Average Error: 29.7 → 0.4
Time: 21.2s
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 r38080 = x;
        double r38081 = 1.0;
        double r38082 = r38080 + r38081;
        double r38083 = cbrt(r38082);
        double r38084 = cbrt(r38080);
        double r38085 = r38083 - r38084;
        return r38085;
}
↓
double f(double x) {
        double r38086 = x;
        double r38087 = -73338.54433713964;
        bool r38088 = r38086 <= r38087;
        double r38089 = 71130.44665790045;
        bool r38090 = r38086 <= r38089;
        double r38091 = !r38090;
        bool r38092 = r38088 || r38091;
        double r38093 = cbrt(r38086);
        double r38094 = -1.0;
        double r38095 = cbrt(r38094);
        double r38096 = -r38086;
        double r38097 = cbrt(r38096);
        double r38098 = r38095 * r38097;
        double r38099 = r38093 - r38098;
        double r38100 = r38093 / r38086;
        double r38101 = 0.1111111111111111;
        double r38102 = r38101 / r38086;
        double r38103 = 0.3333333333333333;
        double r38104 = r38102 - r38103;
        double r38105 = r38100 * r38104;
        double r38106 = r38099 - r38105;
        double r38107 = 1.0;
        double r38108 = r38086 + r38107;
        double r38109 = cbrt(r38108);
        double r38110 = r38109 * r38109;
        double r38111 = cbrt(r38110);
        double r38112 = 3.0;
        double r38113 = pow(r38086, r38112);
        double r38114 = pow(r38107, r38112);
        double r38115 = r38113 + r38114;
        double r38116 = cbrt(r38115);
        double r38117 = cbrt(r38116);
        double r38118 = r38107 - r38086;
        double r38119 = r38107 * r38118;
        double r38120 = r38086 * r38086;
        double r38121 = r38119 + r38120;
        double r38122 = cbrt(r38121);
        double r38123 = cbrt(r38122);
        double r38124 = r38117 / r38123;
        double r38125 = r38111 * r38124;
        double r38126 = r38125 - r38093;
        double r38127 = r38092 ? r38106 : r38126;
        return r38127;
}

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)))