Average Error: 30.5 → 8.6
Time: 5.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4084.8911496338142:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 0.15097238251788841:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4084.8911496338142:\\
\;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\

\mathbf{elif}\;x \le 0.15097238251788841:\\
\;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\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 r64629 = x;
        double r64630 = 1.0;
        double r64631 = r64629 + r64630;
        double r64632 = cbrt(r64631);
        double r64633 = cbrt(r64629);
        double r64634 = r64632 - r64633;
        return r64634;
}
↓
double f(double x) {
        double r64635 = x;
        double r64636 = -4084.891149633814;
        bool r64637 = r64635 <= r64636;
        double r64638 = 0.03292181069958847;
        double r64639 = 1.0;
        double r64640 = 4.0;
        double r64641 = pow(r64635, r64640);
        double r64642 = r64639 / r64641;
        double r64643 = r64638 * r64642;
        double r64644 = 0.037037037037037035;
        double r64645 = 3.0;
        double r64646 = pow(r64635, r64645);
        double r64647 = r64639 / r64646;
        double r64648 = r64644 * r64647;
        double r64649 = r64643 - r64648;
        double r64650 = r64644 / r64635;
        double r64651 = r64650 / r64635;
        double r64652 = r64649 + r64651;
        double r64653 = cbrt(r64652);
        double r64654 = 0.15097238251788841;
        bool r64655 = r64635 <= r64654;
        double r64656 = 1.0;
        double r64657 = r64635 + r64656;
        double r64658 = cbrt(r64657);
        double r64659 = cbrt(r64635);
        double r64660 = r64659 * r64659;
        double r64661 = cbrt(r64660);
        double r64662 = cbrt(r64659);
        double r64663 = r64661 * r64662;
        double r64664 = r64658 - r64663;
        double r64665 = 0.0;
        double r64666 = r64665 + r64656;
        double r64667 = r64658 + r64659;
        double r64668 = r64658 * r64667;
        double r64669 = 0.6666666666666666;
        double r64670 = pow(r64635, r64669);
        double r64671 = r64668 + r64670;
        double r64672 = r64666 / r64671;
        double r64673 = r64655 ? r64664 : r64672;
        double r64674 = r64637 ? r64653 : r64673;
        return r64674;
}

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 < -4084.891149633814

    1. Initial program 60.0

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

    3. Applied add-cbrt-cube60.0

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

    4. Simplified60.0

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

    5. Taylor expanded around inf 30.7

      \[\leadsto \sqrt[3]{\color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{x}^{2}} + 0.0329218106995884732 \cdot \frac{1}{{x}^{4}}\right) - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}}}\]

    6. Simplified29.8

      \[\leadsto \sqrt[3]{\color{blue}{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}}\]

    if -4084.891149633814 < x < 0.15097238251788841

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    if 0.15097238251788841 < x

    1. Initial program 59.7

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

    3. Applied flip3--59.5

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

    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. Simplified4.5

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4084.8911496338142:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 0.15097238251788841:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))