Average Error: 29.8 → 0.4
Time: 20.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 2.5441169569262456 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 2.5441169569262456 \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}:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\

\end{array}
double f(double x) {
        double r4415426 = x;
        double r4415427 = 1.0;
        double r4415428 = r4415426 + r4415427;
        double r4415429 = cbrt(r4415428);
        double r4415430 = cbrt(r4415426);
        double r4415431 = r4415429 - r4415430;
        return r4415431;
}
↓
double f(double x) {
        double r4415432 = 1.0;
        double r4415433 = x;
        double r4415434 = r4415432 + r4415433;
        double r4415435 = cbrt(r4415434);
        double r4415436 = cbrt(r4415433);
        double r4415437 = r4415435 - r4415436;
        double r4415438 = 2.5441169569262456e-05;
        bool r4415439 = r4415437 <= r4415438;
        double r4415440 = -1.0;
        double r4415441 = cbrt(r4415440);
        double r4415442 = -r4415433;
        double r4415443 = cbrt(r4415442);
        double r4415444 = r4415441 * r4415443;
        double r4415445 = r4415436 - r4415444;
        double r4415446 = -0.1111111111111111;
        double r4415447 = r4415446 / r4415433;
        double r4415448 = 0.3333333333333333;
        double r4415449 = r4415447 + r4415448;
        double r4415450 = r4415436 / r4415433;
        double r4415451 = r4415449 * r4415450;
        double r4415452 = r4415445 + r4415451;
        double r4415453 = r4415433 * r4415433;
        double r4415454 = r4415453 + r4415440;
        double r4415455 = cbrt(r4415454);
        double r4415456 = r4415433 - r4415432;
        double r4415457 = cbrt(r4415456);
        double r4415458 = r4415455 / r4415457;
        double r4415459 = r4415458 - r4415436;
        double r4415460 = r4415439 ? r4415452 : r4415459;
        return r4415460;
}

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)) < 2.5441169569262456e-05

    1. Initial program 60.6

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

    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 2.5441169569262456e-05 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.3

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

    3. Applied flip-+0.3

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

    4. Applied cbrt-div0.3

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

    5. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 2.5441169569262456 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))