Average Error: 29.7 → 0.6
Time: 6.5s
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 r118626 = x;
        double r118627 = 1.0;
        double r118628 = r118626 + r118627;
        double r118629 = cbrt(r118628);
        double r118630 = cbrt(r118626);
        double r118631 = r118629 - r118630;
        return r118631;
}
↓
double f(double x) {
        double r118632 = 0.0;
        double r118633 = 1.0;
        double r118634 = r118632 + r118633;
        double r118635 = x;
        double r118636 = r118635 + r118633;
        double r118637 = cbrt(r118636);
        double r118638 = cbrt(r118635);
        double r118639 = r118637 + r118638;
        double r118640 = r118637 * r118639;
        double r118641 = 2.0;
        double r118642 = pow(r118638, r118641);
        double r118643 = cbrt(r118642);
        double r118644 = r118643 * r118643;
        double r118645 = r118644 * r118643;
        double r118646 = r118640 + r118645;
        double r118647 = r118634 / r118646;
        return r118647;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.7

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

  3. Applied flip3--29.6

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

  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. Simplified32.8

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

  7. Applied add-cbrt-cube39.9

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

  8. Simplified15.7

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

  10. Applied add-cube-cbrt15.8

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

  11. Applied unpow-prod-down15.8

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

  12. Applied cbrt-prod7.9

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

  14. Applied unpow-prod-down7.9

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

  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 2019318 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))