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on 11/21/2014 4:00 PM
A while back I came upon a seemingly not-too-difficult programming exercise:
Define a recurrence \(x_n\) by
$$f(y, z) = 108 - \frac{815 - 1500/z}{y}$$
$$x_0 = 4$$
$$x_1 = 4.25$$
$$x_i = f(x_{i-1}, x_{i-2})$$
Compute \(x_{30}\).
This isn’t too hard to code up, using perhaps a recursive function to represent \(x_i\). With normal double-precision floats, as \(i\) increases, the result converges neatly toward 100. Super!
Unfortunately, 100 is not even close to the right answer. This recurrence actually [...]