Average Error: 29.8 → 22.7
Time: 29.7s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.333109823078832 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.058098422757922 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x \cdot n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.333109823078832 \cdot 10^{-05}:\\
\;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 1.058098422757922 \cdot 10^{-12}:\\
\;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x \cdot n}}{x}\right)\\

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

\end{array}
double f(double x, double n) {
        double r2055333 = x;
        double r2055334 = 1.0;
        double r2055335 = r2055333 + r2055334;
        double r2055336 = n;
        double r2055337 = r2055334 / r2055336;
        double r2055338 = pow(r2055335, r2055337);
        double r2055339 = pow(r2055333, r2055337);
        double r2055340 = r2055338 - r2055339;
        return r2055340;
}
↓
double f(double x, double n) {
        double r2055341 = 1.0;
        double r2055342 = n;
        double r2055343 = r2055341 / r2055342;
        double r2055344 = -3.333109823078832e-05;
        bool r2055345 = r2055343 <= r2055344;
        double r2055346 = x;
        double r2055347 = r2055346 + r2055341;
        double r2055348 = pow(r2055347, r2055343);
        double r2055349 = sqrt(r2055348);
        double r2055350 = r2055349 * r2055349;
        double r2055351 = pow(r2055346, r2055343);
        double r2055352 = r2055350 - r2055351;
        double r2055353 = 1.058098422757922e-12;
        bool r2055354 = r2055343 <= r2055353;
        double r2055355 = log(r2055346);
        double r2055356 = r2055355 / r2055346;
        double r2055357 = r2055342 * r2055342;
        double r2055358 = r2055356 / r2055357;
        double r2055359 = r2055343 / r2055346;
        double r2055360 = 0.5;
        double r2055361 = r2055346 * r2055342;
        double r2055362 = r2055360 / r2055361;
        double r2055363 = r2055362 / r2055346;
        double r2055364 = r2055359 - r2055363;
        double r2055365 = r2055358 + r2055364;
        double r2055366 = r2055354 ? r2055365 : r2055352;
        double r2055367 = r2055345 ? r2055352 : r2055366;
        return r2055367;
}

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) < -3.333109823078832e-05 or 1.058098422757922e-12 < (/ 1 n)

    1. Initial program 8.9

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

    3. Applied add-sqr-sqrt8.9

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

    if -3.333109823078832e-05 < (/ 1 n) < 1.058098422757922e-12

    1. Initial program 45.0

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

    3. Applied add-sqr-sqrt45.0

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

    4. Taylor expanded around inf 33.3

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

    5. Simplified32.7

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.333109823078832 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.058098422757922 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\log x}{x}}{n \cdot n} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x \cdot n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))