Average Error: 29.6 → 8.5
Time: 8.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.49221087635387833 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.49221087635387833 \cdot 10^{154}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{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 r65024 = x;
        double r65025 = 1.0;
        double r65026 = r65024 + r65025;
        double r65027 = cbrt(r65026);
        double r65028 = cbrt(r65024);
        double r65029 = r65027 - r65028;
        return r65029;
}
↓
double f(double x) {
        double r65030 = x;
        double r65031 = 3.4922108763538783e+154;
        bool r65032 = r65030 <= r65031;
        double r65033 = 1.0;
        double r65034 = exp(r65033);
        double r65035 = log(r65034);
        double r65036 = r65030 + r65033;
        double r65037 = cbrt(r65036);
        double r65038 = r65037 * r65037;
        double r65039 = cbrt(r65030);
        double r65040 = r65037 + r65039;
        double r65041 = r65039 * r65040;
        double r65042 = r65038 + r65041;
        double r65043 = r65035 / r65042;
        double r65044 = 3.0;
        double r65045 = pow(r65043, r65044);
        double r65046 = cbrt(r65045);
        double r65047 = r65037 * r65040;
        double r65048 = 0.6666666666666666;
        double r65049 = pow(r65030, r65048);
        double r65050 = r65047 + r65049;
        double r65051 = r65033 / r65050;
        double r65052 = r65032 ? r65046 : r65051;
        return r65052;
}

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 < 3.4922108763538783e+154

    1. Initial program 25.2

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

    3. Applied add-log-exp26.5

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

    4. Applied add-log-exp26.5

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

    5. Applied diff-log26.5

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

    6. Simplified25.2

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

    8. Applied add-cbrt-cube25.2

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

    9. Simplified25.2

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

    11. Applied flip3--25.1

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

    12. Simplified24.5

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

    13. Simplified24.5

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

    15. Applied add-log-exp26.9

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

    16. Applied add-log-exp26.9

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

    17. Applied add-log-exp26.9

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

    18. Applied sum-log26.9

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

    19. Applied diff-log26.9

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

    20. Simplified9.0

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

    if 3.4922108763538783e+154 < x

    1. Initial program 61.0

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

    3. Applied flip3--61.0

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

    4. Simplified1.0

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

    5. Simplified5.4

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.49221087635387833 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))