Average Error: 29.9 → 12.0
Time: 6.5s
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 r64730 = x;
        double r64731 = 1.0;
        double r64732 = r64730 + r64731;
        double r64733 = cbrt(r64732);
        double r64734 = cbrt(r64730);
        double r64735 = r64733 - r64734;
        return r64735;
}
↓
double f(double x) {
        double r64736 = x;
        double r64737 = -4.53999343152509e+61;
        bool r64738 = r64736 <= r64737;
        double r64739 = 1.0;
        double r64740 = 2.0;
        double r64741 = pow(r64736, r64740);
        double r64742 = r64739 / r64741;
        double r64743 = 0.3333333333333333;
        double r64744 = pow(r64742, r64743);
        double r64745 = 0.3333333333333333;
        double r64746 = 0.06172839506172839;
        double r64747 = 8.0;
        double r64748 = pow(r64736, r64747);
        double r64749 = r64739 / r64748;
        double r64750 = pow(r64749, r64743);
        double r64751 = r64746 * r64750;
        double r64752 = 0.1111111111111111;
        double r64753 = 5.0;
        double r64754 = pow(r64736, r64753);
        double r64755 = r64739 / r64754;
        double r64756 = pow(r64755, r64743);
        double r64757 = r64752 * r64756;
        double r64758 = r64751 - r64757;
        double r64759 = fma(r64744, r64745, r64758);
        double r64760 = 2677.8139172161887;
        bool r64761 = r64736 <= r64760;
        double r64762 = 1.0;
        double r64763 = r64736 + r64762;
        double r64764 = cbrt(r64763);
        double r64765 = r64764 * r64764;
        double r64766 = cbrt(r64736);
        double r64767 = r64766 * r64766;
        double r64768 = cbrt(r64767);
        double r64769 = cbrt(r64766);
        double r64770 = r64768 * r64769;
        double r64771 = r64766 * r64770;
        double r64772 = r64765 - r64771;
        double r64773 = r64764 + r64766;
        double r64774 = r64772 / r64773;
        double r64775 = 7.0;
        double r64776 = pow(r64736, r64775);
        double r64777 = r64739 / r64776;
        double r64778 = pow(r64777, r64743);
        double r64779 = 0.04938271604938271;
        double r64780 = 0.6666666666666666;
        double r64781 = r64739 / r64736;
        double r64782 = pow(r64781, r64743);
        double r64783 = r64780 * r64782;
        double r64784 = 4.0;
        double r64785 = pow(r64736, r64784);
        double r64786 = r64739 / r64785;
        double r64787 = pow(r64786, r64743);
        double r64788 = r64752 * r64787;
        double r64789 = r64783 - r64788;
        double r64790 = fma(r64778, r64779, r64789);
        double r64791 = r64769 * r64769;
        double r64792 = r64791 * r64769;
        double r64793 = r64764 + r64792;
        double r64794 = r64790 / r64793;
        double r64795 = r64761 ? r64774 : r64794;
        double r64796 = r64738 ? r64759 : r64795;
        return r64796;
}

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)))