Average Error: 29.0 → 22.1
Time: 25.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -57533769617751403819054902804480 \lor \neg \left(n \le 254252678.20771694183349609375\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -57533769617751403819054902804480 \lor \neg \left(n \le 254252678.20771694183349609375\right):\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r60606 = x;
        double r60607 = 1.0;
        double r60608 = r60606 + r60607;
        double r60609 = n;
        double r60610 = r60607 / r60609;
        double r60611 = pow(r60608, r60610);
        double r60612 = pow(r60606, r60610);
        double r60613 = r60611 - r60612;
        return r60613;
}
↓
double f(double x, double n) {
        double r60614 = n;
        double r60615 = -5.75337696177514e+31;
        bool r60616 = r60614 <= r60615;
        double r60617 = 254252678.20771694;
        bool r60618 = r60614 <= r60617;
        double r60619 = !r60618;
        bool r60620 = r60616 || r60619;
        double r60621 = 1.0;
        double r60622 = r60621 / r60614;
        double r60623 = x;
        double r60624 = r60622 / r60623;
        double r60625 = 0.5;
        double r60626 = r60625 / r60614;
        double r60627 = 2.0;
        double r60628 = pow(r60623, r60627);
        double r60629 = r60626 / r60628;
        double r60630 = log(r60623);
        double r60631 = r60630 * r60621;
        double r60632 = pow(r60614, r60627);
        double r60633 = r60623 * r60632;
        double r60634 = r60631 / r60633;
        double r60635 = r60629 - r60634;
        double r60636 = r60624 - r60635;
        double r60637 = r60623 + r60621;
        double r60638 = pow(r60637, r60622);
        double r60639 = pow(r60623, r60622);
        double r60640 = r60638 - r60639;
        double r60641 = exp(r60640);
        double r60642 = log(r60641);
        double r60643 = r60620 ? r60636 : r60642;
        return r60643;
}

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 < -5.75337696177514e+31 or 254252678.20771694 < n

    1. Initial program 44.8

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

    2. Taylor expanded around inf 32.7

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

    3. Simplified32.1

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

    if -5.75337696177514e+31 < n < 254252678.20771694

    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-log-exp10.2

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

    4. Applied add-log-exp10.2

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

    5. Applied diff-log10.2

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

    6. Simplified10.2

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -57533769617751403819054902804480 \lor \neg \left(n \le 254252678.20771694183349609375\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))