Average Error: 29.8 → 0.4
Time: 16.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 8.229480875598937927861697971820831298828 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\sqrt[3]{x}}{x} \cdot 0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798 \cdot \frac{\sqrt[3]{x}}{x}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 8.229480875598937927861697971820831298828 \cdot 10^{-5}:\\
\;\;\;\;\left(\frac{\sqrt[3]{x}}{x} \cdot 0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798 \cdot \frac{\sqrt[3]{x}}{x}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\

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

\end{array}
double f(double x) {
        double r82083 = x;
        double r82084 = 1.0;
        double r82085 = r82083 + r82084;
        double r82086 = cbrt(r82085);
        double r82087 = cbrt(r82083);
        double r82088 = r82086 - r82087;
        return r82088;
}
↓
double f(double x) {
        double r82089 = 1.0;
        double r82090 = x;
        double r82091 = r82089 + r82090;
        double r82092 = cbrt(r82091);
        double r82093 = cbrt(r82090);
        double r82094 = r82092 - r82093;
        double r82095 = 8.229480875598938e-05;
        bool r82096 = r82094 <= r82095;
        double r82097 = r82093 / r82090;
        double r82098 = 0.3333333333333333;
        double r82099 = r82097 * r82098;
        double r82100 = 0.1111111111111111;
        double r82101 = r82100 * r82097;
        double r82102 = r82101 / r82090;
        double r82103 = r82099 - r82102;
        double r82104 = -r82090;
        double r82105 = cbrt(r82104);
        double r82106 = -1.0;
        double r82107 = cbrt(r82106);
        double r82108 = r82105 * r82107;
        double r82109 = r82093 - r82108;
        double r82110 = r82103 + r82109;
        double r82111 = cbrt(r82093);
        double r82112 = r82111 * r82111;
        double r82113 = r82111 * r82112;
        double r82114 = r82092 - r82113;
        double r82115 = r82096 ? r82110 : r82114;
        return r82115;
}

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 (- (cbrt (+ x 1.0)) (cbrt x)) < 8.229480875598938e-05

    1. Initial program 60.5

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

    2. Simplified60.5

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

    4. Applied add-log-exp63.6

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

    5. Applied add-log-exp63.6

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

    6. Applied diff-log63.6

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

    7. Simplified60.5

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

    8. Taylor expanded around -inf 64.0

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

    9. Simplified0.7

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

    if 8.229480875598938e-05 < (- (cbrt (+ x 1.0)) (cbrt x))

    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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 8.229480875598937927861697971820831298828 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{\sqrt[3]{x}}{x} \cdot 0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798 \cdot \frac{\sqrt[3]{x}}{x}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))