Average Error: 29.4 → 22.0
Time: 30.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -74876590.12174678:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{1}{x}}{n} \cdot \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \mathbf{elif}\;n \le 33522299.50529173:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{1}{x}}{n} \cdot \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -74876590.12174678:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{1}{x}}{n} \cdot \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\

\mathbf{elif}\;n \le 33522299.50529173:\\
\;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{1}{x}}{n} \cdot \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\

\end{array}
double f(double x, double n) {
        double r3080852 = x;
        double r3080853 = 1.0;
        double r3080854 = r3080852 + r3080853;
        double r3080855 = n;
        double r3080856 = r3080853 / r3080855;
        double r3080857 = pow(r3080854, r3080856);
        double r3080858 = pow(r3080852, r3080856);
        double r3080859 = r3080857 - r3080858;
        return r3080859;
}
↓
double f(double x, double n) {
        double r3080860 = n;
        double r3080861 = -74876590.12174678;
        bool r3080862 = r3080860 <= r3080861;
        double r3080863 = 1.0;
        double r3080864 = x;
        double r3080865 = r3080863 / r3080864;
        double r3080866 = r3080865 / r3080860;
        double r3080867 = 0.5;
        double r3080868 = r3080867 / r3080864;
        double r3080869 = r3080866 * r3080868;
        double r3080870 = r3080866 - r3080869;
        double r3080871 = log(r3080864);
        double r3080872 = r3080871 / r3080864;
        double r3080873 = r3080860 * r3080860;
        double r3080874 = r3080872 / r3080873;
        double r3080875 = r3080870 + r3080874;
        double r3080876 = 33522299.50529173;
        bool r3080877 = r3080860 <= r3080876;
        double r3080878 = r3080863 + r3080864;
        double r3080879 = r3080863 / r3080860;
        double r3080880 = pow(r3080878, r3080879);
        double r3080881 = r3080880 * r3080880;
        double r3080882 = cbrt(r3080881);
        double r3080883 = pow(r3080864, r3080879);
        double r3080884 = r3080882 - r3080883;
        double r3080885 = r3080877 ? r3080884 : r3080875;
        double r3080886 = r3080862 ? r3080875 : r3080885;
        return r3080886;
}

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 2 regimes

  2. if n < -74876590.12174678 or 33522299.50529173 < n

    1. Initial program 44.8

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

    2. Taylor expanded around inf 32.4

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

    3. Simplified31.8

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

    if -74876590.12174678 < n < 33522299.50529173

    1. Initial program 8.6

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

    3. Applied add-cbrt-cube8.7

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

    4. Taylor expanded around 0 8.7

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -74876590.12174678:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{1}{x}}{n} \cdot \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \mathbf{elif}\;n \le 33522299.50529173:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{1}{x}}{n} \cdot \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\log x}{x}}{n \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))