Average Error: 29.1 → 22.4
Time: 27.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\log x}{n} \cdot \frac{1}{x \cdot n} + \frac{1}{x \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\
\;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\
\;\;\;\;\left(\frac{\log x}{n} \cdot \frac{1}{x \cdot n} + \frac{1}{x \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\

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

\end{array}
double f(double x, double n) {
        double r1353913 = x;
        double r1353914 = 1.0;
        double r1353915 = r1353913 + r1353914;
        double r1353916 = n;
        double r1353917 = r1353914 / r1353916;
        double r1353918 = pow(r1353915, r1353917);
        double r1353919 = pow(r1353913, r1353917);
        double r1353920 = r1353918 - r1353919;
        return r1353920;
}
↓
double f(double x, double n) {
        double r1353921 = 1.0;
        double r1353922 = n;
        double r1353923 = r1353921 / r1353922;
        double r1353924 = -104130347.93584307;
        bool r1353925 = r1353923 <= r1353924;
        double r1353926 = x;
        double r1353927 = r1353926 + r1353921;
        double r1353928 = pow(r1353927, r1353923);
        double r1353929 = pow(r1353926, r1353923);
        double r1353930 = r1353928 - r1353929;
        double r1353931 = log(r1353930);
        double r1353932 = cbrt(r1353931);
        double r1353933 = r1353932 * r1353932;
        double r1353934 = exp(r1353933);
        double r1353935 = pow(r1353934, r1353932);
        double r1353936 = 2.8321253549703665e-07;
        bool r1353937 = r1353923 <= r1353936;
        double r1353938 = log(r1353926);
        double r1353939 = r1353938 / r1353922;
        double r1353940 = r1353926 * r1353922;
        double r1353941 = r1353921 / r1353940;
        double r1353942 = r1353939 * r1353941;
        double r1353943 = r1353942 + r1353941;
        double r1353944 = 0.5;
        double r1353945 = r1353926 * r1353926;
        double r1353946 = r1353922 * r1353945;
        double r1353947 = r1353944 / r1353946;
        double r1353948 = r1353943 - r1353947;
        double r1353949 = r1353937 ? r1353948 : r1353935;
        double r1353950 = r1353925 ? r1353935 : r1353949;
        return r1353950;
}

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) < -104130347.93584307 or 2.8321253549703665e-07 < (/ 1 n)

    1. Initial program 7.7

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

    3. Applied add-exp-log7.7

      \[\leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt7.7

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

    6. Applied exp-prod7.7

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

    if -104130347.93584307 < (/ 1 n) < 2.8321253549703665e-07

    1. Initial program 44.7

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

    2. Taylor expanded around inf 33.1

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

    3. Simplified33.1

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\log x}{n} \cdot \frac{1}{x \cdot n} + \frac{1}{x \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))