Average Error: 29.1 → 19.3
Time: 53.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\frac{\log x}{n}}{x \cdot n} + \frac{1}{x \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\
\;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\
\;\;\;\;\left(\frac{\frac{\log x}{n}}{x \cdot n} + \frac{1}{x \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\

\mathbf{else}:\\
\;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\

\end{array}
double f(double x, double n) {
        double r2552996 = x;
        double r2552997 = 1.0;
        double r2552998 = r2552996 + r2552997;
        double r2552999 = n;
        double r2553000 = r2552997 / r2552999;
        double r2553001 = pow(r2552998, r2553000);
        double r2553002 = pow(r2552996, r2553000);
        double r2553003 = r2553001 - r2553002;
        return r2553003;
}
↓
double f(double x, double n) {
        double r2553004 = 1.0;
        double r2553005 = n;
        double r2553006 = r2553004 / r2553005;
        double r2553007 = -104130347.93584307;
        bool r2553008 = r2553006 <= r2553007;
        double r2553009 = x;
        double r2553010 = r2553009 + r2553004;
        double r2553011 = pow(r2553010, r2553006);
        double r2553012 = pow(r2553009, r2553006);
        double r2553013 = r2553011 - r2553012;
        double r2553014 = log(r2553013);
        double r2553015 = exp(r2553014);
        double r2553016 = 2.8321253549703665e-07;
        bool r2553017 = r2553006 <= r2553016;
        double r2553018 = log(r2553009);
        double r2553019 = r2553018 / r2553005;
        double r2553020 = r2553009 * r2553005;
        double r2553021 = r2553019 / r2553020;
        double r2553022 = r2553004 / r2553020;
        double r2553023 = r2553021 + r2553022;
        double r2553024 = 0.5;
        double r2553025 = r2553020 * r2553009;
        double r2553026 = r2553024 / r2553025;
        double r2553027 = r2553023 - r2553026;
        double r2553028 = log1p(r2553009);
        double r2553029 = r2553028 / r2553005;
        double r2553030 = exp(r2553029);
        double r2553031 = r2553030 - r2553012;
        double r2553032 = r2553017 ? r2553027 : r2553031;
        double r2553033 = r2553008 ? r2553015 : r2553032;
        return r2553033;
}

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 3 regimes

  2. if (/ 1 n) < -104130347.93584307

    1. Initial program 0

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

    3. Applied add-exp-log0

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

    if -104130347.93584307 < (/ 1 n) < 2.8321253549703665e-07

    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 33.1

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

    3. Simplified33.0

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

    if 2.8321253549703665e-07 < (/ 1 n)

    1. Initial program 23.4

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

    3. Applied add-exp-log23.5

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

    4. Applied pow-exp23.5

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

    5. Simplified1.1

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\frac{\log x}{n}}{x \cdot n} + \frac{1}{x \cdot n}\right) - \frac{\frac{1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))