Average Error: 29.6 → 9.0
Time: 5.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -137893.79262274364:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 4.63592369012709109 \cdot 10^{-48}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x + 1\right) - x}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -137893.79262274364:\\
\;\;\;\;\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 4.63592369012709109 \cdot 10^{-48}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\left(x + 1\right) - x}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)}^{3}}\\

\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 r57283 = x;
        double r57284 = 1.0;
        double r57285 = r57283 + r57284;
        double r57286 = cbrt(r57285);
        double r57287 = cbrt(r57283);
        double r57288 = r57286 - r57287;
        return r57288;
}
↓
double f(double x) {
        double r57289 = x;
        double r57290 = -137893.79262274364;
        bool r57291 = r57289 <= r57290;
        double r57292 = 0.03292181069958847;
        double r57293 = 1.0;
        double r57294 = 4.0;
        double r57295 = pow(r57289, r57294);
        double r57296 = r57293 / r57295;
        double r57297 = r57292 * r57296;
        double r57298 = 0.037037037037037035;
        double r57299 = 3.0;
        double r57300 = pow(r57289, r57299);
        double r57301 = r57293 / r57300;
        double r57302 = r57298 * r57301;
        double r57303 = r57297 - r57302;
        double r57304 = r57298 / r57289;
        double r57305 = r57304 / r57289;
        double r57306 = r57303 + r57305;
        double r57307 = cbrt(r57306);
        double r57308 = 4.635923690127091e-48;
        bool r57309 = r57289 <= r57308;
        double r57310 = 1.0;
        double r57311 = r57289 + r57310;
        double r57312 = r57311 - r57289;
        double r57313 = cbrt(r57289);
        double r57314 = r57313 * r57313;
        double r57315 = cbrt(r57314);
        double r57316 = cbrt(r57313);
        double r57317 = r57315 * r57316;
        double r57318 = cbrt(r57311);
        double r57319 = r57317 + r57318;
        double r57320 = r57317 * r57319;
        double r57321 = r57318 * r57318;
        double r57322 = r57320 + r57321;
        double r57323 = r57312 / r57322;
        double r57324 = pow(r57323, r57299);
        double r57325 = cbrt(r57324);
        double r57326 = 0.0;
        double r57327 = r57326 + r57310;
        double r57328 = r57318 + r57313;
        double r57329 = r57318 * r57328;
        double r57330 = 0.6666666666666666;
        double r57331 = pow(r57289, r57330);
        double r57332 = r57329 + r57331;
        double r57333 = r57327 / r57332;
        double r57334 = r57309 ? r57325 : r57333;
        double r57335 = r57291 ? r57307 : r57334;
        return r57335;
}

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 < -137893.79262274364

    1. Initial program 60.3

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

    3. Applied add-cbrt-cube60.3

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

    4. Simplified60.3

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

    5. Taylor expanded around inf 31.7

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

    6. Simplified31.1

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

    if -137893.79262274364 < x < 4.635923690127091e-48

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied cbrt-prod0.1

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

    9. Applied flip3--0.1

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

    10. Simplified0.1

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

    11. Simplified0.1

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

    if 4.635923690127091e-48 < x

    1. Initial program 51.3

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

    3. Applied flip3--51.3

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

    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. Simplified3.9

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -137893.79262274364:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 4.63592369012709109 \cdot 10^{-48}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x + 1\right) - x}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}} + \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))