Average Error: 29.6 → 0.7
Time: 20.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.40871634357608854770660400390625 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.40871634357608854770660400390625 \cdot 10^{-8}:\\
\;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\

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

\end{array}
double f(double x) {
        double r46599 = x;
        double r46600 = 1.0;
        double r46601 = r46599 + r46600;
        double r46602 = cbrt(r46601);
        double r46603 = cbrt(r46599);
        double r46604 = r46602 - r46603;
        return r46604;
}
↓
double f(double x) {
        double r46605 = x;
        double r46606 = 1.0;
        double r46607 = r46605 + r46606;
        double r46608 = cbrt(r46607);
        double r46609 = cbrt(r46605);
        double r46610 = r46608 - r46609;
        double r46611 = 1.4087163435760885e-08;
        bool r46612 = r46610 <= r46611;
        double r46613 = r46609 / r46605;
        double r46614 = 0.3333333333333333;
        double r46615 = 0.1111111111111111;
        double r46616 = r46615 / r46605;
        double r46617 = r46614 - r46616;
        double r46618 = r46613 * r46617;
        double r46619 = -1.0;
        double r46620 = cbrt(r46619);
        double r46621 = -r46605;
        double r46622 = cbrt(r46621);
        double r46623 = r46620 * r46622;
        double r46624 = r46609 - r46623;
        double r46625 = r46618 + r46624;
        double r46626 = r46608 * r46608;
        double r46627 = cbrt(r46626);
        double r46628 = cbrt(r46608);
        double r46629 = r46627 * r46628;
        double r46630 = cbrt(r46609);
        double r46631 = r46630 * r46630;
        double r46632 = r46631 * r46630;
        double r46633 = r46629 - r46632;
        double r46634 = r46612 ? r46625 : r46633;
        return r46634;
}

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)) < 1.4087163435760885e-08

    1. Initial program 61.0

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

    3. Applied add-cube-cbrt61.1

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

    4. Applied cbrt-prod61.3

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

    6. Applied add-cube-cbrt61.3

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

    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. Simplified0.6

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

    if 1.4087163435760885e-08 < (- (cbrt (+ x 1.0)) (cbrt x))

    1. Initial program 0.7

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

    3. Applied add-cube-cbrt0.7

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

    4. Applied cbrt-prod0.7

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

    6. Applied add-cube-cbrt0.7

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \le 1.40871634357608854770660400390625 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333148296162562473909929395 - \frac{0.1111111111111111049432054187491303309798}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))