Average Error: 30.0 → 11.8
Time: 6.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4690392002297262 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.00897341644053555217:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4690392002297262 \cdot 10^{61}:\\
\;\;\;\;\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}}\\

\mathbf{elif}\;x \le 0.00897341644053555217:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r69696 = x;
        double r69697 = 1.0;
        double r69698 = r69696 + r69697;
        double r69699 = cbrt(r69698);
        double r69700 = cbrt(r69696);
        double r69701 = r69699 - r69700;
        return r69701;
}
↓
double f(double x) {
        double r69702 = x;
        double r69703 = -4.469039200229726e+61;
        bool r69704 = r69702 <= r69703;
        double r69705 = 0.3333333333333333;
        double r69706 = 1.0;
        double r69707 = 2.0;
        double r69708 = pow(r69702, r69707);
        double r69709 = r69706 / r69708;
        double r69710 = 0.3333333333333333;
        double r69711 = pow(r69709, r69710);
        double r69712 = r69705 * r69711;
        double r69713 = 0.06172839506172839;
        double r69714 = 8.0;
        double r69715 = pow(r69702, r69714);
        double r69716 = r69706 / r69715;
        double r69717 = pow(r69716, r69710);
        double r69718 = r69713 * r69717;
        double r69719 = r69712 + r69718;
        double r69720 = 0.1111111111111111;
        double r69721 = 5.0;
        double r69722 = pow(r69702, r69721);
        double r69723 = r69706 / r69722;
        double r69724 = pow(r69723, r69710);
        double r69725 = r69720 * r69724;
        double r69726 = r69719 - r69725;
        double r69727 = 0.008973416440535552;
        bool r69728 = r69702 <= r69727;
        double r69729 = r69702 * r69702;
        double r69730 = 1.0;
        double r69731 = r69730 * r69730;
        double r69732 = r69729 - r69731;
        double r69733 = cbrt(r69732);
        double r69734 = r69702 - r69730;
        double r69735 = cbrt(r69734);
        double r69736 = r69733 / r69735;
        double r69737 = cbrt(r69702);
        double r69738 = r69736 - r69737;
        double r69739 = 0.0;
        double r69740 = r69739 + r69730;
        double r69741 = r69702 + r69730;
        double r69742 = cbrt(r69741);
        double r69743 = r69742 + r69737;
        double r69744 = r69742 * r69743;
        double r69745 = 0.6666666666666666;
        double r69746 = pow(r69702, r69745);
        double r69747 = r69744 + r69746;
        double r69748 = r69740 / r69747;
        double r69749 = r69728 ? r69738 : r69748;
        double r69750 = r69704 ? r69726 : r69749;
        return r69750;
}

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 < -4.469039200229726e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.0

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

    if -4.469039200229726e+61 < x < 0.008973416440535552

    1. Initial program 5.1

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

    3. Applied flip-+5.1

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

    4. Applied cbrt-div5.0

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

    if 0.008973416440535552 < x

    1. Initial program 59.0

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

    3. Applied flip3--58.8

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

    4. Simplified1.0

      \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

    5. Simplified4.4

      \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4690392002297262 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.00897341644053555217:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))