Trapezoidal rule that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as "Monkhorst-Pack integration". For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpson's rule. The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has
As-if rule two exceptions concern the copying of objects, and are called copy elision and the return value optimization. The effect of the as-if rule depends on the specific compiler implementation. As an example, in the Microsoft C++ compiler, it causes omission of certain optimizations such as instruction reordering around calls to library functions, since such calls may cause input/output actions or accesses to memory locations marked , and changes in the order of those change observable behavior. The as-if rule is not specific to C++; many other programming languages, including C, Rust , Go, Ocaml, etc... -implementable in compilers- have a