Average Error: 30.2 → 9.1
Time: 8.1s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4234.57245693455206:\\ \;\;\;\;\left(0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \le 0.00213870861947561752:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4234.57245693455206:\\
\;\;\;\;\left(0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\

\mathbf{elif}\;x \le 0.00213870861947561752:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r55561 = x;
        double r55562 = 1.0;
        double r55563 = r55561 + r55562;
        double r55564 = cbrt(r55563);
        double r55565 = cbrt(r55561);
        double r55566 = r55564 - r55565;
        return r55566;
}
↓
double f(double x) {
        double r55567 = x;
        double r55568 = -4234.572456934552;
        bool r55569 = r55567 <= r55568;
        double r55570 = 0.3333333333333333;
        double r55571 = 1.0;
        double r55572 = 2.0;
        double r55573 = pow(r55567, r55572);
        double r55574 = r55571 / r55573;
        double r55575 = cbrt(r55574);
        double r55576 = r55570 * r55575;
        double r55577 = 8.0;
        double r55578 = pow(r55567, r55577);
        double r55579 = r55571 / r55578;
        double r55580 = cbrt(r55579);
        double r55581 = 0.06172839506172839;
        double r55582 = r55580 * r55581;
        double r55583 = r55576 + r55582;
        double r55584 = 0.1111111111111111;
        double r55585 = 5.0;
        double r55586 = pow(r55567, r55585);
        double r55587 = r55571 / r55586;
        double r55588 = cbrt(r55587);
        double r55589 = r55584 * r55588;
        double r55590 = r55583 - r55589;
        double r55591 = 0.0021387086194756175;
        bool r55592 = r55567 <= r55591;
        double r55593 = r55567 * r55567;
        double r55594 = 1.0;
        double r55595 = r55594 * r55594;
        double r55596 = r55593 - r55595;
        double r55597 = cbrt(r55596);
        double r55598 = r55567 - r55594;
        double r55599 = cbrt(r55598);
        double r55600 = r55597 / r55599;
        double r55601 = cbrt(r55567);
        double r55602 = r55600 - r55601;
        double r55603 = r55567 - r55567;
        double r55604 = r55594 + r55603;
        double r55605 = r55567 + r55594;
        double r55606 = cbrt(r55605);
        double r55607 = r55601 + r55606;
        double r55608 = r55606 * r55607;
        double r55609 = 0.6666666666666666;
        double r55610 = pow(r55567, r55609);
        double r55611 = r55608 + r55610;
        double r55612 = r55604 / r55611;
        double r55613 = r55592 ? r55602 : r55612;
        double r55614 = r55569 ? r55590 : r55613;
        return r55614;
}

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 < -4234.572456934552

    1. Initial program 60.1

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

    3. Applied add-cube-cbrt60.4

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

    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. Taylor expanded around inf 44.9

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    6. Simplified31.2

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}}\]

    if -4234.572456934552 < x < 0.0021387086194756175

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied cbrt-div0.1

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

    if 0.0021387086194756175 < x

    1. Initial program 58.9

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

    3. Applied add-cube-cbrt59.4

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\]
    4. Using strategy rm

    5. Applied flip3--59.4

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

    6. Simplified1.4

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

    7. Simplified4.4

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4234.57245693455206:\\ \;\;\;\;\left(0.333333333333333315 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \le 0.00213870861947561752:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))