Average Error: 30.1 → 0.6
Time: 12.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}
double f(double x) {
        double r37170 = x;
        double r37171 = 1.0;
        double r37172 = r37170 + r37171;
        double r37173 = cbrt(r37172);
        double r37174 = cbrt(r37170);
        double r37175 = r37173 - r37174;
        return r37175;
}
↓
double f(double x) {
        double r37176 = 1.0;
        double r37177 = x;
        double r37178 = r37176 + r37177;
        double r37179 = cbrt(r37178);
        double r37180 = cbrt(r37177);
        double r37181 = r37179 + r37180;
        double r37182 = r37179 * r37181;
        double r37183 = r37180 * r37180;
        double r37184 = cbrt(r37183);
        double r37185 = cbrt(r37180);
        double r37186 = r37180 * r37185;
        double r37187 = r37184 * r37186;
        double r37188 = r37182 + r37187;
        double r37189 = r37176 / r37188;
        return r37189;
}

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

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

  5. Simplified32.7

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

  7. Applied sqr-pow32.7

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

  8. Simplified32.5

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

  9. Simplified0.5

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

  11. Applied add-cube-cbrt0.6

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

  12. Applied cbrt-prod0.6

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

  13. Applied associate-*l*0.6

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

  14. Simplified0.6

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

  15. Final simplification0.6

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

Reproduce

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