Domain-to-range ratio

Consider the function isEven() below, which checks the parity of an unsigned short number ${\displaystyle x}$ , any value between ${\displaystyle 0}$ and ${\displaystyle 65,536}$ , and yields a boolean value which corresponds to whether ${\displaystyle x}$ is even or odd. This solution takes advantage of the fact that integer division in programming typically rounds towards zero.

bool isEven(unsigned short x) {    return (x / 2) == ((x + 3)/2 - 1);}

Because ${\displaystyle x}$ can be any value from ${\displaystyle 0}$ to ${\displaystyle 65,535}$ , the function's domain has a cardinality of ${\displaystyle 65,536}$ . The function yields ${\displaystyle 0}$ , if ${\displaystyle x}$ is even, or ${\displaystyle 1}$ , if ${\displaystyle x}$ is odd. This is expressed as the range ${\displaystyle \{0;1\}}$ , which has a cardinality of ${\displaystyle 2}$ . Therefore, the domain-to-range ratio of isEven() is given by:$${\displaystyle DRR={65,536 \over 2}=32,768}$$ Here, the domain-to-range ratio indicates that this function would require a comparatively large number of tests to find errors. If a test program attempts every possible value of ${\displaystyle x}$ in order from ${\displaystyle 0}$ to ${\displaystyle 65,535}$ , the program would have to perform ${\displaystyle 32,768}$ tests for each of the two possible outputs in order to find errors or edge cases. Because errors in functions with a high domain-to-range ratio are difficult to identify via manual testing or methods which reduce the number of tested inputs, such as orthogonal array testing or all-pairs testing, more computationally complex techniques may be used, such as fuzzing or static program analysis, to find errors.

• Software testing
• Formal verification