Define two sets of intervals to be equivalent relative to a set of numbers (the base) if both sets' intervals cover the exact same set of numbers in the base. For example, given the base [0, 1, 3, 4, 6], {[0, 1], [3, 6]} is equivalent to {[0, 3], [4, 6]} but not to {[1, 3], [4, 6]} because the latter doesn't cover 0.
Define a set of intervals to be maximal relative to a base if there's no equivalent set with fewer intervals. Equivalently, a set of intervals is maximal iff, for any two intervals in the set, there exists at least one number in the base between the upper bound of one interval and the lower bound of the other.
For example, given the base [0, 1, 3, 4, 6], the set of ranges {[0, 1], [4, 6]} is maximal because 3 is between 1 and 4, but {[0, 1], [3, 6]} is not because there's nothing in the base between 1 and 3.
You are given a (potentially very large) ordered list of floats—the base—and a list of intervals, each of which may be open or closed.