Average Error: 29.8 → 8.9
Time: 4.9s
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 r77719 = x;
        double r77720 = 1.0;
        double r77721 = r77719 + r77720;
        double r77722 = cbrt(r77721);
        double r77723 = cbrt(r77719);
        double r77724 = r77722 - r77723;
        return r77724;
}
↓
double f(double x) {
        double r77725 = x;
        double r77726 = -3618.8027686995374;
        bool r77727 = r77725 <= r77726;
        double r77728 = 0.03292181069958847;
        double r77729 = 1.0;
        double r77730 = 4.0;
        double r77731 = pow(r77725, r77730);
        double r77732 = r77729 / r77731;
        double r77733 = r77728 * r77732;
        double r77734 = 0.037037037037037035;
        double r77735 = 3.0;
        double r77736 = pow(r77725, r77735);
        double r77737 = r77729 / r77736;
        double r77738 = r77734 * r77737;
        double r77739 = r77733 - r77738;
        double r77740 = r77734 / r77725;
        double r77741 = r77740 / r77725;
        double r77742 = r77739 + r77741;
        double r77743 = cbrt(r77742);
        double r77744 = 2.8739991244123034e-09;
        bool r77745 = r77725 <= r77744;
        double r77746 = 1.0;
        double r77747 = r77725 + r77746;
        double r77748 = cbrt(r77747);
        double r77749 = r77748 * r77748;
        double r77750 = cbrt(r77749);
        double r77751 = cbrt(r77748);
        double r77752 = r77750 * r77751;
        double r77753 = cbrt(r77725);
        double r77754 = r77752 - r77753;
        double r77755 = 0.0;
        double r77756 = r77755 + r77746;
        double r77757 = r77748 + r77753;
        double r77758 = r77748 * r77757;
        double r77759 = 0.6666666666666666;
        double r77760 = pow(r77725, r77759);
        double r77761 = r77758 + r77760;
        double r77762 = r77756 / r77761;
        double r77763 = r77745 ? r77754 : r77762;
        double r77764 = r77727 ? r77743 : r77763;
        return r77764;
}

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)))