Average Error: 30.0 → 0.6
Time: 7.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}
double f(double x) {
        double r56752 = x;
        double r56753 = 1.0;
        double r56754 = r56752 + r56753;
        double r56755 = cbrt(r56754);
        double r56756 = cbrt(r56752);
        double r56757 = r56755 - r56756;
        return r56757;
}
↓
double f(double x) {
        double r56758 = 0.0;
        double r56759 = 1.0;
        double r56760 = r56758 + r56759;
        double r56761 = x;
        double r56762 = r56761 + r56759;
        double r56763 = cbrt(r56762);
        double r56764 = cbrt(r56761);
        double r56765 = r56763 + r56764;
        double r56766 = r56763 * r56765;
        double r56767 = 2.0;
        double r56768 = pow(r56764, r56767);
        double r56769 = cbrt(r56768);
        double r56770 = r56769 * r56769;
        double r56771 = r56770 * r56769;
        double r56772 = r56766 + r56771;
        double r56773 = r56760 / r56772;
        return r56773;
}

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

    \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified33.2

    \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube40.2

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\sqrt[3]{\left({x}^{\frac{2}{3}} \cdot {x}^{\frac{2}{3}}\right) \cdot {x}^{\frac{2}{3}}}}}\]

  8. Simplified15.5

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{{x}^{2}}}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt15.5

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}}}\]

  11. Applied unpow-prod-down15.5

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}}}}\]

  12. Applied cbrt-prod8.2

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\sqrt[3]{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}}\]
  13. Using strategy rm

  14. Applied unpow-prod-down8.2

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]

  15. Applied cbrt-prod0.6

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]

  16. Final simplification0.6

    \[\leadsto \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]

Reproduce

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