  Building on a classification of zeros of cubic equations due to the $12$-th
century Persian mathematician Sharaf al-Din Tusi, together with Smale's theory
of {\it point estimation}, we derive an efficient recipe for computing
high-precision approximation to a real root of an arbitrary real cubic
equation. First, via reversible transformations we reduce any real cubic
equation into one of four canonical forms with $0$, $\pm 1$ coefficients,
except for the constant term as $\pm q$, $q \geq 0$. Next, given any form, if
$\rho_q$ is an approximation to $\sqrt[3]{q}$ to within a relative error of
five percent, we prove a {\it seed} $x_0$ in $\{ \rho_q, \pm .95 \rho_q,
-\frac{1}{3}, 1 \}$ can be selected such that in $t$ Newton iterations $|x_t -
\theta_q| \leq \sqrt[3]{q}\cdot 2^{-2^{t}}$ for some real root $\theta_q$.
While computing a good seed, even for approximation of $\sqrt[3]{q}$, is
considered to be ``somewhat of black art'' (see Wikipedia), as we justify,
$\rho_q$ is readily computable from {\it mantissa} and {\it exponent} of $q$.
It follows that the above approach gives a simple recipe for numerical
approximation of solutions of real cubic equations independent of Cardano's
formula.

Numerical Analysis;Numerical Analysis