Average Error: 29.2 → 8.9
Time: 12.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4221.179793202217297221068292856216430664:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212 - \left(\sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798 - 0.3333333333333333148296162562473909929395 \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right)\\ \mathbf{elif}\;x \le 2.627646657846368341371692551461380915612 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4221.179793202217297221068292856216430664:\\
\;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212 - \left(\sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798 - 0.3333333333333333148296162562473909929395 \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right)\\

\mathbf{elif}\;x \le 2.627646657846368341371692551461380915612 \cdot 10^{-7}:\\
\;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\

\end{array}
double f(double x) {
        double r36745 = x;
        double r36746 = 1.0;
        double r36747 = r36745 + r36746;
        double r36748 = cbrt(r36747);
        double r36749 = cbrt(r36745);
        double r36750 = r36748 - r36749;
        return r36750;
}
↓
double f(double x) {
        double r36751 = x;
        double r36752 = -4221.179793202217;
        bool r36753 = r36751 <= r36752;
        double r36754 = 1.0;
        double r36755 = 8.0;
        double r36756 = pow(r36751, r36755);
        double r36757 = r36754 / r36756;
        double r36758 = cbrt(r36757);
        double r36759 = 0.06172839506172839;
        double r36760 = r36758 * r36759;
        double r36761 = 5.0;
        double r36762 = pow(r36751, r36761);
        double r36763 = r36754 / r36762;
        double r36764 = cbrt(r36763);
        double r36765 = 0.1111111111111111;
        double r36766 = r36764 * r36765;
        double r36767 = 0.3333333333333333;
        double r36768 = r36751 * r36751;
        double r36769 = r36754 / r36768;
        double r36770 = cbrt(r36769);
        double r36771 = r36767 * r36770;
        double r36772 = r36766 - r36771;
        double r36773 = r36760 - r36772;
        double r36774 = 2.6276466578463683e-07;
        bool r36775 = r36751 <= r36774;
        double r36776 = 1.0;
        double r36777 = r36776 + r36751;
        double r36778 = cbrt(r36777);
        double r36779 = cbrt(r36751);
        double r36780 = r36778 - r36779;
        double r36781 = 0.6666666666666666;
        double r36782 = pow(r36751, r36781);
        double r36783 = r36778 + r36779;
        double r36784 = r36778 * r36783;
        double r36785 = r36782 + r36784;
        double r36786 = r36776 / r36785;
        double r36787 = r36775 ? r36780 : r36786;
        double r36788 = r36753 ? r36773 : r36787;
        return r36788;
}

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 < -4221.179793202217

    1. Initial program 60.1

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

    3. Applied add-cube-cbrt60.3

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

    4. Applied cbrt-prod60.6

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

    5. Simplified60.6

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

    6. Simplified60.6

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

    8. Applied add-cube-cbrt60.7

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

    9. Taylor expanded around inf 45.5

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

    10. Simplified31.7

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

    if -4221.179793202217 < x < 2.6276466578463683e-07

    1. Initial program 0.1

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

    if 2.6276466578463683e-07 < x

    1. Initial program 58.4

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

    3. Applied flip3--58.3

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

    4. Simplified1.0

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

    5. Simplified4.4

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4221.179793202217297221068292856216430664:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212 - \left(\sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798 - 0.3333333333333333148296162562473909929395 \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right)\\ \mathbf{elif}\;x \le 2.627646657846368341371692551461380915612 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))