Average Error: 29.6 → 22.2
Time: 10.7s
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 r52700 = x;
        double r52701 = 1.0;
        double r52702 = r52700 + r52701;
        double r52703 = n;
        double r52704 = r52701 / r52703;
        double r52705 = pow(r52702, r52704);
        double r52706 = pow(r52700, r52704);
        double r52707 = r52705 - r52706;
        return r52707;
}
↓
double f(double x, double n) {
        double r52708 = n;
        double r52709 = -3.5791419874615427;
        bool r52710 = r52708 <= r52709;
        double r52711 = 2.37986048177438e+24;
        bool r52712 = r52708 <= r52711;
        double r52713 = !r52712;
        bool r52714 = r52710 || r52713;
        double r52715 = 1.0;
        double r52716 = r52715 / r52708;
        double r52717 = x;
        double r52718 = r52716 / r52717;
        double r52719 = 0.5;
        double r52720 = r52719 / r52708;
        double r52721 = 2.0;
        double r52722 = pow(r52717, r52721);
        double r52723 = r52720 / r52722;
        double r52724 = log(r52717);
        double r52725 = r52724 * r52715;
        double r52726 = pow(r52708, r52721);
        double r52727 = r52717 * r52726;
        double r52728 = r52725 / r52727;
        double r52729 = r52723 - r52728;
        double r52730 = r52718 - r52729;
        double r52731 = r52717 + r52715;
        double r52732 = pow(r52731, r52716);
        double r52733 = pow(r52717, r52716);
        double r52734 = r52732 - r52733;
        double r52735 = exp(r52734);
        double r52736 = log(r52735);
        double r52737 = r52714 ? r52730 : r52736;
        return r52737;
}

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))))