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.

翻译：基于12世纪波斯数学家Sharaf al-Din Tusi对三次方程零点分类的研究，结合Smale的“点估计”理论，我们推导出一种高效方法，可计算任意实系数三次方程实根的高精度近似值。首先，通过可逆变换将任意实系数三次方程简化为四种标准形式之一，其系数仅为$0$、$\pm 1$，常数项为$\pm q$（$q \geq 0$）。其次，对任一标准形式，若$\rho_q$是$\sqrt[3]{q}$的相对误差不超过5%的近似值，我们证明可在$\{ \rho_q, \pm .95 \rho_q, -\frac{1}{3}, 1 \}$中选择一个“初始值”$x_0$，使得经过$t$次牛顿迭代后，存在实根$\theta_q$满足$|x_t - \theta_q| \leq \sqrt[3]{q}\cdot 2^{-2^{t}}$。尽管计算$\sqrt[3]{q}$的良好初始值常被视为“某种黑魔法”（参见维基百科），但我们论证了$\rho_q$可直接由$q$的“尾数”和“指数”计算得出。因此，上述方法提供了一种无需依赖Cardano公式即可数值逼近实数三次方程解的简洁方案。