Number::Interval and Set::Window seem to be concerned with intervals on a number line. Neither module has defined the basic arithmetic operators (+, -, · and ÷) or mathematical functions. I need these along with comparison functions to test the 18 (eep!) relations between two intervals.

For my application an interval represents a value with a known amount of error. The interval [4, 6] means that the real answer is somewhere between 4 and 6, but we don't know exactly where. Because floating point calculations aren't exact, the results of interval calculations are rounded out by rounding the lower bound down and the upper bound up to the next representable numbers. The end result gives you a lower and upper bound that contain the real answer.

The basic operators for intervals are defined as:

x op y = [ min{ lower(x) op lower(y), lower(x) op upper(y),
                upper(x) op lower(y), upper(x) op upper(y) },
           max{ lower(x) op lower(y), lower(x) op upper(y),
                upper(x) op lower(y), upper(x) op upper(y) } ]
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With a few proofs we can eliminate some of the operations. So, to add two intervals you would add the lower bounds together and the upper bounds together. For example, x + y = [lower(x) + lower(y), upper(x) + upper(y)]. So your example of add([1, 4], [9, 12]) would return [10, 16], which does contain 10.

Thank you for your reply.