Average Error: 32.0 → 23.4
Time: 12.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -71002.377540986956 \lor \neg \left(n \le 104178.387061829286\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{2}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -71002.377540986956 \lor \neg \left(n \le 104178.387061829286\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}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{2}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\\

\end{array}
double f(double x, double n) {
        double r74001 = x;
        double r74002 = 1.0;
        double r74003 = r74001 + r74002;
        double r74004 = n;
        double r74005 = r74002 / r74004;
        double r74006 = pow(r74003, r74005);
        double r74007 = pow(r74001, r74005);
        double r74008 = r74006 - r74007;
        return r74008;
}
↓
double f(double x, double n) {
        double r74009 = n;
        double r74010 = -71002.37754098696;
        bool r74011 = r74009 <= r74010;
        double r74012 = 104178.38706182929;
        bool r74013 = r74009 <= r74012;
        double r74014 = !r74013;
        bool r74015 = r74011 || r74014;
        double r74016 = 1.0;
        double r74017 = r74016 / r74009;
        double r74018 = x;
        double r74019 = r74017 / r74018;
        double r74020 = 0.5;
        double r74021 = r74020 / r74009;
        double r74022 = 2.0;
        double r74023 = pow(r74018, r74022);
        double r74024 = r74021 / r74023;
        double r74025 = log(r74018);
        double r74026 = r74025 * r74016;
        double r74027 = pow(r74009, r74022);
        double r74028 = r74018 * r74027;
        double r74029 = r74026 / r74028;
        double r74030 = r74024 - r74029;
        double r74031 = r74019 - r74030;
        double r74032 = r74018 + r74016;
        double r74033 = pow(r74032, r74017);
        double r74034 = r74017 / r74022;
        double r74035 = pow(r74023, r74034);
        double r74036 = r74033 - r74035;
        double r74037 = r74015 ? r74031 : r74036;
        return r74037;
}

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 < -71002.37754098696 or 104178.38706182929 < n

    1. Initial program 44.0

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

    2. Taylor expanded around inf 32.2

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

    3. Simplified31.6

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

    if -71002.37754098696 < n < 104178.38706182929

    1. Initial program 2.7

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

    3. Applied add-sqr-sqrt2.7

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

    5. Applied sqrt-pow12.7

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

    6. Applied sqrt-pow12.7

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

    7. Applied pow-prod-down3.4

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

    8. Simplified3.4

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

  4. Final simplification23.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -71002.377540986956 \lor \neg \left(n \le 104178.387061829286\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{2}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))