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 As-if rule The standard for the C++ programming language allows compilers for this language to apply any optimizing transformation to a program during compilation, provided that such optimizations make no change in the "observable behavior" of the program, as specified in the standard; this mostly means that any actions the program performs on its environment occur in the specified order. This rule is commonly referred to as the as-if rule. The rule has three main exceptions. The first is that programs exhibiting undefined behavior are exempt; since the observable behavior is not well-defined anyway, "any" transformation is valid. The other