Average Error: 30.1 → 0.6
Time: 15.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}
double f(double x) {
        double r58118 = x;
        double r58119 = 1.0;
        double r58120 = r58118 + r58119;
        double r58121 = cbrt(r58120);
        double r58122 = cbrt(r58118);
        double r58123 = r58121 - r58122;
        return r58123;
}
↓
double f(double x) {
        double r58124 = 1.0;
        double r58125 = x;
        double r58126 = cbrt(r58125);
        double r58127 = r58126 * r58126;
        double r58128 = cbrt(r58127);
        double r58129 = cbrt(r58126);
        double r58130 = r58126 * r58129;
        double r58131 = r58128 * r58130;
        double r58132 = r58125 + r58124;
        double r58133 = cbrt(r58132);
        double r58134 = r58133 + r58126;
        double r58135 = r58133 * r58134;
        double r58136 = r58131 + r58135;
        double r58137 = r58124 / r58136;
        return r58137;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.1

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

  3. Applied flip3--30.1

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

  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. Simplified32.7

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

  7. Applied sqr-pow32.7

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

  8. Simplified32.5

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

  9. Simplified0.5

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

  11. Applied add-cube-cbrt0.6

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

  12. Applied cbrt-prod0.6

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

  13. Applied associate-*l*0.6

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

  14. Simplified0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))