Here's some simpler code that computes the same result -- I'm too tired and dumb...

Here's some simpler code that computes the same result -- I'm too tired and dumb to work out if it will for any number:

    def rationalizations(x):
        """Generate good rational approximations of x in order of
        increasing denominator."""
        assert 0 <= x
        ix = int(x)
        yield ix, 1
        if x != ix:
            for numer, denom in rationalizations(1.0/(x-ix)):
                yield denom + ix * numer, numer

    import itertools, math
    def show(x, n): return list(itertools.islice(rationalizations(x), n))

    print show(math.pi, 11)
    # [(3, 1), (22, 7), (333, 106), (355, 113), (103993, 33102),
    # (104348, 33215), (208341, 66317), (312689, 99532),
    # (833719, 265381), (1146408, 364913), (4272943, 1360120)]

From https://github.com/darius/sketchbook/blob/master/misc/ration...