Average Error: 29.0 → 21.9
Time: 17.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -80925173752378880 \lor \neg \left(n \le 196016184.31866574\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -80925173752378880 \lor \neg \left(n \le 196016184.31866574\right):\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\

\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 r72333 = x;
        double r72334 = 1.0;
        double r72335 = r72333 + r72334;
        double r72336 = n;
        double r72337 = r72334 / r72336;
        double r72338 = pow(r72335, r72337);
        double r72339 = pow(r72333, r72337);
        double r72340 = r72338 - r72339;
        return r72340;
}
↓
double f(double x, double n) {
        double r72341 = n;
        double r72342 = -8.092517375237888e+16;
        bool r72343 = r72341 <= r72342;
        double r72344 = 196016184.31866574;
        bool r72345 = r72341 <= r72344;
        double r72346 = !r72345;
        bool r72347 = r72343 || r72346;
        double r72348 = 1.0;
        double r72349 = x;
        double r72350 = r72348 / r72349;
        double r72351 = 1.0;
        double r72352 = r72351 / r72341;
        double r72353 = log(r72349);
        double r72354 = -r72353;
        double r72355 = 2.0;
        double r72356 = pow(r72341, r72355);
        double r72357 = r72354 / r72356;
        double r72358 = r72352 - r72357;
        double r72359 = r72350 * r72358;
        double r72360 = 0.5;
        double r72361 = pow(r72349, r72355);
        double r72362 = r72361 * r72341;
        double r72363 = r72360 / r72362;
        double r72364 = r72359 - r72363;
        double r72365 = r72349 + r72348;
        double r72366 = r72348 / r72341;
        double r72367 = pow(r72365, r72366);
        double r72368 = pow(r72349, r72366);
        double r72369 = sqrt(r72368);
        double r72370 = r72369 * r72369;
        double r72371 = r72367 - r72370;
        double r72372 = r72347 ? r72364 : r72371;
        return r72372;
}

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 < -8.092517375237888e+16 or 196016184.31866574 < n

    1. Initial program 44.4

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified31.6

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

    if -8.092517375237888e+16 < n < 196016184.31866574

    1. Initial program 9.6

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

    3. Applied add-sqr-sqrt9.7

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

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -80925173752378880 \lor \neg \left(n \le 196016184.31866574\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))