As a rigorous statistical approach, statistical Taylor expansion extends the conventional Taylor expansion by replacing precise input variables with random variables of known distributions, to compute means and standard deviations of the results. Statistical Taylor expansion traces the dependency of the input uncertainties in the intermediate steps, so that the variables in the intermediate analytic expressions can no longer be regarded as independent of each other, and the result of the analytic expression is path independent. Thus, it differs fundamentally from the conventional common approaches in applied mathematics which optimize execution path for each calculation. In fact, statistical Taylor expansion may standardize numerical calculations for analytic expressions. Its statistical nature allows religious testing of its result when the sample size is large enough. This paper also introduces an implementation of statistical Taylor expansion called variance arithmetic and presents corresponding test results in a very wide range of mathematical applications. Another important conclusion of this paper is that the numerical errors in the library function can have significant effects on the result. For example, the periodic numerical errors in the trigonometric library functions can resonate with periodic signals, producing large numerical errors in the results.

翻译：作为一种严格的统计方法，统计泰勒展开通过将精确输入变量替换为已知分布的随机变量，扩展了传统泰勒展开，以计算结果的均值和标准差。统计泰勒展开追踪了输入不确定性在中间步骤中的依赖性，使得中间解析表达式中的变量不再被视为彼此独立，且解析表达式的结果是路径无关的。因此，它与应用数学中为每次计算优化执行路径的传统常用方法存在根本区别。实际上，统计泰勒展开可能实现解析表达式数值计算的标准化。其统计特性允许在样本量足够大时对其结果进行严格检验。本文还介绍了一种称为方差算术的统计泰勒展开实现，并在广泛的数学应用中展示了相应的测试结果。本文的另一个重要结论是，库函数中的数值误差可能对结果产生显著影响。例如，三角函数库函数中的周期性数值误差可能与周期性信号产生共振，从而在结果中引入较大的数值误差。