Average Error: 29.4 → 21.7
Time: 25.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -27546108091290.3828125 \lor \neg \left(n \le 28958019.0281741321086883544921875\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -27546108091290.3828125 \lor \neg \left(n \le 28958019.0281741321086883544921875\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}:\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

\end{array}
double f(double x, double n) {
        double r74646 = x;
        double r74647 = 1.0;
        double r74648 = r74646 + r74647;
        double r74649 = n;
        double r74650 = r74647 / r74649;
        double r74651 = pow(r74648, r74650);
        double r74652 = pow(r74646, r74650);
        double r74653 = r74651 - r74652;
        return r74653;
}
↓
double f(double x, double n) {
        double r74654 = n;
        double r74655 = -27546108091290.383;
        bool r74656 = r74654 <= r74655;
        double r74657 = 28958019.028174132;
        bool r74658 = r74654 <= r74657;
        double r74659 = !r74658;
        bool r74660 = r74656 || r74659;
        double r74661 = 1.0;
        double r74662 = r74661 / r74654;
        double r74663 = x;
        double r74664 = r74662 / r74663;
        double r74665 = 0.5;
        double r74666 = r74665 / r74654;
        double r74667 = 2.0;
        double r74668 = pow(r74663, r74667);
        double r74669 = r74666 / r74668;
        double r74670 = log(r74663);
        double r74671 = r74670 * r74661;
        double r74672 = pow(r74654, r74667);
        double r74673 = r74663 * r74672;
        double r74674 = r74671 / r74673;
        double r74675 = r74669 - r74674;
        double r74676 = r74664 - r74675;
        double r74677 = r74663 + r74661;
        double r74678 = pow(r74677, r74662);
        double r74679 = 3.0;
        double r74680 = pow(r74678, r74679);
        double r74681 = cbrt(r74680);
        double r74682 = pow(r74663, r74662);
        double r74683 = r74681 - r74682;
        double r74684 = r74660 ? r74676 : r74683;
        return r74684;
}

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 < -27546108091290.383 or 28958019.028174132 < n

    1. Initial program 45.0

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

    2. Taylor expanded around inf 31.9

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

    3. Simplified31.3

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

    if -27546108091290.383 < n < 28958019.028174132

    1. Initial program 9.3

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

    3. Applied add-cbrt-cube9.4

      \[\leadsto \color{blue}{\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)}\]

    4. Simplified9.4

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

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -27546108091290.3828125 \lor \neg \left(n \le 28958019.0281741321086883544921875\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))