Average Error: 29.9 → 8.9
Time: 10.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3435.29454990893873:\\ \;\;\;\;\left(0.061728395061728392 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + 0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \le 0.10532502913683747:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3435.29454990893873:\\
\;\;\;\;\left(0.061728395061728392 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + 0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\

\mathbf{elif}\;x \le 0.10532502913683747:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{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 r57266 = x;
        double r57267 = 1.0;
        double r57268 = r57266 + r57267;
        double r57269 = cbrt(r57268);
        double r57270 = cbrt(r57266);
        double r57271 = r57269 - r57270;
        return r57271;
}
↓
double f(double x) {
        double r57272 = x;
        double r57273 = -3435.2945499089387;
        bool r57274 = r57272 <= r57273;
        double r57275 = 0.06172839506172839;
        double r57276 = 1.0;
        double r57277 = 8.0;
        double r57278 = pow(r57272, r57277);
        double r57279 = r57276 / r57278;
        double r57280 = cbrt(r57279);
        double r57281 = r57275 * r57280;
        double r57282 = 0.3333333333333333;
        double r57283 = 2.0;
        double r57284 = pow(r57272, r57283);
        double r57285 = r57276 / r57284;
        double r57286 = cbrt(r57285);
        double r57287 = r57282 * r57286;
        double r57288 = r57281 + r57287;
        double r57289 = 0.1111111111111111;
        double r57290 = 5.0;
        double r57291 = pow(r57272, r57290);
        double r57292 = r57276 / r57291;
        double r57293 = cbrt(r57292);
        double r57294 = r57289 * r57293;
        double r57295 = r57288 - r57294;
        double r57296 = 0.10532502913683747;
        bool r57297 = r57272 <= r57296;
        double r57298 = r57272 * r57272;
        double r57299 = 1.0;
        double r57300 = r57299 * r57299;
        double r57301 = r57298 - r57300;
        double r57302 = cbrt(r57301);
        double r57303 = r57272 - r57299;
        double r57304 = cbrt(r57303);
        double r57305 = r57302 / r57304;
        double r57306 = cbrt(r57272);
        double r57307 = r57305 - r57306;
        double r57308 = r57272 + r57299;
        double r57309 = cbrt(r57308);
        double r57310 = r57309 + r57306;
        double r57311 = r57309 * r57310;
        double r57312 = 0.6666666666666666;
        double r57313 = pow(r57272, r57312);
        double r57314 = r57311 + r57313;
        double r57315 = r57299 / r57314;
        double r57316 = r57297 ? r57307 : r57315;
        double r57317 = r57274 ? r57295 : r57316;
        return r57317;
}

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 < -3435.2945499089387

    1. Initial program 60.2

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

    3. Applied add-cube-cbrt60.4

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

    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. Using strategy rm

    6. Applied add-cube-cbrt60.7

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

    7. Applied cbrt-prod60.6

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

    8. Applied associate-*l*60.7

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

    9. Simplified60.8

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

    10. Taylor expanded around inf 44.9

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    11. Simplified30.7

      \[\leadsto \color{blue}{\left(0.061728395061728392 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + 0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}}\]

    if -3435.2945499089387 < x < 0.10532502913683747

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied cbrt-div0.1

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

    if 0.10532502913683747 < x

    1. Initial program 59.5

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

    3. Applied flip3--59.5

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

    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 simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3435.29454990893873:\\ \;\;\;\;\left(0.061728395061728392 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + 0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \le 0.10532502913683747:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))