Average Error: 29.9 → 12.0
Time: 5.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4537552488591828 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.0030493216515878772:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4537552488591828 \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 0.0030493216515878772:\\
\;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\

\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 r61788 = x;
        double r61789 = 1.0;
        double r61790 = r61788 + r61789;
        double r61791 = cbrt(r61790);
        double r61792 = cbrt(r61788);
        double r61793 = r61791 - r61792;
        return r61793;
}
↓
double f(double x) {
        double r61794 = x;
        double r61795 = -4.453755248859183e+61;
        bool r61796 = r61794 <= r61795;
        double r61797 = 0.3333333333333333;
        double r61798 = 1.0;
        double r61799 = 2.0;
        double r61800 = pow(r61794, r61799);
        double r61801 = r61798 / r61800;
        double r61802 = 0.3333333333333333;
        double r61803 = pow(r61801, r61802);
        double r61804 = r61797 * r61803;
        double r61805 = 0.06172839506172839;
        double r61806 = 8.0;
        double r61807 = pow(r61794, r61806);
        double r61808 = r61798 / r61807;
        double r61809 = pow(r61808, r61802);
        double r61810 = r61805 * r61809;
        double r61811 = r61804 + r61810;
        double r61812 = 0.1111111111111111;
        double r61813 = 5.0;
        double r61814 = pow(r61794, r61813);
        double r61815 = r61798 / r61814;
        double r61816 = pow(r61815, r61802);
        double r61817 = r61812 * r61816;
        double r61818 = r61811 - r61817;
        double r61819 = 0.003049321651587877;
        bool r61820 = r61794 <= r61819;
        double r61821 = 3.0;
        double r61822 = pow(r61794, r61821);
        double r61823 = 1.0;
        double r61824 = pow(r61823, r61821);
        double r61825 = r61822 + r61824;
        double r61826 = cbrt(r61825);
        double r61827 = r61794 * r61794;
        double r61828 = r61823 * r61823;
        double r61829 = r61794 * r61823;
        double r61830 = r61828 - r61829;
        double r61831 = r61827 + r61830;
        double r61832 = cbrt(r61831);
        double r61833 = r61826 / r61832;
        double r61834 = cbrt(r61794);
        double r61835 = r61833 - r61834;
        double r61836 = 0.0;
        double r61837 = r61836 + r61823;
        double r61838 = r61794 + r61823;
        double r61839 = cbrt(r61838);
        double r61840 = r61839 + r61834;
        double r61841 = r61839 * r61840;
        double r61842 = 0.6666666666666666;
        double r61843 = pow(r61794, r61842);
        double r61844 = r61841 + r61843;
        double r61845 = r61837 / r61844;
        double r61846 = r61820 ? r61835 : r61845;
        double r61847 = r61796 ? r61818 : r61846;
        return r61847;
}

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 < -4.453755248859183e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.2

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

    if -4.453755248859183e+61 < x < 0.003049321651587877

    1. Initial program 4.8

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

    3. Applied flip3-+4.7

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

    4. Applied cbrt-div4.7

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

    if 0.003049321651587877 < x

    1. Initial program 59.1

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

    3. Applied flip3--59.0

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

    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. Simplified4.5

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4537552488591828 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.0030493216515878772:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))