Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide very pessimistic overestimates, causing unnecessary verification failure. We have developed a new approach called Symbolic Taylor Expansions that avoids these problems, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. FPTaylor emits per-instance analysis certificates in the form of HOL Light proofs that can be machine checked. In this article, we present the basic ideas behind Symbolic Taylor Expansions in detail. We also survey as well as thoroughly evaluate six tool families, namely, Gappa (two tool options studied), Fluctuat, PRECiSA, Real2Float, Rosa, and FPTaylor (two tool options studied) on 24 examples, running on the same machine, and taking care to find the best options for running each of these tools. This study demonstrates that FPTaylor estimates round-off errors within much tighter bounds compared to other tools on a significant number of case studies. We also release FPTaylor along with our benchmarks, thus contributing to future studies and tool development in this area.

中文翻译：

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用符号泰勒展开对浮点舍入误差的严格估计

对最大浮点舍入误差的严格估计是许多形式验证工具的核心功能。不幸的是，该任务的可用技术通常会提供非常悲观的高估，从而导致不必要的验证失败。我们开发了一种新方法，称为符号泰勒展开式它避免了这些问题，并实施了一个名为 FPTaylor 的新工具来体现这种方法。我们方法的关键是使用严格的全局优化，而不是更熟悉的区间算术、仿射算术和/或 SMT 求解器。FPTaylor 以 HOL Light 证明的形式发出每个实例的分析证书，可以进行机器检查。在本文中，我们详细介绍了符号泰勒展开背后的基本思想。我们还在 24 个示例上调查并彻底评估了六个工具系列，即 Gappa（研究了两个工具选项）、Fluctuat、PRECiSA、Real2Float、Rosa 和 FPTaylor（研究了两个工具选项），在同一台机器上运行，并采用注意找到运行这些工具的最佳选项。这项研究表明，与大量案例研究中的其他工具相比，FPTaylor 在更严格的范围内估计舍入误差。我们还将 FPTaylor 与我们的基准一起发布，从而为该领域的未来研究和工具开发做出贡献。