= Solution
\b[About five steps are certified by the uniform worst-case estimate; the actual iteration reaches double-precision accuracy in about four.] Put $K=15/14$. The initial enclosure gives $e_0=|x_0-r|<1/5$, and repeated squaring yields
$$
e_k\le\frac1K(Ke_0)^{2^k}
<\frac{14}{15}\left(\frac3{14}\right)^{2^k}.
$$
For $k=4$ this is about $1.84\times10^{-11}$, while for $k=5$ it is below $4\times10^{-22}$. Thus five ideal-arithmetic steps push this bound well below the approximately $10^{-16}$ relative resolution of <double-precision floating-point format>.
The bound is conservative. Factoring $f(x)=(x^2+1)(x^4-6x^2+6)$ identifies
$$
r=\sqrt{3-\sqrt3}\approx1.126032500610494238.
$$
High-precision evaluation of the actual successive errors gives approximately $7.40\times10^{-2}$, $3.56\times10^{-3}$, $9.91\times10^{-6}$, $7.73\times10^{-11}$ and $4.70\times10^{-21}$. Thus the fourth step would already exceed double-precision accuracy in exact arithmetic. In floating-point evaluation, cancellation in the residual and other <roundoff errors> prevent the subsequent theoretical digits from being realized.
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