Average Error: 29.6 → 11.7
Time: 5.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4771775817553536 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 6.76969302521690079 \cdot 10^{-12}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4771775817553536 \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 6.76969302521690079 \cdot 10^{-12}:\\
\;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\frac{1}{3}} \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 r58582 = x;
        double r58583 = 1.0;
        double r58584 = r58582 + r58583;
        double r58585 = cbrt(r58584);
        double r58586 = cbrt(r58582);
        double r58587 = r58585 - r58586;
        return r58587;
}
↓
double f(double x) {
        double r58588 = x;
        double r58589 = -4.4771775817553536e+61;
        bool r58590 = r58588 <= r58589;
        double r58591 = 0.3333333333333333;
        double r58592 = 1.0;
        double r58593 = 2.0;
        double r58594 = pow(r58588, r58593);
        double r58595 = r58592 / r58594;
        double r58596 = 0.3333333333333333;
        double r58597 = pow(r58595, r58596);
        double r58598 = r58591 * r58597;
        double r58599 = 0.06172839506172839;
        double r58600 = 8.0;
        double r58601 = pow(r58588, r58600);
        double r58602 = r58592 / r58601;
        double r58603 = pow(r58602, r58596);
        double r58604 = r58599 * r58603;
        double r58605 = r58598 + r58604;
        double r58606 = 0.1111111111111111;
        double r58607 = 5.0;
        double r58608 = pow(r58588, r58607);
        double r58609 = r58592 / r58608;
        double r58610 = pow(r58609, r58596);
        double r58611 = r58606 * r58610;
        double r58612 = r58605 - r58611;
        double r58613 = 6.769693025216901e-12;
        bool r58614 = r58588 <= r58613;
        double r58615 = 1.0;
        double r58616 = r58588 + r58615;
        double r58617 = cbrt(r58616);
        double r58618 = r58617 * r58617;
        double r58619 = pow(r58618, r58596);
        double r58620 = cbrt(r58617);
        double r58621 = r58619 * r58620;
        double r58622 = cbrt(r58588);
        double r58623 = r58621 - r58622;
        double r58624 = 0.0;
        double r58625 = r58624 + r58615;
        double r58626 = r58617 + r58622;
        double r58627 = r58617 * r58626;
        double r58628 = 0.6666666666666666;
        double r58629 = pow(r58588, r58628);
        double r58630 = r58627 + r58629;
        double r58631 = r58625 / r58630;
        double r58632 = r58614 ? r58623 : r58631;
        double r58633 = r58590 ? r58612 : r58632;
        return r58633;
}

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 < -4.4771775817553536e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 39.8

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

    if -4.4771775817553536e+61 < x < 6.769693025216901e-12

    1. Initial program 5.3

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

    3. Applied add-cube-cbrt5.3

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

    4. Applied cbrt-prod5.3

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

    6. Applied pow1/35.0

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

    if 6.769693025216901e-12 < x

    1. Initial program 56.8

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

    3. Applied flip3--56.6

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

    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. Simplified4.3

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4771775817553536 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 6.76969302521690079 \cdot 10^{-12}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))