所以结果是近似的。我们表明，选择合适的基数可以操纵分布以减少平均误差。其次，我们表明，低精度LNS添加和减法可以有效地在逻辑中有效地实现，而不是常用的ROM查找表，可以通过适当的基础选择来减少其复杂性。当算术结果比输入具有更高的精度时，也会显示类似的效果。第三，如果来自外部来源的输入数据预计不会在 LNS 中，我们可以通过选择 LNS 基来匹配输入的预期分布来减少转换误差。因此，没有一个碱基可以提供全局最优值，碱基选择是不同因素之间的权衡。第四，我们展示了在 LNS 中实现的电路对于短字长需要更低的面积和功耗。碱基选择是不同因素之间的权衡。第四，我们展示了在 LNS 中实现的电路对于短字长需要更低的面积和功耗。碱基选择是不同因素之间的权衡。第四，我们展示了在 LNS 中实现的电路对于短字长需要更低的面积和功耗。