Average Error: 29.9 → 0.5
Time: 6.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{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)}
double f(double x) {
        double r84738 = x;
        double r84739 = 1.0;
        double r84740 = r84738 + r84739;
        double r84741 = cbrt(r84740);
        double r84742 = cbrt(r84738);
        double r84743 = r84741 - r84742;
        return r84743;
}
↓
double f(double x) {
        double r84744 = 1.0;
        double r84745 = x;
        double r84746 = r84745 + r84744;
        double r84747 = cbrt(r84746);
        double r84748 = r84747 * r84747;
        double r84749 = cbrt(r84745);
        double r84750 = r84749 * r84749;
        double r84751 = r84747 * r84749;
        double r84752 = r84750 + r84751;
        double r84753 = r84748 + r84752;
        double r84754 = r84744 / r84753;
        return r84754;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

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

  3. Applied add-exp-log29.9

    \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}}\]
  4. Using strategy rm

  5. Applied flip3--29.9

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

  6. Applied log-div29.9

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

  7. Taylor expanded around 0 2.6

    \[\leadsto e^{\color{blue}{\log 1} - \log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}\]
  8. Using strategy rm

  9. Applied diff-log2.6

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

  10. Applied rem-exp-log0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))