Average Error: 29.5 → 0.6
Time: 19.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.153901576955718155659269541501998901367 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.153901576955718155659269541501998901367 \cdot 10^{-4}:\\
\;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left({\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\right)}\\

\end{array}
double f(double x) {
        double r40279 = x;
        double r40280 = 1.0;
        double r40281 = r40279 + r40280;
        double r40282 = cbrt(r40281);
        double r40283 = cbrt(r40279);
        double r40284 = r40282 - r40283;
        return r40284;
}
↓
double f(double x) {
        double r40285 = x;
        double r40286 = 1.0;
        double r40287 = r40285 + r40286;
        double r40288 = cbrt(r40287);
        double r40289 = cbrt(r40285);
        double r40290 = r40288 - r40289;
        double r40291 = 0.00011539015769557182;
        bool r40292 = r40290 <= r40291;
        double r40293 = r40289 / r40285;
        double r40294 = 0.3333333333333333;
        double r40295 = 0.1111111111111111;
        double r40296 = r40295 / r40285;
        double r40297 = r40294 - r40296;
        double r40298 = r40293 * r40297;
        double r40299 = -1.0;
        double r40300 = cbrt(r40299);
        double r40301 = -r40285;
        double r40302 = cbrt(r40301);
        double r40303 = r40300 * r40302;
        double r40304 = r40289 - r40303;
        double r40305 = r40298 + r40304;
        double r40306 = 0.3333333333333333;
        double r40307 = pow(r40287, r40306);
        double r40308 = r40307 - r40289;
        double r40309 = log(r40308);
        double r40310 = exp(r40309);
        double r40311 = r40292 ? r40305 : r40310;
        return r40311;
}

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 1.0)) (cbrt x)) < 0.00011539015769557182

    1. Initial program 60.4

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

    3. Applied add-cube-cbrt60.7

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

    4. Taylor expanded around -inf 64.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x} + e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}\right) - \left(0.1111111111111111049432054187491303309798 \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}} + {\left(-1 \cdot x\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1}\right)}\]

    5. Simplified0.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)}\]

    if 0.00011539015769557182 < (- (cbrt (+ x 1.0)) (cbrt x))

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}}\]
    4. Using strategy rm

    5. Applied pow1/30.6

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.153901576955718155659269541501998901367 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))