Average Error: 29.6 → 22.2
Time: 10.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.579141987461542662174451834289357066154 \lor \neg \left(n \le 2379860481774380090654720\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -3.579141987461542662174451834289357066154 \lor \neg \left(n \le 2379860481774380090654720\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 r52591 = x;
        double r52592 = 1.0;
        double r52593 = r52591 + r52592;
        double r52594 = n;
        double r52595 = r52592 / r52594;
        double r52596 = pow(r52593, r52595);
        double r52597 = pow(r52591, r52595);
        double r52598 = r52596 - r52597;
        return r52598;
}
↓
double f(double x, double n) {
        double r52599 = n;
        double r52600 = -3.5791419874615427;
        bool r52601 = r52599 <= r52600;
        double r52602 = 2.37986048177438e+24;
        bool r52603 = r52599 <= r52602;
        double r52604 = !r52603;
        bool r52605 = r52601 || r52604;
        double r52606 = 1.0;
        double r52607 = r52606 / r52599;
        double r52608 = x;
        double r52609 = r52607 / r52608;
        double r52610 = 0.5;
        double r52611 = r52610 / r52599;
        double r52612 = 2.0;
        double r52613 = pow(r52608, r52612);
        double r52614 = r52611 / r52613;
        double r52615 = log(r52608);
        double r52616 = r52615 * r52606;
        double r52617 = pow(r52599, r52612);
        double r52618 = r52608 * r52617;
        double r52619 = r52616 / r52618;
        double r52620 = r52614 - r52619;
        double r52621 = r52609 - r52620;
        double r52622 = r52608 + r52606;
        double r52623 = pow(r52622, r52607);
        double r52624 = pow(r52608, r52607);
        double r52625 = r52623 - r52624;
        double r52626 = exp(r52625);
        double r52627 = log(r52626);
        double r52628 = r52605 ? r52621 : r52627;
        return r52628;
}

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 < -3.5791419874615427 or 2.37986048177438e+24 < n

    1. Initial program 44.7

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

    2. Taylor expanded around inf 32.3

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

    3. Simplified31.7

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

    if -3.5791419874615427 < n < 2.37986048177438e+24

    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-log-exp9.8

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

    4. Applied add-log-exp9.7

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

    5. Applied diff-log9.7

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

    6. Simplified9.7

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.579141987461542662174451834289357066154 \lor \neg \left(n \le 2379860481774380090654720\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))