Average Error: 29.7 → 12.0
Time: 5.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4628901123669468 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.9364214566131179 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1} \cdot \sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4628901123669468 \cdot 10^{61}:\\
\;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\

\mathbf{elif}\;x \le 1.9364214566131179 \cdot 10^{-4}:\\
\;\;\;\;\sqrt[3]{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1} \cdot \sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 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 r68686 = x;
        double r68687 = 1.0;
        double r68688 = r68686 + r68687;
        double r68689 = cbrt(r68688);
        double r68690 = cbrt(r68686);
        double r68691 = r68689 - r68690;
        return r68691;
}
↓
double f(double x) {
        double r68692 = x;
        double r68693 = -4.462890112366947e+61;
        bool r68694 = r68692 <= r68693;
        double r68695 = 0.3333333333333333;
        double r68696 = 1.0;
        double r68697 = 2.0;
        double r68698 = pow(r68692, r68697);
        double r68699 = r68696 / r68698;
        double r68700 = 0.3333333333333333;
        double r68701 = pow(r68699, r68700);
        double r68702 = r68695 * r68701;
        double r68703 = 0.06172839506172839;
        double r68704 = 8.0;
        double r68705 = pow(r68692, r68704);
        double r68706 = r68696 / r68705;
        double r68707 = pow(r68706, r68700);
        double r68708 = r68703 * r68707;
        double r68709 = r68702 + r68708;
        double r68710 = 0.1111111111111111;
        double r68711 = 5.0;
        double r68712 = pow(r68692, r68711);
        double r68713 = r68696 / r68712;
        double r68714 = pow(r68713, r68700);
        double r68715 = r68710 * r68714;
        double r68716 = r68709 - r68715;
        double r68717 = 0.0001936421456613118;
        bool r68718 = r68692 <= r68717;
        double r68719 = r68692 * r68692;
        double r68720 = 1.0;
        double r68721 = r68720 * r68720;
        double r68722 = r68719 - r68721;
        double r68723 = cbrt(r68722);
        double r68724 = r68723 * r68723;
        double r68725 = r68692 - r68720;
        double r68726 = cbrt(r68725);
        double r68727 = r68726 * r68726;
        double r68728 = r68724 / r68727;
        double r68729 = cbrt(r68728);
        double r68730 = r68692 + r68720;
        double r68731 = cbrt(r68730);
        double r68732 = cbrt(r68731);
        double r68733 = r68729 * r68732;
        double r68734 = cbrt(r68692);
        double r68735 = r68733 - r68734;
        double r68736 = 0.0;
        double r68737 = r68736 + r68720;
        double r68738 = r68731 + r68734;
        double r68739 = r68731 * r68738;
        double r68740 = 0.6666666666666666;
        double r68741 = pow(r68692, r68740);
        double r68742 = r68739 + r68741;
        double r68743 = r68737 / r68742;
        double r68744 = r68718 ? r68735 : r68743;
        double r68745 = r68694 ? r68716 : r68744;
        return r68745;
}

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 < -4.462890112366947e+61

    1. Initial program 61.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

    2. Taylor expanded around inf 40.6

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    if -4.462890112366947e+61 < x < 0.0001936421456613118

    1. Initial program 4.5

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

    3. Applied add-cube-cbrt4.5

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

    4. Applied cbrt-prod4.5

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

    6. Applied flip-+4.5

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

    7. Applied cbrt-div4.5

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

    8. Applied flip-+4.5

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

    9. Applied cbrt-div4.5

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

    10. Applied frac-times4.5

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

    if 0.0001936421456613118 < x

    1. Initial program 58.7

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

    3. Applied flip3--58.5

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

    4. Simplified1.0

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

    5. Simplified4.4

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4628901123669468 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.9364214566131179 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1} \cdot \sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))