## Abstract

The top image is the machine-precision trajectory of 0.1 under the quadratic map 4*x* (1 − *x*). It suffers from roundoff error and the terms beyond the 60th are not the same as the values in the true trajectory of \(\frac{1}{{10}}\) shown just below it. But it turns out that the noisy trajectory can be shadowed, meaning that there is a value near \(\frac{1}{{10}}\)— it turns out to be 0.09999999999999998884314845320503 — whose true trajectory under *f* matches the noisy trajectory very closely (to within 10^{−15}). Finding this value is a true needle-in-a-gigantic-haystack problem, but sophisticated optimization algorithms are capable of getting it in under one second, and with only a few lines of code. The third image shows the absolute difference between the noisy trajectory and true trajectories in the vicinity of the shadow value; this gives an indication of the difficulty of finding this value, since the spike that defines it is very narrow.