Machine contouring must not introduce information which is not present in the data. The one‐dimensional spline fit has well defined smoothness properties. These are duplicated for two‐dimensional interpolation in this paper, by solving the corresponding differential equation. Finite difference equations are deduced from a principle of minimum total curvature, and an iterative method of solution is outlined. Observations do not have to lie on a regular grid. Gravity and aeromagnetic surveys provide examples which compare favorably with the work of draftsmen.

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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.

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