Average Error: 29.8 → 12.1
Time: 5.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.45528114356724952 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.0528046201104133 \cdot 10^{-14}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.45528114356724952 \cdot 10^{61}:\\
\;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\

\mathbf{elif}\;x \le 1.0528046201104133 \cdot 10^{-14}:\\
\;\;\;\;\log \left(e^{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r67762 = x;
        double r67763 = 1.0;
        double r67764 = r67762 + r67763;
        double r67765 = cbrt(r67764);
        double r67766 = cbrt(r67762);
        double r67767 = r67765 - r67766;
        return r67767;
}
↓
double f(double x) {
        double r67768 = x;
        double r67769 = -4.4552811435672495e+61;
        bool r67770 = r67768 <= r67769;
        double r67771 = 0.3333333333333333;
        double r67772 = 1.0;
        double r67773 = 2.0;
        double r67774 = pow(r67768, r67773);
        double r67775 = r67772 / r67774;
        double r67776 = 0.3333333333333333;
        double r67777 = pow(r67775, r67776);
        double r67778 = r67771 * r67777;
        double r67779 = 0.06172839506172839;
        double r67780 = 8.0;
        double r67781 = pow(r67768, r67780);
        double r67782 = r67772 / r67781;
        double r67783 = pow(r67782, r67776);
        double r67784 = r67779 * r67783;
        double r67785 = r67778 + r67784;
        double r67786 = 0.1111111111111111;
        double r67787 = 5.0;
        double r67788 = pow(r67768, r67787);
        double r67789 = r67772 / r67788;
        double r67790 = pow(r67789, r67776);
        double r67791 = r67786 * r67790;
        double r67792 = r67785 - r67791;
        double r67793 = 1.0528046201104133e-14;
        bool r67794 = r67768 <= r67793;
        double r67795 = 1.0;
        double r67796 = r67768 + r67795;
        double r67797 = cbrt(r67796);
        double r67798 = r67797 * r67797;
        double r67799 = cbrt(r67798);
        double r67800 = cbrt(r67797);
        double r67801 = r67799 * r67800;
        double r67802 = cbrt(r67768);
        double r67803 = r67801 - r67802;
        double r67804 = exp(r67803);
        double r67805 = log(r67804);
        double r67806 = 0.0;
        double r67807 = r67806 + r67795;
        double r67808 = r67797 + r67802;
        double r67809 = r67797 * r67808;
        double r67810 = 0.6666666666666666;
        double r67811 = pow(r67768, r67810);
        double r67812 = r67809 + r67811;
        double r67813 = r67807 / r67812;
        double r67814 = r67794 ? r67805 : r67813;
        double r67815 = r67770 ? r67792 : r67814;
        return r67815;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4.4552811435672495e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 41.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    if -4.4552811435672495e+61 < x < 1.0528046201104133e-14

    1. Initial program 5.2

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

    3. Applied add-log-exp5.7

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

    4. Applied add-log-exp5.7

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

    5. Applied diff-log5.7

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

    6. Simplified5.2

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

    8. Applied add-cube-cbrt5.2

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

    9. Applied cbrt-prod5.2

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

    if 1.0528046201104133e-14 < x

    1. Initial program 56.9

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

    3. Applied flip3--56.8

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

    4. Simplified1.0

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

    5. Simplified4.3

      \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.45528114356724952 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.0528046201104133 \cdot 10^{-14}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))