Average Error: 30.2 → 0.4
Time: 17.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -107349.3406070635537616908550262451171875 \lor \neg \left(x \le 49305.12015159700240474194288253784179688\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -107349.3406070635537616908550262451171875 \lor \neg \left(x \le 49305.12015159700240474194288253784179688\right):\\
\;\;\;\;\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}:\\
\;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\\

\end{array}
double f(double x) {
        double r49291 = x;
        double r49292 = 1.0;
        double r49293 = r49291 + r49292;
        double r49294 = cbrt(r49293);
        double r49295 = cbrt(r49291);
        double r49296 = r49294 - r49295;
        return r49296;
}
↓
double f(double x) {
        double r49297 = x;
        double r49298 = -107349.34060706355;
        bool r49299 = r49297 <= r49298;
        double r49300 = 49305.120151597;
        bool r49301 = r49297 <= r49300;
        double r49302 = !r49301;
        bool r49303 = r49299 || r49302;
        double r49304 = cbrt(r49297);
        double r49305 = r49304 / r49297;
        double r49306 = 0.3333333333333333;
        double r49307 = 0.1111111111111111;
        double r49308 = r49307 / r49297;
        double r49309 = r49306 - r49308;
        double r49310 = r49305 * r49309;
        double r49311 = -1.0;
        double r49312 = cbrt(r49311);
        double r49313 = -r49297;
        double r49314 = cbrt(r49313);
        double r49315 = r49312 * r49314;
        double r49316 = r49304 - r49315;
        double r49317 = r49310 + r49316;
        double r49318 = 1.0;
        double r49319 = r49297 + r49318;
        double r49320 = cbrt(r49319);
        double r49321 = cbrt(r49304);
        double r49322 = r49321 * r49321;
        double r49323 = r49322 * r49321;
        double r49324 = r49320 - r49323;
        double r49325 = exp(r49324);
        double r49326 = log(r49325);
        double r49327 = r49303 ? r49317 : r49326;
        return r49327;
}

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 x < -107349.34060706355 or 49305.120151597 < x

    1. Initial program 60.5

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

    3. Applied add-cube-cbrt60.6

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

    4. Applied cbrt-prod60.7

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

    5. Taylor expanded around -inf 64.0

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

    6. Simplified0.6

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

    if -107349.34060706355 < x < 49305.120151597

    1. Initial program 0.2

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

    3. Applied add-log-exp0.2

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

    4. Applied add-log-exp0.2

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

    5. Applied diff-log0.2

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

    6. Simplified0.2

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

    8. Applied add-cube-cbrt0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -107349.3406070635537616908550262451171875 \lor \neg \left(x \le 49305.12015159700240474194288253784179688\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))