Newton-Raphson Cube Root

Problem

Implement Newton-Raphson root finding to compute the cube root of a positive number $a$.

Define $f(x) = x^3 - a$. The root of $f$ is $x = a^{1/3}$. Newton's method iterates:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{x_n^3 - a}{3x_n^2}$

Starting from initial guess $x_0$, run exactly num_iterations iterations and output the final $x$.

Input Format

Three space-separated values: a x0 num_iterations

Output Format

The final approximation of $a^{1/3}$, to 4 decimal places.