Average Error: 29.7 → 0.6
Time: 18.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}\]
\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]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}
double f(double x) {
        double r40729 = x;
        double r40730 = 1.0;
        double r40731 = r40729 + r40730;
        double r40732 = cbrt(r40731);
        double r40733 = cbrt(r40729);
        double r40734 = r40732 - r40733;
        return r40734;
}
↓
double f(double x) {
        double r40735 = 0.0;
        double r40736 = 1.0;
        double r40737 = r40735 + r40736;
        double r40738 = x;
        double r40739 = r40738 + r40736;
        double r40740 = cbrt(r40739);
        double r40741 = cbrt(r40738);
        double r40742 = r40740 + r40741;
        double r40743 = r40740 * r40742;
        double r40744 = r40741 * r40741;
        double r40745 = cbrt(r40744);
        double r40746 = r40741 * r40745;
        double r40747 = cbrt(r40741);
        double r40748 = r40747 * r40747;
        double r40749 = cbrt(r40748);
        double r40750 = cbrt(r40747);
        double r40751 = r40749 * r40750;
        double r40752 = r40746 * r40751;
        double r40753 = r40743 + r40752;
        double r40754 = r40737 / r40753;
        return r40754;
}

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--29.7

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

  4. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied cbrt-prod0.6

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

  9. Applied associate-*r*0.6

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

  11. Applied add-cube-cbrt0.6

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

  12. Applied cbrt-prod0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))