Average Error: 29.8 → 11.9
Time: 5.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.43142291971707334 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.464806713910809466:\\ \;\;\;\;\left(\sqrt{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\frac{1}{3}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\right) \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.43142291971707334 \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 0.464806713910809466:\\
\;\;\;\;\left(\sqrt{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\frac{1}{3}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\right) \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \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 r61843 = x;
        double r61844 = 1.0;
        double r61845 = r61843 + r61844;
        double r61846 = cbrt(r61845);
        double r61847 = cbrt(r61843);
        double r61848 = r61846 - r61847;
        return r61848;
}
↓
double f(double x) {
        double r61849 = x;
        double r61850 = -4.4314229197170733e+61;
        bool r61851 = r61849 <= r61850;
        double r61852 = 0.3333333333333333;
        double r61853 = 1.0;
        double r61854 = 2.0;
        double r61855 = pow(r61849, r61854);
        double r61856 = r61853 / r61855;
        double r61857 = 0.3333333333333333;
        double r61858 = pow(r61856, r61857);
        double r61859 = r61852 * r61858;
        double r61860 = 0.06172839506172839;
        double r61861 = 8.0;
        double r61862 = pow(r61849, r61861);
        double r61863 = r61853 / r61862;
        double r61864 = pow(r61863, r61857);
        double r61865 = r61860 * r61864;
        double r61866 = r61859 + r61865;
        double r61867 = 0.1111111111111111;
        double r61868 = 5.0;
        double r61869 = pow(r61849, r61868);
        double r61870 = r61853 / r61869;
        double r61871 = pow(r61870, r61857);
        double r61872 = r61867 * r61871;
        double r61873 = r61866 - r61872;
        double r61874 = 0.46480671391080947;
        bool r61875 = r61849 <= r61874;
        double r61876 = 1.0;
        double r61877 = r61849 + r61876;
        double r61878 = cbrt(r61877);
        double r61879 = r61878 * r61878;
        double r61880 = pow(r61879, r61857);
        double r61881 = sqrt(r61880);
        double r61882 = cbrt(r61879);
        double r61883 = sqrt(r61882);
        double r61884 = r61881 * r61883;
        double r61885 = cbrt(r61878);
        double r61886 = r61884 * r61885;
        double r61887 = cbrt(r61849);
        double r61888 = r61886 - r61887;
        double r61889 = 0.0;
        double r61890 = r61889 + r61876;
        double r61891 = r61878 + r61887;
        double r61892 = r61878 * r61891;
        double r61893 = 0.6666666666666666;
        double r61894 = pow(r61849, r61893);
        double r61895 = r61892 + r61894;
        double r61896 = r61890 / r61895;
        double r61897 = r61875 ? r61888 : r61896;
        double r61898 = r61851 ? r61873 : r61897;
        return r61898;
}

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 < -4.4314229197170733e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 41.1

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

    if -4.4314229197170733e+61 < x < 0.46480671391080947

    1. Initial program 4.6

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

    3. Applied add-cube-cbrt4.6

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

    4. Applied cbrt-prod4.6

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

    6. Applied add-sqr-sqrt4.6

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

    8. Applied pow1/34.4

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

    if 0.46480671391080947 < x

    1. Initial program 59.6

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

    3. Applied flip3--59.5

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

    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 simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.43142291971707334 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.464806713910809466:\\ \;\;\;\;\left(\sqrt{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\frac{1}{3}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\right) \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))