Average Error: 29.9 → 12.2
Time: 5.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4788638807140642 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.0127415927447725574:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4788638807140642 \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 0.0127415927447725574:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\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 r74740 = x;
        double r74741 = 1.0;
        double r74742 = r74740 + r74741;
        double r74743 = cbrt(r74742);
        double r74744 = cbrt(r74740);
        double r74745 = r74743 - r74744;
        return r74745;
}
↓
double f(double x) {
        double r74746 = x;
        double r74747 = -4.478863880714064e+61;
        bool r74748 = r74746 <= r74747;
        double r74749 = 0.3333333333333333;
        double r74750 = 1.0;
        double r74751 = 2.0;
        double r74752 = pow(r74746, r74751);
        double r74753 = r74750 / r74752;
        double r74754 = 0.3333333333333333;
        double r74755 = pow(r74753, r74754);
        double r74756 = r74749 * r74755;
        double r74757 = 0.06172839506172839;
        double r74758 = 8.0;
        double r74759 = pow(r74746, r74758);
        double r74760 = r74750 / r74759;
        double r74761 = pow(r74760, r74754);
        double r74762 = r74757 * r74761;
        double r74763 = r74756 + r74762;
        double r74764 = 0.1111111111111111;
        double r74765 = 5.0;
        double r74766 = pow(r74746, r74765);
        double r74767 = r74750 / r74766;
        double r74768 = pow(r74767, r74754);
        double r74769 = r74764 * r74768;
        double r74770 = r74763 - r74769;
        double r74771 = 0.012741592744772557;
        bool r74772 = r74746 <= r74771;
        double r74773 = r74746 * r74746;
        double r74774 = 1.0;
        double r74775 = r74774 * r74774;
        double r74776 = r74773 - r74775;
        double r74777 = cbrt(r74776);
        double r74778 = r74746 - r74774;
        double r74779 = cbrt(r74778);
        double r74780 = r74777 / r74779;
        double r74781 = cbrt(r74746);
        double r74782 = r74780 - r74781;
        double r74783 = 0.0;
        double r74784 = r74783 + r74774;
        double r74785 = r74746 + r74774;
        double r74786 = cbrt(r74785);
        double r74787 = r74786 + r74781;
        double r74788 = r74786 * r74787;
        double r74789 = 0.6666666666666666;
        double r74790 = pow(r74746, r74789);
        double r74791 = r74788 + r74790;
        double r74792 = r74784 / r74791;
        double r74793 = r74772 ? r74782 : r74792;
        double r74794 = r74748 ? r74770 : r74793;
        return r74794;
}

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 < -4.478863880714064e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 41.2

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

    if -4.478863880714064e+61 < x < 0.012741592744772557

    1. Initial program 4.9

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

    3. Applied flip-+4.9

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

    4. Applied cbrt-div4.8

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

    if 0.012741592744772557 < x

    1. Initial program 58.9

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

    3. Applied flip3--58.8

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

    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 simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4788638807140642 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.0127415927447725574:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))