Average Error: 29.6 → 11.6
Time: 4.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.450208027756156638035805139753243266011 \cdot 10^{61}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 6.899568930112678520481222913841856669848 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.450208027756156638035805139753243266011 \cdot 10^{61}:\\
\;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\

\mathbf{elif}\;x \le 6.899568930112678520481222913841856669848 \cdot 10^{-9}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\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 r61727 = x;
        double r61728 = 1.0;
        double r61729 = r61727 + r61728;
        double r61730 = cbrt(r61729);
        double r61731 = cbrt(r61727);
        double r61732 = r61730 - r61731;
        return r61732;
}
↓
double f(double x) {
        double r61733 = x;
        double r61734 = -4.4502080277561566e+61;
        bool r61735 = r61733 <= r61734;
        double r61736 = 0.3333333333333333;
        double r61737 = 1.0;
        double r61738 = 2.0;
        double r61739 = pow(r61733, r61738);
        double r61740 = r61737 / r61739;
        double r61741 = 0.3333333333333333;
        double r61742 = pow(r61740, r61741);
        double r61743 = r61736 * r61742;
        double r61744 = 0.06172839506172839;
        double r61745 = 8.0;
        double r61746 = pow(r61733, r61745);
        double r61747 = r61737 / r61746;
        double r61748 = pow(r61747, r61741);
        double r61749 = r61744 * r61748;
        double r61750 = r61743 + r61749;
        double r61751 = 0.1111111111111111;
        double r61752 = 5.0;
        double r61753 = pow(r61733, r61752);
        double r61754 = r61737 / r61753;
        double r61755 = pow(r61754, r61741);
        double r61756 = r61751 * r61755;
        double r61757 = r61750 - r61756;
        double r61758 = 6.8995689301126785e-09;
        bool r61759 = r61733 <= r61758;
        double r61760 = 1.0;
        double r61761 = r61733 + r61760;
        double r61762 = cbrt(r61761);
        double r61763 = r61762 * r61762;
        double r61764 = cbrt(r61763);
        double r61765 = cbrt(r61762);
        double r61766 = r61764 * r61765;
        double r61767 = cbrt(r61733);
        double r61768 = r61766 - r61767;
        double r61769 = 0.0;
        double r61770 = r61769 + r61760;
        double r61771 = r61762 + r61767;
        double r61772 = r61762 * r61771;
        double r61773 = 0.6666666666666666;
        double r61774 = pow(r61733, r61773);
        double r61775 = r61772 + r61774;
        double r61776 = r61770 / r61775;
        double r61777 = r61759 ? r61768 : r61776;
        double r61778 = r61735 ? r61757 : r61777;
        return r61778;
}

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 < -4.4502080277561566e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 39.5

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    if -4.4502080277561566e+61 < x < 6.8995689301126785e-09

    1. Initial program 4.8

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

    3. Applied add-cube-cbrt4.8

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

    4. Applied cbrt-prod4.8

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

    if 6.8995689301126785e-09 < x

    1. Initial program 57.4

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

    3. Applied flip3--57.2

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

    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. Simplified4.3

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.450208027756156638035805139753243266011 \cdot 10^{61}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 6.899568930112678520481222913841856669848 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))