Average Error: 29.6 → 21.7
Time: 30.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2161219111661097 \cdot 10^{-07}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 4.362000941685765 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.2161219111661097 \cdot 10^{-07}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\

\mathbf{elif}\;\frac{1}{n} \le 4.362000941685765 \cdot 10^{-12}:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\\

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

\end{array}
double f(double x, double n) {
        double r2148874 = x;
        double r2148875 = 1.0;
        double r2148876 = r2148874 + r2148875;
        double r2148877 = n;
        double r2148878 = r2148875 / r2148877;
        double r2148879 = pow(r2148876, r2148878);
        double r2148880 = pow(r2148874, r2148878);
        double r2148881 = r2148879 - r2148880;
        return r2148881;
}
↓
double f(double x, double n) {
        double r2148882 = 1.0;
        double r2148883 = n;
        double r2148884 = r2148882 / r2148883;
        double r2148885 = -1.2161219111661097e-07;
        bool r2148886 = r2148884 <= r2148885;
        double r2148887 = x;
        double r2148888 = r2148887 + r2148882;
        double r2148889 = pow(r2148888, r2148884);
        double r2148890 = pow(r2148887, r2148884);
        double r2148891 = sqrt(r2148890);
        double r2148892 = r2148891 * r2148891;
        double r2148893 = r2148889 - r2148892;
        double r2148894 = 4.362000941685765e-12;
        bool r2148895 = r2148884 <= r2148894;
        double r2148896 = r2148884 / r2148887;
        double r2148897 = log(r2148887);
        double r2148898 = r2148897 / r2148883;
        double r2148899 = r2148896 * r2148898;
        double r2148900 = r2148896 + r2148899;
        double r2148901 = 0.5;
        double r2148902 = r2148887 * r2148883;
        double r2148903 = r2148887 * r2148902;
        double r2148904 = r2148901 / r2148903;
        double r2148905 = r2148900 - r2148904;
        double r2148906 = r2148895 ? r2148905 : r2148893;
        double r2148907 = r2148886 ? r2148893 : r2148906;
        return r2148907;
}

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 (/ 1 n) < -1.2161219111661097e-07 or 4.362000941685765e-12 < (/ 1 n)

    1. Initial program 8.7

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

    3. Applied add-sqr-sqrt8.8

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

    if -1.2161219111661097e-07 < (/ 1 n) < 4.362000941685765e-12

    1. Initial program 45.1

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

    3. Applied add-sqr-sqrt45.1

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

    4. Taylor expanded around inf 32.0

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

    5. Simplified31.4

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

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2161219111661097 \cdot 10^{-07}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 4.362000941685765 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{1}{n}}{x} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))