Programming with Numerical Uncertainties

Numerical software, common in scientific computing or embedded systems, inevitably uses an approximation of the real arithmetic in which most algorithms are designed. In many domains, roundoff errors are not the only source of inaccuracy and measurement as well as truncation errors further increase the uncertainty of the computed results. Adequate tools are needed to help users select suitable approximations (data types and algorithms) which satisfy their accuracy requirements, especially for safetycritical applications. Determining that a computation produces accurate results is challenging. Roundoff errors and error propagation depend on the ranges of variables in complex and non-obvious ways; even determining these ranges accurately for nonlinear programs poses a significant challenge. In numerical loops, roundoff errors grow, in general, unboundedly. Finally, due to numerical errors, the control flow in the finite-precision implementation may diverge from the ideal real-valued one by taking a different branch and produce a result that is far-off of the expected one. In this thesis, we present techniques and tools for automated and sound analysis, verification and synthesis of numerical programs. We focus on numerical errors due to roundoff from floating-point and fixed-point arithmetic, external input uncertainties or truncation errors. Our work uses interval or affine arithmetic together with Satisfiability Modulo Theories (SMT) technology as well as analytical properties of the underlying mathematical problems. This combination of techniques enables us to compute sound and yet accurate error bounds for nonlinear computations, determine closed-form symbolic invariants for unbounded loops and quantify the effects of discontinuities on numerical errors. We can furthermore certify the results of self-correcting iterative algorithms. Accuracy usually comes at the expense of resource efficiency: more precise data types need more time, space and energy. We propose a programming model where the scientist writes his or her numerical program in a real-valued specification language with explicit error annotations. It is then the task of our verifying compiler to select a suitable floating-point or fixed-point data type which guarantees the needed accuracy. Sometimes accuracy can be gained by simply re-arranging the non-associative finite-precision computation. We present a scalable technique that searches for a more optimal evaluation order and show that the gains can be substantial. We have implemented all our techniques and evaluated them on a number of benchmarks from scientific computing and embedded systems, with promising results.