Average Error: 30.0 → 0.5
Time: 22.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}
double f(double x) {
        double r47799 = x;
        double r47800 = 1.0;
        double r47801 = r47799 + r47800;
        double r47802 = cbrt(r47801);
        double r47803 = cbrt(r47799);
        double r47804 = r47802 - r47803;
        return r47804;
}
↓
double f(double x) {
        double r47805 = 1.0;
        double r47806 = x;
        double r47807 = r47806 + r47805;
        double r47808 = cbrt(r47807);
        double r47809 = cbrt(r47806);
        double r47810 = r47808 + r47809;
        double r47811 = r47808 * r47810;
        double r47812 = r47809 * r47809;
        double r47813 = r47811 + r47812;
        double r47814 = r47805 / r47813;
        return r47814;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.0

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied flip3--30.0

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}}\]

  4. Simplified0.5

    \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified33.2

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  6. Using strategy rm

  7. Applied sqr-pow33.2

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{{x}^{\left(\frac{\frac{2}{3}}{2}\right)} \cdot {x}^{\left(\frac{\frac{2}{3}}{2}\right)}}}\]

  8. Simplified33.0

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\sqrt[3]{x}} \cdot {x}^{\left(\frac{\frac{2}{3}}{2}\right)}}\]

  9. Simplified0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{x}}}\]

  10. Final simplification0.5

    \[\leadsto \frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))