Average Error: 30.0 → 22.8
Time: 11.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.059000321787014 \cdot 10^{-6} \lor \neg \left(\frac{1}{n} \le 5.0494628055209996 \cdot 10^{-25}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.059000321787014 \cdot 10^{-6} \lor \neg \left(\frac{1}{n} \le 5.0494628055209996 \cdot 10^{-25}\right):\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\end{array}
double f(double x, double n) {
        double r52854 = x;
        double r52855 = 1.0;
        double r52856 = r52854 + r52855;
        double r52857 = n;
        double r52858 = r52855 / r52857;
        double r52859 = pow(r52856, r52858);
        double r52860 = pow(r52854, r52858);
        double r52861 = r52859 - r52860;
        return r52861;
}
↓
double f(double x, double n) {
        double r52862 = 1.0;
        double r52863 = n;
        double r52864 = r52862 / r52863;
        double r52865 = -1.059000321787014e-06;
        bool r52866 = r52864 <= r52865;
        double r52867 = 5.049462805521e-25;
        bool r52868 = r52864 <= r52867;
        double r52869 = !r52868;
        bool r52870 = r52866 || r52869;
        double r52871 = x;
        double r52872 = r52871 + r52862;
        double r52873 = pow(r52872, r52864);
        double r52874 = 3.0;
        double r52875 = pow(r52873, r52874);
        double r52876 = cbrt(r52875);
        double r52877 = pow(r52871, r52864);
        double r52878 = r52876 - r52877;
        double r52879 = r52864 / r52871;
        double r52880 = 0.5;
        double r52881 = r52880 / r52863;
        double r52882 = 2.0;
        double r52883 = pow(r52871, r52882);
        double r52884 = r52881 / r52883;
        double r52885 = log(r52871);
        double r52886 = r52885 * r52862;
        double r52887 = pow(r52863, r52882);
        double r52888 = r52871 * r52887;
        double r52889 = r52886 / r52888;
        double r52890 = r52884 - r52889;
        double r52891 = r52879 - r52890;
        double r52892 = r52870 ? r52878 : r52891;
        return r52892;
}

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 (/ 1.0 n) < -1.059000321787014e-06 or 5.049462805521e-25 < (/ 1.0 n)

    1. Initial program 10.1

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

    3. Applied add-cbrt-cube10.1

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

    4. Simplified10.1

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

    if -1.059000321787014e-06 < (/ 1.0 n) < 5.049462805521e-25

    1. Initial program 45.3

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

    2. Taylor expanded around inf 33.0

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

    3. Simplified32.4

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.059000321787014 \cdot 10^{-6} \lor \neg \left(\frac{1}{n} \le 5.0494628055209996 \cdot 10^{-25}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))