Average Error: 29.9 → 8.7
Time: 8.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-4545286189804307}{4503599627370496}:\\ \;\;\;\;\sqrt[3]{\left(\frac{4744532940768917}{144115188075855872} \cdot \frac{1}{{x}^{4}} - \frac{667199944795629}{18014398509481984} \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{\frac{667199944795629}{18014398509481984}}{x}}{x}}\\ \mathbf{elif}\;x \le \frac{2935373004731401}{2305843009213693952}:\\ \;\;\;\;{\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-4545286189804307}{4503599627370496}:\\
\;\;\;\;\sqrt[3]{\left(\frac{4744532940768917}{144115188075855872} \cdot \frac{1}{{x}^{4}} - \frac{667199944795629}{18014398509481984} \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{\frac{667199944795629}{18014398509481984}}{x}}{x}}\\

\mathbf{elif}\;x \le \frac{2935373004731401}{2305843009213693952}:\\
\;\;\;\;{\left(x + 1\right)}^{\frac{1}{3}} - \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 r39171 = x;
        double r39172 = 1.0;
        double r39173 = r39171 + r39172;
        double r39174 = cbrt(r39173);
        double r39175 = cbrt(r39171);
        double r39176 = r39174 - r39175;
        return r39176;
}
↓
double f(double x) {
        double r39177 = x;
        double r39178 = -4545286189804307.0;
        double r39179 = 4503599627370496.0;
        double r39180 = r39178 / r39179;
        bool r39181 = r39177 <= r39180;
        double r39182 = 4744532940768917.0;
        double r39183 = 1.4411518807585587e+17;
        double r39184 = r39182 / r39183;
        double r39185 = 1.0;
        double r39186 = 4.0;
        double r39187 = pow(r39177, r39186);
        double r39188 = r39185 / r39187;
        double r39189 = r39184 * r39188;
        double r39190 = 667199944795629.0;
        double r39191 = 18014398509481984.0;
        double r39192 = r39190 / r39191;
        double r39193 = 3.0;
        double r39194 = pow(r39177, r39193);
        double r39195 = r39185 / r39194;
        double r39196 = r39192 * r39195;
        double r39197 = r39189 - r39196;
        double r39198 = r39192 / r39177;
        double r39199 = r39198 / r39177;
        double r39200 = r39197 + r39199;
        double r39201 = cbrt(r39200);
        double r39202 = 2935373004731401.0;
        double r39203 = 2.305843009213694e+18;
        double r39204 = r39202 / r39203;
        bool r39205 = r39177 <= r39204;
        double r39206 = 1.0;
        double r39207 = r39177 + r39206;
        double r39208 = 0.3333333333333333;
        double r39209 = pow(r39207, r39208);
        double r39210 = cbrt(r39177);
        double r39211 = r39209 - r39210;
        double r39212 = cbrt(r39207);
        double r39213 = r39212 + r39210;
        double r39214 = r39212 * r39213;
        double r39215 = 0.6666666666666666;
        double r39216 = pow(r39177, r39215);
        double r39217 = r39214 + r39216;
        double r39218 = r39206 / r39217;
        double r39219 = r39205 ? r39211 : r39218;
        double r39220 = r39181 ? r39201 : r39219;
        return r39220;
}

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 < -1.0092562762862982

    1. Initial program 59.3

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

    3. Applied add-cbrt-cube59.3

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

    4. Simplified59.3

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

    5. Taylor expanded around inf 30.9

      \[\leadsto \sqrt[3]{\color{blue}{\left(0.03703703703703703498106847291637677699327 \cdot \frac{1}{{x}^{2}} + 0.03292181069958847322576289684548100922257 \cdot \frac{1}{{x}^{4}}\right) - 0.03703703703703703498106847291637677699327 \cdot \frac{1}{{x}^{3}}}}\]

    6. Simplified29.9

      \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{4744532940768917}{144115188075855872} \cdot \frac{1}{{x}^{4}} - \frac{667199944795629}{18014398509481984} \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{\frac{667199944795629}{18014398509481984}}{x}}{x}}}\]

    if -1.0092562762862982 < x < 0.0012730151155140351

    1. Initial program 0.0

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

    3. Applied pow1/30.0

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

    if 0.0012730151155140351 < x

    1. Initial program 58.9

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

    3. Applied flip3--58.8

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

    4. Simplified1.0

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

    5. Simplified4.4

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-4545286189804307}{4503599627370496}:\\ \;\;\;\;\sqrt[3]{\left(\frac{4744532940768917}{144115188075855872} \cdot \frac{1}{{x}^{4}} - \frac{667199944795629}{18014398509481984} \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{\frac{667199944795629}{18014398509481984}}{x}}{x}}\\ \mathbf{elif}\;x \le \frac{2935373004731401}{2305843009213693952}:\\ \;\;\;\;{\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))