Average Error: 30.2 → 0.6
Time: 16.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\left(\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}
double f(double x) {
        double r77953 = x;
        double r77954 = 1.0;
        double r77955 = r77953 + r77954;
        double r77956 = cbrt(r77955);
        double r77957 = cbrt(r77953);
        double r77958 = r77956 - r77957;
        return r77958;
}
↓
double f(double x) {
        double r77959 = 1.0;
        double r77960 = x;
        double r77961 = r77960 + r77959;
        double r77962 = cbrt(r77961);
        double r77963 = cbrt(r77960);
        double r77964 = cbrt(r77963);
        double r77965 = r77964 * r77964;
        double r77966 = r77965 * r77964;
        double r77967 = r77962 + r77966;
        double r77968 = r77967 * r77963;
        double r77969 = r77962 * r77962;
        double r77970 = r77968 + r77969;
        double r77971 = r77959 / r77970;
        return r77971;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.2

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

  2. Simplified30.2

    \[\leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube30.3

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

  5. Simplified30.3

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

  7. Applied flip3--30.2

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

  8. Simplified15.5

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

  9. Simplified15.5

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

  11. Applied cube-div16.0

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

  12. Applied cbrt-div16.0

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

  13. Simplified16.0

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

  14. Simplified0.5

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

  16. Applied add-cube-cbrt0.6

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

  17. Final simplification0.6

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))