Average Error: 29.7 → 0.4
Time: 15.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 1.0947415319151332 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 1.0947415319151332 \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]{x}\\

\end{array}
double f(double x) {
        double r2302373 = x;
        double r2302374 = 1.0;
        double r2302375 = r2302373 + r2302374;
        double r2302376 = cbrt(r2302375);
        double r2302377 = cbrt(r2302373);
        double r2302378 = r2302376 - r2302377;
        return r2302378;
}
↓
double f(double x) {
        double r2302379 = 1.0;
        double r2302380 = x;
        double r2302381 = r2302379 + r2302380;
        double r2302382 = cbrt(r2302381);
        double r2302383 = cbrt(r2302380);
        double r2302384 = r2302382 - r2302383;
        double r2302385 = 1.0947415319151332e-05;
        bool r2302386 = r2302384 <= r2302385;
        double r2302387 = -1.0;
        double r2302388 = cbrt(r2302387);
        double r2302389 = -r2302380;
        double r2302390 = cbrt(r2302389);
        double r2302391 = r2302388 * r2302390;
        double r2302392 = r2302383 - r2302391;
        double r2302393 = -0.1111111111111111;
        double r2302394 = r2302393 / r2302380;
        double r2302395 = 0.3333333333333333;
        double r2302396 = r2302394 + r2302395;
        double r2302397 = r2302383 / r2302380;
        double r2302398 = r2302396 * r2302397;
        double r2302399 = r2302392 + r2302398;
        double r2302400 = r2302386 ? r2302399 : r2302384;
        return r2302400;
}

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)) < 1.0947415319151332e-05

    1. Initial program 60.5

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

    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 1.0947415319151332e-05 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.3

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

    2. Taylor expanded around 0 30.9

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

    3. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 1.0947415319151332 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))