DouKaku?

public static double CubeRoot(double x)
{
  double a0 = (double)(-x);
  double[] a = { a0, 0.0, 0.0, 1.0 };
  return SolveAlgebraicEquationByNewtonMethod(a, 1e-12);
}

delegate double MathFunc(double x);

static double SolveAlgebraicEquationByNewtonMethod(double[] a, double delta)
{
  MathFunc f = CreateFunc(a);
  MathFunc fPrime = CreateFunc(Derivative(a));
  double x = 1.0;
  while (NearlyEquals(fPrime(x), 0.0, double.Epsilon))
    x += 1.0;
  double betterX = 0;
  while (true)
  {
    betterX = Newton(f, fPrime, x);
    if (NearlyEquals(x, betterX, delta))
      break;
    x = betterX;
  }
  return betterX;
}

static MathFunc CreateFunc(double[] a)
{
  return delegate(double x)
  {
    double fx = 0.0;
    for (int i = 0; i < a.Length; i++)
    {
      double d = a[i];
      if (NearlyEquals(d, 0.0, double.Epsilon))
        continue;
      for (int j = 0; j < i; j++)
        d *= x;
      fx += d;
    }
    return fx;
  };
}

static double[] Derivative(double[] a)
{
  double[] aPrime = new double[a.Length - 1];
  for (int i = 1; i < a.Length; i++)
  {
    aPrime[i - 1] = a[i] * (double)i;
  }
  return aPrime;
}

static double Newton(MathFunc f, MathFunc fPrime, double x)
{
  return x - (f(x) / fPrime(x));
}

static bool NearlyEquals(double d1, double d2, double delta)
{
  return (((d1 - d2) <= delta) && ((d2 - d1) <= delta));
}