Average Error: 29.9 → 12.0
Time: 6.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.539993431525089741023971091851498224407 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 2677.813917216188656311715021729469299316:\\ \;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.539993431525089741023971091851498224407 \cdot 10^{61}:\\
\;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\

\mathbf{elif}\;x \le 2677.813917216188656311715021729469299316:\\
\;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\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}}}\\

\end{array}
double f(double x) {
        double r57793 = x;
        double r57794 = 1.0;
        double r57795 = r57793 + r57794;
        double r57796 = cbrt(r57795);
        double r57797 = cbrt(r57793);
        double r57798 = r57796 - r57797;
        return r57798;
}
↓
double f(double x) {
        double r57799 = x;
        double r57800 = -4.53999343152509e+61;
        bool r57801 = r57799 <= r57800;
        double r57802 = 1.0;
        double r57803 = 2.0;
        double r57804 = pow(r57799, r57803);
        double r57805 = r57802 / r57804;
        double r57806 = 0.3333333333333333;
        double r57807 = pow(r57805, r57806);
        double r57808 = 0.3333333333333333;
        double r57809 = 0.06172839506172839;
        double r57810 = 8.0;
        double r57811 = pow(r57799, r57810);
        double r57812 = r57802 / r57811;
        double r57813 = pow(r57812, r57806);
        double r57814 = r57809 * r57813;
        double r57815 = 0.1111111111111111;
        double r57816 = 5.0;
        double r57817 = pow(r57799, r57816);
        double r57818 = r57802 / r57817;
        double r57819 = pow(r57818, r57806);
        double r57820 = r57815 * r57819;
        double r57821 = r57814 - r57820;
        double r57822 = fma(r57807, r57808, r57821);
        double r57823 = 2677.8139172161887;
        bool r57824 = r57799 <= r57823;
        double r57825 = 1.0;
        double r57826 = r57799 + r57825;
        double r57827 = cbrt(r57826);
        double r57828 = r57827 * r57827;
        double r57829 = cbrt(r57799);
        double r57830 = r57829 * r57829;
        double r57831 = cbrt(r57830);
        double r57832 = cbrt(r57829);
        double r57833 = r57831 * r57832;
        double r57834 = r57829 * r57833;
        double r57835 = r57828 - r57834;
        double r57836 = r57827 + r57829;
        double r57837 = r57835 / r57836;
        double r57838 = 7.0;
        double r57839 = pow(r57799, r57838);
        double r57840 = r57802 / r57839;
        double r57841 = pow(r57840, r57806);
        double r57842 = 0.04938271604938271;
        double r57843 = 0.6666666666666666;
        double r57844 = r57802 / r57799;
        double r57845 = pow(r57844, r57806);
        double r57846 = r57843 * r57845;
        double r57847 = 4.0;
        double r57848 = pow(r57799, r57847);
        double r57849 = r57802 / r57848;
        double r57850 = pow(r57849, r57806);
        double r57851 = r57815 * r57850;
        double r57852 = r57846 - r57851;
        double r57853 = fma(r57841, r57842, r57852);
        double r57854 = r57832 * r57832;
        double r57855 = r57854 * r57832;
        double r57856 = r57827 + r57855;
        double r57857 = r57853 / r57856;
        double r57858 = r57824 ? r57837 : r57857;
        double r57859 = r57801 ? r57822 : r57858;
        return r57859;
}

Error

Bits error versus x

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. Taylor expanded around inf 40.0

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

    3. Simplified40.0

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

    if -4.53999343152509e+61 < x < 2677.8139172161887

    1. Initial program 4.7

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

    3. Applied flip--4.7

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

    5. Applied add-cube-cbrt4.7

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

    6. Applied cbrt-prod4.6

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

    if 2677.8139172161887 < x

    1. Initial program 60.2

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

    3. Applied flip--60.2

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

    5. Applied add-cube-cbrt60.2

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

    6. Taylor expanded around inf 5.1

      \[\leadsto \frac{\color{blue}{\left(0.0493827160493827133080912972218357026577 \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\]

    7. Simplified5.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.539993431525089741023971091851498224407 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 2677.813917216188656311715021729469299316:\\ \;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \end{array}\]

Reproduce

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