Average Error: 29.7 → 0.4
Time: 20.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -73338.5443371396395377814769744873046875 \lor \neg \left(x \le 71130.44665790045110043138265609741210938\right):\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) - \frac{\sqrt[3]{x}}{x} \cdot \left(\frac{0.1111111111111111049432054187491303309798}{x} - 0.3333333333333333148296162562473909929395\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{\sqrt[3]{{x}^{3} + {1}^{3}}}}{\sqrt[3]{\sqrt[3]{1 \cdot \left(1 - x\right) + x \cdot x}}} - \sqrt[3]{x}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -73338.5443371396395377814769744873046875 \lor \neg \left(x \le 71130.44665790045110043138265609741210938\right):\\
\;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) - \frac{\sqrt[3]{x}}{x} \cdot \left(\frac{0.1111111111111111049432054187491303309798}{x} - 0.3333333333333333148296162562473909929395\right)\\

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

\end{array}
double f(double x) {
        double r38130 = x;
        double r38131 = 1.0;
        double r38132 = r38130 + r38131;
        double r38133 = cbrt(r38132);
        double r38134 = cbrt(r38130);
        double r38135 = r38133 - r38134;
        return r38135;
}
↓
double f(double x) {
        double r38136 = x;
        double r38137 = -73338.54433713964;
        bool r38138 = r38136 <= r38137;
        double r38139 = 71130.44665790045;
        bool r38140 = r38136 <= r38139;
        double r38141 = !r38140;
        bool r38142 = r38138 || r38141;
        double r38143 = cbrt(r38136);
        double r38144 = -1.0;
        double r38145 = cbrt(r38144);
        double r38146 = -r38136;
        double r38147 = cbrt(r38146);
        double r38148 = r38145 * r38147;
        double r38149 = r38143 - r38148;
        double r38150 = r38143 / r38136;
        double r38151 = 0.1111111111111111;
        double r38152 = r38151 / r38136;
        double r38153 = 0.3333333333333333;
        double r38154 = r38152 - r38153;
        double r38155 = r38150 * r38154;
        double r38156 = r38149 - r38155;
        double r38157 = 1.0;
        double r38158 = r38136 + r38157;
        double r38159 = cbrt(r38158);
        double r38160 = r38159 * r38159;
        double r38161 = cbrt(r38160);
        double r38162 = 3.0;
        double r38163 = pow(r38136, r38162);
        double r38164 = pow(r38157, r38162);
        double r38165 = r38163 + r38164;
        double r38166 = cbrt(r38165);
        double r38167 = cbrt(r38166);
        double r38168 = r38157 - r38136;
        double r38169 = r38157 * r38168;
        double r38170 = r38136 * r38136;
        double r38171 = r38169 + r38170;
        double r38172 = cbrt(r38171);
        double r38173 = cbrt(r38172);
        double r38174 = r38167 / r38173;
        double r38175 = r38161 * r38174;
        double r38176 = r38175 - r38143;
        double r38177 = r38142 ? r38156 : r38176;
        return r38177;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -73338.54433713964 or 71130.44665790045 < x

    1. Initial program 60.3

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

    3. Applied add-cube-cbrt60.5

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

    4. Applied cbrt-prod60.6

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

    6. Applied add-cube-cbrt60.7

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

    7. Taylor expanded around -inf 64.0

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

    8. Simplified0.7

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

    if -73338.54433713964 < x < 71130.44665790045

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    4. Applied cbrt-prod0.2

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

    6. Applied flip3-+0.2

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

    7. Applied cbrt-div0.2

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

    8. Applied cbrt-div0.2

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

    9. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -73338.5443371396395377814769744873046875 \lor \neg \left(x \le 71130.44665790045110043138265609741210938\right):\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) - \frac{\sqrt[3]{x}}{x} \cdot \left(\frac{0.1111111111111111049432054187491303309798}{x} - 0.3333333333333333148296162562473909929395\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{\sqrt[3]{{x}^{3} + {1}^{3}}}}{\sqrt[3]{\sqrt[3]{1 \cdot \left(1 - x\right) + x \cdot x}}} - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))