Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations should be performed as if their results were first computed with an infinite precision and then rounded to the target format. A consequence is that floating-point arithmetic satisfies the ‘standard model’ that is often used for analysing the accuracy of floating-point algorithms. But that is only scraping the surface, and floating-point arithmetic offers much more.In this survey we recall the history of floating-point arithmetic as well as its specification mandated by the IEEE-754 Standard. We also recall what properties it entails and what every programmer should know when designing a floating-point algorithm. We provide various basic blocks that can be implemented with floating-point arithmetic. In particular, one can actually compute the rounding error caused by some floating-point operations, which paves the way to designing more accurate algorithms. More generally, properties of floating-point arithmetic make it possible to extend the accuracy of computations beyond working precision.

中文翻译：

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浮点运算

当涉及基于物理的数值计算时，浮点数具有直观的含义，因此它们已成为计算机中逼近实数的最常用方法。IEEE-754 标准早在 1985 年就以严格但有用的方式指定其语义，在使浮点运算今天无处不在方面发挥了重要作用。特别是，浮点运算应该像其结果首先被执行一样执行以无限精度计算，然后四舍五入为目标格式。结果是浮点运算满足通常用于分析浮点算法精度的“标准模型”。但这只是冰山一角，浮点运算提供的远不止这些。在本次调查中，我们回顾了浮点运算的历史以及 IEEE-754 标准规定的规范。我们还回顾了它包含的属性以及每个程序员在设计浮点算法时应该知道的内容。我们提供各种可以用浮点运算实现的基本块。特别是，可以实际计算由某些浮点运算引起的舍入误差，这为设计更精确的算法铺平了道路。更一般地说，浮点运算的特性使得将计算精度扩展到工作精度之外成为可能。我们提供各种可以用浮点运算实现的基本块。特别是，可以实际计算由某些浮点运算引起的舍入误差，这为设计更精确的算法铺平了道路。更一般地说，浮点运算的特性使得将计算精度扩展到工作精度之外成为可能。我们提供各种可以用浮点运算实现的基本块。特别是，可以实际计算由某些浮点运算引起的舍入误差，这为设计更精确的算法铺平了道路。更一般地说，浮点运算的特性使得将计算精度扩展到工作精度之外成为可能。