The logistic map is a quadratic map defined as *f*(*x*) = *Ax*(1-*x*) where
*A* is a real constant. For some values of *A*, iterating the function
produce chaotic behaviour (see [7] p.130-139). For the
value 4, the behaviour is chaotic, and the function maps the
interval [0,1] onto itself. The chaotic nature of the system
produced has the effect that when computed using floating point
arithmetic we rapidly get erroneous results. This is illustrated in
section 2.1.1.

We examine the lookahead required to compute this using our
implementation of exact real arithmetic supposing *a* is exactly equal
to 4. Let *f*(*x*) = 4*x*(1-*x*). Observe that

so we could compute this using either the stream representation alone
or the (mantissa, exponent). In fact, the lookahead can be smaller for
the (mantissa, exponent) case if we implement multiplication by four
by simply adding two to the exponent. This is a hollow saving,
however, because in order to obtain the same precision we must evaluate
more of the exponent. For example, if the exponent is *n*, and we
require *m* signed binary digits of the result past the binary point,
we must in fact evaluate *m*+*n* digits of the mantissa.