Average Error: 29.9 → 0.4
Time: 18.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -78626.83968822401948273181915283203125:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right) - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)\\ \mathbf{elif}\;x \le 58824.09509360689844470471143722534179688:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -78626.83968822401948273181915283203125:\\
\;\;\;\;\left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right) - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)\\

\mathbf{elif}\;x \le 58824.09509360689844470471143722534179688:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)\\

\end{array}
double f(double x) {
        double r42139 = x;
        double r42140 = 1.0;
        double r42141 = r42139 + r42140;
        double r42142 = cbrt(r42141);
        double r42143 = cbrt(r42139);
        double r42144 = r42142 - r42143;
        return r42144;
}
↓
double f(double x) {
        double r42145 = x;
        double r42146 = -78626.83968822402;
        bool r42147 = r42145 <= r42146;
        double r42148 = 0.3333333333333333;
        double r42149 = 0.1111111111111111;
        double r42150 = r42149 / r42145;
        double r42151 = r42148 - r42150;
        double r42152 = cbrt(r42145);
        double r42153 = 1.0;
        double r42154 = r42153 / r42145;
        double r42155 = r42152 * r42154;
        double r42156 = r42151 * r42155;
        double r42157 = -1.0;
        double r42158 = cbrt(r42157);
        double r42159 = -r42145;
        double r42160 = cbrt(r42159);
        double r42161 = r42158 * r42160;
        double r42162 = r42161 - r42152;
        double r42163 = r42156 - r42162;
        double r42164 = 58824.0950936069;
        bool r42165 = r42145 <= r42164;
        double r42166 = 1.0;
        double r42167 = r42145 + r42166;
        double r42168 = cbrt(r42167);
        double r42169 = r42168 * r42168;
        double r42170 = cbrt(r42169);
        double r42171 = cbrt(r42168);
        double r42172 = r42170 * r42171;
        double r42173 = r42172 - r42152;
        double r42174 = r42152 / r42145;
        double r42175 = r42151 * r42174;
        double r42176 = r42175 - r42162;
        double r42177 = r42165 ? r42173 : r42176;
        double r42178 = r42147 ? r42163 : r42177;
        return r42178;
}

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 < -78626.83968822402

    1. Initial program 60.4

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

    2. Simplified60.4

      \[\leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt60.5

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

    5. Taylor expanded around -inf 64.0

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

    6. Simplified0.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)}\]
    7. Using strategy rm

    8. Applied div-inv0.7

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

    if -78626.83968822402 < x < 58824.0950936069

    1. Initial program 0.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

    2. Simplified0.2

      \[\leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.2

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

    5. Applied cbrt-prod0.2

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

    if 58824.0950936069 < x

    1. Initial program 60.3

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

    2. Simplified60.3

      \[\leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt60.8

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

    5. Taylor expanded around -inf 64.0

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

    6. Simplified0.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -78626.83968822401948273181915283203125:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right) - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)\\ \mathbf{elif}\;x \le 58824.09509360689844470471143722534179688:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) \cdot \frac{\sqrt[3]{x}}{x} - \left(\sqrt[3]{-1} \cdot \sqrt[3]{-x} - \sqrt[3]{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))