Before revision:  We provide a framework where we study algorithmic problems under smoothed analysis of the input, the relaxation of the problem requirements, or the witness of a recognition problem . Our framework specifies a widely applicable set of prerequisites that make real RAM algorithms suitable for smoothed analysis. We prove that suitable algorithms can (under smoothed analysis) be robustly executed with expected logarithmic bit-precision. This shows in a formal way that inputs which need high bit-precision are contrived and that these algorithms are likely robust for realistic input. Interestingly our techniques generalize to problems with a natural notion of resource augmentation (geometric packing, the art gallery problem ) and recognition problems (recognition of realizable order types or disk intersection graphs).
After revision:  We provide a framework where we study algorithmic problems under smoothed analysis of the input, the relaxation of the problem ETR and if it can be written as an ETR formula. Traditionally, these problems are studied in the real RAM, a model of computation that assumes that the storage and comparison of real-valued numbers can be done in constant space and time, with infinite precision. The complexity class ER is often called a real RAM analogue of NP, since the problem ETR can be viewed as the real-valued variant of SAT. In this paper we prove a real RAM analogue to the art gallery problem ) and recognition problems (recognition of realizable order types or disk intersection graphs).