Average Error: 29.4 → 8.4
Time: 5.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5417.7010773935963:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 1.6325139996809951 \cdot 10^{-9}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5417.7010773935963:\\
\;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\

\mathbf{elif}\;x \le 1.6325139996809951 \cdot 10^{-9}:\\
\;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r51317 = x;
        double r51318 = 1.0;
        double r51319 = r51317 + r51318;
        double r51320 = cbrt(r51319);
        double r51321 = cbrt(r51317);
        double r51322 = r51320 - r51321;
        return r51322;
}
↓
double f(double x) {
        double r51323 = x;
        double r51324 = -5417.701077393596;
        bool r51325 = r51323 <= r51324;
        double r51326 = 0.03292181069958847;
        double r51327 = 1.0;
        double r51328 = 4.0;
        double r51329 = pow(r51323, r51328);
        double r51330 = r51327 / r51329;
        double r51331 = r51326 * r51330;
        double r51332 = 0.037037037037037035;
        double r51333 = 3.0;
        double r51334 = pow(r51323, r51333);
        double r51335 = r51327 / r51334;
        double r51336 = r51332 * r51335;
        double r51337 = r51331 - r51336;
        double r51338 = r51332 / r51323;
        double r51339 = r51338 / r51323;
        double r51340 = r51337 + r51339;
        double r51341 = cbrt(r51340);
        double r51342 = 1.6325139996809951e-09;
        bool r51343 = r51323 <= r51342;
        double r51344 = 1.0;
        double r51345 = r51323 + r51344;
        double r51346 = cbrt(r51345);
        double r51347 = cbrt(r51323);
        double r51348 = r51346 - r51347;
        double r51349 = exp(r51348);
        double r51350 = log(r51349);
        double r51351 = 0.0;
        double r51352 = r51351 + r51344;
        double r51353 = r51346 + r51347;
        double r51354 = r51346 * r51353;
        double r51355 = 0.6666666666666666;
        double r51356 = pow(r51323, r51355);
        double r51357 = r51354 + r51356;
        double r51358 = r51352 / r51357;
        double r51359 = r51343 ? r51350 : r51358;
        double r51360 = r51325 ? r51341 : r51359;
        return r51360;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -5417.701077393596

    1. Initial program 60.0

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

    3. Applied add-cbrt-cube60.0

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

    4. Simplified60.0

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

    5. Taylor expanded around inf 30.6

      \[\leadsto \sqrt[3]{\color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{x}^{2}} + 0.0329218106995884732 \cdot \frac{1}{{x}^{4}}\right) - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}}}\]

    6. Simplified29.6

      \[\leadsto \sqrt[3]{\color{blue}{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}}\]

    if -5417.701077393596 < x < 1.6325139996809951e-09

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

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

    if 1.6325139996809951e-09 < x

    1. Initial program 57.8

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

    3. Applied flip3--57.7

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

    4. Simplified1.0

      \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

    5. Simplified4.4

      \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5417.7010773935963:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 1.6325139996809951 \cdot 10^{-9}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))