Average Error: 29.8 → 8.9
Time: 5.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3618.8027686995374:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 2.87399912441230341 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3618.8027686995374:\\
\;\;\;\;\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 2.87399912441230341 \cdot 10^{-9}:\\
\;\;\;\;\sqrt[3]{\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 r52624 = x;
        double r52625 = 1.0;
        double r52626 = r52624 + r52625;
        double r52627 = cbrt(r52626);
        double r52628 = cbrt(r52624);
        double r52629 = r52627 - r52628;
        return r52629;
}
↓
double f(double x) {
        double r52630 = x;
        double r52631 = -3618.8027686995374;
        bool r52632 = r52630 <= r52631;
        double r52633 = 0.03292181069958847;
        double r52634 = 1.0;
        double r52635 = 4.0;
        double r52636 = pow(r52630, r52635);
        double r52637 = r52634 / r52636;
        double r52638 = r52633 * r52637;
        double r52639 = 0.037037037037037035;
        double r52640 = 3.0;
        double r52641 = pow(r52630, r52640);
        double r52642 = r52634 / r52641;
        double r52643 = r52639 * r52642;
        double r52644 = r52638 - r52643;
        double r52645 = r52639 / r52630;
        double r52646 = r52645 / r52630;
        double r52647 = r52644 + r52646;
        double r52648 = cbrt(r52647);
        double r52649 = 2.8739991244123034e-09;
        bool r52650 = r52630 <= r52649;
        double r52651 = 1.0;
        double r52652 = r52630 + r52651;
        double r52653 = cbrt(r52652);
        double r52654 = r52653 * r52653;
        double r52655 = cbrt(r52654);
        double r52656 = cbrt(r52653);
        double r52657 = r52655 * r52656;
        double r52658 = cbrt(r52630);
        double r52659 = r52657 - r52658;
        double r52660 = 0.0;
        double r52661 = r52660 + r52651;
        double r52662 = r52653 + r52658;
        double r52663 = r52653 * r52662;
        double r52664 = 0.6666666666666666;
        double r52665 = pow(r52630, r52664);
        double r52666 = r52663 + r52665;
        double r52667 = r52661 / r52666;
        double r52668 = r52650 ? r52659 : r52667;
        double r52669 = r52632 ? r52648 : r52668;
        return r52669;
}

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 < -3618.8027686995374

    1. Initial program 60.3

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

    3. Applied add-cbrt-cube60.3

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

    4. Simplified60.3

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

    5. Taylor expanded around inf 31.8

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

    6. Simplified31.0

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

    if -3618.8027686995374 < x < 2.8739991244123034e-09

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    if 2.8739991244123034e-09 < x

    1. Initial program 57.6

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

    3. Applied flip3--57.5

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

    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. Simplified4.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3618.8027686995374:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 2.87399912441230341 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))