Average Error: 28.9 → 22.1
Time: 13.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1469491090180712 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.2787797536218279 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot {\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.1469491090180712 \cdot 10^{-6}:\\
\;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.2787797536218279 \cdot 10^{-20}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r88835 = x;
        double r88836 = 1.0;
        double r88837 = r88835 + r88836;
        double r88838 = n;
        double r88839 = r88836 / r88838;
        double r88840 = pow(r88837, r88839);
        double r88841 = pow(r88835, r88839);
        double r88842 = r88840 - r88841;
        return r88842;
}
↓
double f(double x, double n) {
        double r88843 = 1.0;
        double r88844 = n;
        double r88845 = r88843 / r88844;
        double r88846 = -1.1469491090180712e-06;
        bool r88847 = r88845 <= r88846;
        double r88848 = x;
        double r88849 = r88848 + r88843;
        double r88850 = cbrt(r88849);
        double r88851 = r88850 * r88850;
        double r88852 = pow(r88851, r88845);
        double r88853 = pow(r88850, r88845);
        double r88854 = r88852 * r88853;
        double r88855 = pow(r88848, r88845);
        double r88856 = cbrt(r88855);
        double r88857 = r88856 * r88856;
        double r88858 = r88857 * r88856;
        double r88859 = r88854 - r88858;
        double r88860 = 1.2787797536218279e-20;
        bool r88861 = r88845 <= r88860;
        double r88862 = r88845 / r88848;
        double r88863 = 0.5;
        double r88864 = r88863 / r88844;
        double r88865 = 2.0;
        double r88866 = pow(r88848, r88865);
        double r88867 = r88864 / r88866;
        double r88868 = log(r88848);
        double r88869 = r88868 * r88843;
        double r88870 = pow(r88844, r88865);
        double r88871 = r88848 * r88870;
        double r88872 = r88869 / r88871;
        double r88873 = r88867 - r88872;
        double r88874 = r88862 - r88873;
        double r88875 = 0.6666666666666666;
        double r88876 = r88875 * r88845;
        double r88877 = pow(r88849, r88876);
        double r88878 = cbrt(r88877);
        double r88879 = r88878 * r88878;
        double r88880 = r88879 * r88878;
        double r88881 = cbrt(r88845);
        double r88882 = r88881 * r88881;
        double r88883 = pow(r88850, r88882);
        double r88884 = pow(r88883, r88881);
        double r88885 = r88880 * r88884;
        double r88886 = r88885 - r88855;
        double r88887 = r88861 ? r88874 : r88886;
        double r88888 = r88847 ? r88859 : r88887;
        return r88888;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -1.1469491090180712e-06

    1. Initial program 0.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.6

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

    4. Applied unpow-prod-down0.6

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

    6. Applied add-cube-cbrt0.6

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

    if -1.1469491090180712e-06 < (/ 1.0 n) < 1.2787797536218279e-20

    1. Initial program 44.4

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

    2. Taylor expanded around inf 32.7

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

    3. Simplified32.1

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\]

    if 1.2787797536218279e-20 < (/ 1.0 n)

    1. Initial program 27.1

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt27.1

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

    4. Applied unpow-prod-down27.2

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

    6. Applied pow1/327.2

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

    7. Applied pow1/327.2

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

    8. Applied pow-prod-up27.2

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

    9. Applied pow-pow27.1

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

    10. Simplified27.1

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt27.2

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

    13. Applied pow-unpow27.2

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)} \cdot \color{blue}{{\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}} - {x}^{\left(\frac{1}{n}\right)}\]
    14. Using strategy rm

    15. Applied add-cube-cbrt27.2

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1469491090180712 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.2787797536218279 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot {\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))