Average Error: 29.3 → 0.6
Time: 19.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x} \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 r2120732 = x;
        double r2120733 = 1.0;
        double r2120734 = r2120732 + r2120733;
        double r2120735 = cbrt(r2120734);
        double r2120736 = cbrt(r2120732);
        double r2120737 = r2120735 - r2120736;
        return r2120737;
}
↓
double f(double x) {
        double r2120738 = 1.0;
        double r2120739 = x;
        double r2120740 = r2120739 + r2120738;
        double r2120741 = cbrt(r2120740);
        double r2120742 = r2120741 * r2120741;
        double r2120743 = cbrt(r2120739);
        double r2120744 = r2120743 * r2120743;
        double r2120745 = cbrt(r2120744);
        double r2120746 = r2120741 + r2120743;
        double r2120747 = cbrt(r2120743);
        double r2120748 = r2120746 * r2120747;
        double r2120749 = r2120745 * r2120748;
        double r2120750 = r2120742 + r2120749;
        double r2120751 = r2120738 / r2120750;
        return r2120751;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.3

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

  3. Applied flip3--29.2

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

  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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))