5.2 Algorithm Conventions

The specification often uses a numbered list to specify steps in an algorithm. These algorithms are used to clarify semantics. In practice, there may be more efficient algorithms available to implement a given feature.

When an algorithm is to produce a value as a result, the directive "return x" is used to indicate that the result of the algorithm is the value of x and that the algorithm should terminate. The notation Result(n) is used as shorthand for "the result of step n". Type(x) is used as shorthand for "the type of x".

Mathematical operations such as addition, subtraction, negation, multiplication, division, and the mathematical functions defined later in this section should always be understood as computing exact mathematical results on mathematical real numbers, which do not include infinities and do not include a negative zero that is distinguished from positive zero. Algorithms in this standard that model floating-point arithmetic include explicit steps, where necessary, to handle infinities and signed zero and to perform rounding. If a mathematical operation or function is applied to a floating-point number, it should be understood as being applied to the exact mathematical value represented by that floating-point number; such a floating-point number must be finite, and if it is +0 or -0 then the corresponding mathematical value is simply 0.

The mathematical function abs(x) yields the absolute value of x, which is -x if x is negative (less than zero) and otherwise is x itself.

The mathematical function sign(x) yields 1 if x is positive and -1 if x is negative. The sign function is not used in this standard for cases when x is zero.

The notation "x modulo y" (y must be finite and nonzero) computes a value k of the same sign as y (or zero) such that abs(k)< abs(y) and x-k = q * y for some integer q.

The mathematical function floor(x) yields the largest integer (closest to positive infinity) that is not larger than x.

NOTE
floor(x) = x-(x modulo 1).

If an algorithm is defined to "throw an exception", execution of the algorithm is terminated and no result is returned. The calling algorithms are also terminated, until an algorithm step is reached that explicitly deals with the exception, using terminology such as "If an exception was thrown...". Once such an algorithm step has been encountered the exception is no longer considered to have occurred.