Average Error: 29.5 → 0.4
Time: 20.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -56162.19239214984554564580321311950683594 \lor \neg \left(x \le 70939.05875514661602210253477096557617188\right):\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -56162.19239214984554564580321311950683594 \lor \neg \left(x \le 70939.05875514661602210253477096557617188\right):\\
\;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\

\end{array}
double f(double x) {
        double r45838 = x;
        double r45839 = 1.0;
        double r45840 = r45838 + r45839;
        double r45841 = cbrt(r45840);
        double r45842 = cbrt(r45838);
        double r45843 = r45841 - r45842;
        return r45843;
}
↓
double f(double x) {
        double r45844 = x;
        double r45845 = -56162.192392149846;
        bool r45846 = r45844 <= r45845;
        double r45847 = 70939.05875514662;
        bool r45848 = r45844 <= r45847;
        double r45849 = !r45848;
        bool r45850 = r45846 || r45849;
        double r45851 = cbrt(r45844);
        double r45852 = -1.0;
        double r45853 = cbrt(r45852);
        double r45854 = -r45844;
        double r45855 = cbrt(r45854);
        double r45856 = r45853 * r45855;
        double r45857 = r45851 - r45856;
        double r45858 = r45851 / r45844;
        double r45859 = 0.3333333333333333;
        double r45860 = 0.1111111111111111;
        double r45861 = r45860 / r45844;
        double r45862 = r45859 - r45861;
        double r45863 = r45858 * r45862;
        double r45864 = r45857 + r45863;
        double r45865 = 1.0;
        double r45866 = r45844 + r45865;
        double r45867 = cbrt(r45866);
        double r45868 = r45867 * r45867;
        double r45869 = cbrt(r45868);
        double r45870 = cbrt(r45867);
        double r45871 = r45869 * r45870;
        double r45872 = r45871 - r45851;
        double r45873 = r45850 ? r45864 : r45872;
        return r45873;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -56162.192392149846 or 70939.05875514662 < x

    1. Initial program 60.4

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

    3. Applied add-cube-cbrt60.5

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

    4. Applied cbrt-prod60.6

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]

    5. Taylor expanded around -inf 64.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x} + e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}\right) - \left(0.1111111111111111049432054187491303309798 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}} + {\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}\right)}\]

    6. Simplified0.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)}\]

    if -56162.192392149846 < x < 70939.05875514662

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    4. Applied cbrt-prod0.2

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -56162.19239214984554564580321311950683594 \lor \neg \left(x \le 70939.05875514661602210253477096557617188\right):\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))