Average Error: 30.0 → 0.4
Time: 15.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 3.87694688299689 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 3.87694688299689 \cdot 10^{-05}:\\
\;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\

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

\end{array}
double f(double x) {
        double r2081133 = x;
        double r2081134 = 1.0;
        double r2081135 = r2081133 + r2081134;
        double r2081136 = cbrt(r2081135);
        double r2081137 = cbrt(r2081133);
        double r2081138 = r2081136 - r2081137;
        return r2081138;
}
↓
double f(double x) {
        double r2081139 = 1.0;
        double r2081140 = x;
        double r2081141 = r2081139 + r2081140;
        double r2081142 = cbrt(r2081141);
        double r2081143 = cbrt(r2081140);
        double r2081144 = r2081142 - r2081143;
        double r2081145 = 3.87694688299689e-05;
        bool r2081146 = r2081144 <= r2081145;
        double r2081147 = -1.0;
        double r2081148 = cbrt(r2081147);
        double r2081149 = -r2081140;
        double r2081150 = cbrt(r2081149);
        double r2081151 = r2081148 * r2081150;
        double r2081152 = r2081143 - r2081151;
        double r2081153 = -0.1111111111111111;
        double r2081154 = r2081153 / r2081140;
        double r2081155 = 0.3333333333333333;
        double r2081156 = r2081154 + r2081155;
        double r2081157 = r2081143 / r2081140;
        double r2081158 = r2081156 * r2081157;
        double r2081159 = r2081152 + r2081158;
        double r2081160 = cbrt(r2081143);
        double r2081161 = r2081160 * r2081160;
        double r2081162 = r2081160 * r2081161;
        double r2081163 = r2081142 - r2081162;
        double r2081164 = r2081146 ? r2081159 : r2081163;
        return r2081164;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (cbrt (+ x 1)) (cbrt x)) < 3.87694688299689e-05

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \left({\left(x \cdot -1\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1} + \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)}\]

    if 3.87694688299689e-05 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 3.87694688299689 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))