Average Error: 30.0 → 0.6
Time: 27.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2988.405682367412:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\ \mathbf{elif}\;x \le 3712.7024122827197:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2988.405682367412:\\
\;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\

\mathbf{elif}\;x \le 3712.7024122827197:\\
\;\;\;\;\frac{\log \left(e^{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\

\end{array}
double f(double x) {
        double r2613783 = x;
        double r2613784 = 1.0;
        double r2613785 = r2613783 + r2613784;
        double r2613786 = cbrt(r2613785);
        double r2613787 = cbrt(r2613783);
        double r2613788 = r2613786 - r2613787;
        return r2613788;
}
↓
double f(double x) {
        double r2613789 = x;
        double r2613790 = -2988.405682367412;
        bool r2613791 = r2613789 <= r2613790;
        double r2613792 = 1.0;
        double r2613793 = r2613792 / r2613789;
        double r2613794 = r2613793 / r2613789;
        double r2613795 = r2613789 * r2613789;
        double r2613796 = r2613794 / r2613795;
        double r2613797 = cbrt(r2613796);
        double r2613798 = -0.1111111111111111;
        double r2613799 = 0.6666666666666666;
        double r2613800 = cbrt(r2613793);
        double r2613801 = 7.0;
        double r2613802 = pow(r2613789, r2613801);
        double r2613803 = r2613792 / r2613802;
        double r2613804 = cbrt(r2613803);
        double r2613805 = 0.04938271604938271;
        double r2613806 = r2613804 * r2613805;
        double r2613807 = fma(r2613799, r2613800, r2613806);
        double r2613808 = fma(r2613797, r2613798, r2613807);
        double r2613809 = r2613792 + r2613789;
        double r2613810 = cbrt(r2613809);
        double r2613811 = cbrt(r2613789);
        double r2613812 = r2613811 * r2613811;
        double r2613813 = r2613812 * r2613811;
        double r2613814 = cbrt(r2613813);
        double r2613815 = r2613810 + r2613814;
        double r2613816 = r2613808 / r2613815;
        double r2613817 = 3712.7024122827197;
        bool r2613818 = r2613789 <= r2613817;
        double r2613819 = r2613810 * r2613810;
        double r2613820 = r2613814 * r2613814;
        double r2613821 = r2613819 - r2613820;
        double r2613822 = exp(r2613821);
        double r2613823 = log(r2613822);
        double r2613824 = r2613823 / r2613815;
        double r2613825 = r2613818 ? r2613824 : r2613816;
        double r2613826 = r2613791 ? r2613816 : r2613825;
        return r2613826;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2988.405682367412 or 3712.7024122827197 < x

    1. Initial program 60.3

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

    3. Applied add-cbrt-cube60.4

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

    5. Applied flip--60.4

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

    6. Taylor expanded around inf 34.0

      \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]

    7. Simplified1.1

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}}{\sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]

    if -2988.405682367412 < x < 3712.7024122827197

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    5. Applied flip--0.2

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

    7. Applied add-log-exp0.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2988.405682367412:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\ \mathbf{elif}\;x \le 3712.7024122827197:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))