Average Error: 29.9 → 11.8
Time: 6.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.539993431525089741023971091851498224407 \cdot 10^{61}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left(\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \mathbf{elif}\;x \le 1.389772997309203874017778310157034127315 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.539993431525089741023971091851498224407 \cdot 10^{61}:\\
\;\;\;\;\frac{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left(\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\

\mathbf{elif}\;x \le 1.389772997309203874017778310157034127315 \cdot 10^{-6}:\\
\;\;\;\;\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{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\

\end{array}
double f(double x) {
        double r69872 = x;
        double r69873 = 1.0;
        double r69874 = r69872 + r69873;
        double r69875 = cbrt(r69874);
        double r69876 = cbrt(r69872);
        double r69877 = r69875 - r69876;
        return r69877;
}
↓
double f(double x) {
        double r69878 = x;
        double r69879 = -4.53999343152509e+61;
        bool r69880 = r69878 <= r69879;
        double r69881 = 1.0;
        double r69882 = r69878 + r69881;
        double r69883 = cbrt(r69882);
        double r69884 = cbrt(r69878);
        double r69885 = r69883 + r69884;
        double r69886 = 0.3333333333333333;
        double r69887 = 1.0;
        double r69888 = 2.0;
        double r69889 = pow(r69878, r69888);
        double r69890 = r69887 / r69889;
        double r69891 = 0.3333333333333333;
        double r69892 = pow(r69890, r69891);
        double r69893 = r69886 * r69892;
        double r69894 = 0.06172839506172839;
        double r69895 = 8.0;
        double r69896 = pow(r69878, r69895);
        double r69897 = r69887 / r69896;
        double r69898 = pow(r69897, r69891);
        double r69899 = r69894 * r69898;
        double r69900 = r69893 + r69899;
        double r69901 = 0.1111111111111111;
        double r69902 = 5.0;
        double r69903 = pow(r69878, r69902);
        double r69904 = r69887 / r69903;
        double r69905 = pow(r69904, r69891);
        double r69906 = r69901 * r69905;
        double r69907 = r69900 - r69906;
        double r69908 = r69885 * r69907;
        double r69909 = cbrt(r69884);
        double r69910 = r69909 * r69909;
        double r69911 = r69910 * r69909;
        double r69912 = r69883 + r69911;
        double r69913 = r69908 / r69912;
        double r69914 = 1.3897729973092039e-06;
        bool r69915 = r69878 <= r69914;
        double r69916 = 3.0;
        double r69917 = pow(r69878, r69916);
        double r69918 = pow(r69881, r69916);
        double r69919 = r69917 + r69918;
        double r69920 = cbrt(r69919);
        double r69921 = r69878 * r69878;
        double r69922 = r69881 * r69881;
        double r69923 = r69878 * r69881;
        double r69924 = r69922 - r69923;
        double r69925 = r69921 + r69924;
        double r69926 = cbrt(r69925);
        double r69927 = r69920 / r69926;
        double r69928 = r69927 - r69884;
        double r69929 = 0.0;
        double r69930 = r69929 + r69881;
        double r69931 = r69883 * r69885;
        double r69932 = 0.6666666666666666;
        double r69933 = pow(r69878, r69932);
        double r69934 = r69931 + r69933;
        double r69935 = r69930 / r69934;
        double r69936 = r69885 * r69935;
        double r69937 = r69936 / r69912;
        double r69938 = r69915 ? r69928 : r69937;
        double r69939 = r69880 ? r69913 : r69938;
        return r69939;
}

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 < -4.53999343152509e+61

    1. Initial program 61.2

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

    3. Applied flip--61.2

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

    5. Applied add-cube-cbrt61.2

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

    7. Applied difference-of-squares61.2

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

    8. Taylor expanded around inf 40.0

      \[\leadsto \frac{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \color{blue}{\left(\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\]

    if -4.53999343152509e+61 < x < 1.3897729973092039e-06

    1. Initial program 4.7

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
    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-div4.6

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

    if 1.3897729973092039e-06 < x

    1. Initial program 58.6

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

    3. Applied flip--58.6

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

    5. Applied add-cube-cbrt58.6

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

    7. Applied difference-of-squares58.6

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

    9. Applied flip3--58.5

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

    10. Simplified1.2

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

    11. Simplified4.4

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.539993431525089741023971091851498224407 \cdot 10^{61}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left(\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \mathbf{elif}\;x \le 1.389772997309203874017778310157034127315 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))