Average Error: 29.7 → 22.5
Time: 10.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.421397779664377473457544443272669576483 \cdot 10^{-10} \lor \neg \left(\frac{1}{n} \le 1.820657228495789802655210734750766340961 \cdot 10^{-11}\right):\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -6.421397779664377473457544443272669576483 \cdot 10^{-10} \lor \neg \left(\frac{1}{n} \le 1.820657228495789802655210734750766340961 \cdot 10^{-11}\right):\\
\;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\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 r53317 = x;
        double r53318 = 1.0;
        double r53319 = r53317 + r53318;
        double r53320 = n;
        double r53321 = r53318 / r53320;
        double r53322 = pow(r53319, r53321);
        double r53323 = pow(r53317, r53321);
        double r53324 = r53322 - r53323;
        return r53324;
}
↓
double f(double x, double n) {
        double r53325 = 1.0;
        double r53326 = n;
        double r53327 = r53325 / r53326;
        double r53328 = -6.421397779664377e-10;
        bool r53329 = r53327 <= r53328;
        double r53330 = 1.8206572284957898e-11;
        bool r53331 = r53327 <= r53330;
        double r53332 = !r53331;
        bool r53333 = r53329 || r53332;
        double r53334 = x;
        double r53335 = r53334 + r53325;
        double r53336 = pow(r53335, r53327);
        double r53337 = pow(r53334, r53327);
        double r53338 = r53336 - r53337;
        double r53339 = exp(r53338);
        double r53340 = log(r53339);
        double r53341 = r53327 / r53334;
        double r53342 = 0.5;
        double r53343 = r53342 / r53326;
        double r53344 = 2.0;
        double r53345 = pow(r53334, r53344);
        double r53346 = r53343 / r53345;
        double r53347 = log(r53334);
        double r53348 = r53347 * r53325;
        double r53349 = pow(r53326, r53344);
        double r53350 = r53334 * r53349;
        double r53351 = r53348 / r53350;
        double r53352 = r53346 - r53351;
        double r53353 = r53341 - r53352;
        double r53354 = r53333 ? r53340 : r53353;
        return r53354;
}

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) < -6.421397779664377e-10 or 1.8206572284957898e-11 < (/ 1.0 n)

    1. Initial program 9.1

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

    3. Applied add-log-exp9.3

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

    4. Applied add-log-exp9.3

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

    5. Applied diff-log9.3

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

    6. Simplified9.3

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

    if -6.421397779664377e-10 < (/ 1.0 n) < 1.8206572284957898e-11

    1. Initial program 45.6

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

    2. Taylor expanded around inf 33.3

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

    3. Simplified32.7

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.421397779664377473457544443272669576483 \cdot 10^{-10} \lor \neg \left(\frac{1}{n} \le 1.820657228495789802655210734750766340961 \cdot 10^{-11}\right):\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))