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Comment by mci | Hacker News Reader

See slides 43–60 and 72. A node emits ∞ as the partial quotient after 100 resultless iterations. The threshold is configurable. In particular, see lines 184–187 of cf.py:

    # Idealistically inclined folks, who would prefer their
    # computations to hang rather than to yield a heuristic result,
    # can achieve this effect by setting accuracy to a negative
    # value.

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Seems reasonable, but then this is not "exact" precision as you claim, but "very large" or "dynamic precision". Saying that the precision of a real number is exact is very misleading, as it evokes inevitably an automatic theorem-proving context.

Thanks, "dynamic" is a good term. "Very large" would not describe it well: for non-contrived irrational numbers, the chance of an improper rounding to a rational is much smaller than the chance of a meteorite strike in your room within the next five minutes (slide 60).

but then again, the set of "non-contrived irrational numbers" is certainly of measure zero

This discussion would really go better if you followed the links. See lines 92–96 of cf.py:

      By the
    # Gauss-Kuzmin theorem [5, 6], a partial quotient in the
    # continued fraction expansion of almost every number exceeds
    # 1e+31 with probability log2(1+1e-31) =~= 1e-31/ln(2) =~=
    # 1.5e-31.

I'm somewhat familiar with Gauss-Kuzmin theorem. If I understand it well, it means that large partial quotients are rare in most numbers. This is fairly intuitive, and conforming to the experiences of anybody who has done practical computations of continued fractions, as you certainly have.

Notice that this result does not say that most numbers have bounded partial quotients. In fact, a classical result in diophantine approximation theory, if I recall correctly, says that the set of numbers whose partial fractions are bounded (called badly aproximable numbers) is of Lebesgue measure zero. Thus, if your system represents real numbers by sequences of bounded partial quotients, then it can only represent a negligible (but uncountable) set of real numbers. A "random" number in [0,1] will certainly have an unbounded sequence of partial quotients (i.e., with probability 1).

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