Average Error: 29.8 → 0.6
Time: 18.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}
double f(double x) {
        double r1209609 = x;
        double r1209610 = 1.0;
        double r1209611 = r1209609 + r1209610;
        double r1209612 = cbrt(r1209611);
        double r1209613 = cbrt(r1209609);
        double r1209614 = r1209612 - r1209613;
        return r1209614;
}
↓
double f(double x) {
        double r1209615 = 1.0;
        double r1209616 = x;
        double r1209617 = r1209616 + r1209615;
        double r1209618 = cbrt(r1209617);
        double r1209619 = r1209618 * r1209618;
        double r1209620 = cbrt(r1209616);
        double r1209621 = r1209620 * r1209620;
        double r1209622 = cbrt(r1209621);
        double r1209623 = cbrt(r1209620);
        double r1209624 = r1209622 * r1209623;
        double r1209625 = r1209624 * r1209620;
        double r1209626 = cbrt(r1209625);
        double r1209627 = r1209618 + r1209620;
        double r1209628 = r1209627 * r1209623;
        double r1209629 = r1209626 * r1209628;
        double r1209630 = r1209619 + r1209629;
        double r1209631 = r1209615 / r1209630;
        return r1209631;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.8

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

  3. Applied flip3--29.7

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

  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. Simplified0.5

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

  7. Applied add-cube-cbrt0.6

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

  8. Applied cbrt-prod0.6

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

  9. Applied associate-*l*0.6

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

  11. Applied add-cube-cbrt0.6

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

  12. Applied cbrt-prod0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))