On Sound Relative Error Bounds for Floating-Point Arithmetic

State-of-the-art static analysis tools for verifying finite-precision code compute worst-case absolute error bounds on numerical errors. These are, however, often not a good estimate of accuracy as they do not take into account the magnitude of the computed values. Relative errors, which compute errors relative to the value's magnitude, are thus preferable. While today's tools do report relative error bounds, these are merely computed via absolute errors and thus not necessarily tight or more informative. Furthermore, whenever the computed value is close to zero on part of the domain, the tools do not report any relative error estimate at all. Surprisingly, the quality of relative error bounds computed by today's tools has not been systematically studied or reported to date. In this paper, we investigate how state-of-the-art static techniques for computing sound absolute error bounds can be used, extended and combined for the computation of relative errors. Our experiments on a standard benchmark set show that computing relative errors directly, as opposed to via absolute errors, is often beneficial and can provide error estimates up to six orders of magnitude tighter, i.e. more accurate. We also show that interval subdivision, another commonly used technique to reduce over-approximations, has less benefit when computing relative errors directly, but it can help to alleviate the effects of the inherent issue of relative error estimates close to zero

A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4

Being able to soundly estimate roundoff errors of finite-precision computations is important for many applications in embedded systems and scientific computing. Due to the discrepancy between continuous reals and discrete finite-precision values, automated static analysis tools are highly valuable to estimate roundoff errors. The results, however, are only as correct as the implementations of the static analysis tools. This paper presents a formally verified and modular tool which fully automatically checks the correctness of finite-precision roundoff error bounds encoded in a certificate. We present implementations of certificate generation and checking for both Coq and HOL4 and evaluate it on a number of examples from the literature. The experiments use both in-logic evaluation of Coq and HOL4, and execution of extracted code outside of the logics: we benchmark Coq extracted unverified OCaml code and a CakeML-generated verified binary

Evaluation of synthetic and experimental training data in supervised machine learning applied to charge state detection of quantum dots

Automated tuning of gate-defined quantum dots is a requirement for large scale semiconductor based qubit initialisation. An essential step of these tuning procedures is charge state detection based on charge stability diagrams. Using supervised machine learning to perform this task requires a large dataset for models to train on. In order to avoid hand labelling experimental data, synthetic data has been explored as an alternative. While providing a significant increase in the size of the training dataset compared to using experimental data, using synthetic data means that classifiers are trained on data sourced from a different distribution than the experimental data that is part of the tuning process. Here we evaluate the prediction accuracy of a range of machine learning models trained on simulated and experimental data and their ability to generalise to experimental charge stability diagrams in two dimensional electron gas and nanowire devices. We find that classifiers perform best on either purely experimental or a combination of synthetic and experimental training data, and that adding common experimental noise signatures to the synthetic data does not dramatically improve the classification accuracy. These results suggest that experimental training data as well as realistic quantum dot simulations and noise models are essential in charge state detection using supervised machine learning

Inferring Interval-Valued Floating-Point Preconditions

Aggregated roundoff errors caused by floating-point arithmetic can make numerical code highly unreliable. Verified postconditions for floating-point functions can guarantee the accuracy of their results under specific preconditions on the function inputs, but how to systematically find an adequate precondition for a desired error bound has not been explored so far. We present two novel techniques for automatically synthesizing preconditions for floating-point functions that guarantee that user-provided accuracy requirements are satisfied. Our evaluation on a standard benchmark set shows that our approaches are complementary and able to find accurate preconditions in reasonable time