Average Error: 29.5 → 22.3
Time: 29.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -550033767897.8672:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.948562593498665 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -550033767897.8672:\\
\;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 4.948562593498665 \cdot 10^{-24}:\\
\;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right) + \frac{\frac{1}{n}}{x}\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\

\end{array}
double f(double x, double n) {
        double r2128567 = x;
        double r2128568 = 1.0;
        double r2128569 = r2128567 + r2128568;
        double r2128570 = n;
        double r2128571 = r2128568 / r2128570;
        double r2128572 = pow(r2128569, r2128571);
        double r2128573 = pow(r2128567, r2128571);
        double r2128574 = r2128572 - r2128573;
        return r2128574;
}
↓
double f(double x, double n) {
        double r2128575 = 1.0;
        double r2128576 = n;
        double r2128577 = r2128575 / r2128576;
        double r2128578 = -550033767897.8672;
        bool r2128579 = r2128577 <= r2128578;
        double r2128580 = x;
        double r2128581 = r2128580 + r2128575;
        double r2128582 = 2.0;
        double r2128583 = r2128577 / r2128582;
        double r2128584 = pow(r2128581, r2128583);
        double r2128585 = pow(r2128580, r2128583);
        double r2128586 = r2128584 - r2128585;
        double r2128587 = log(r2128586);
        double r2128588 = exp(r2128587);
        double r2128589 = r2128585 + r2128584;
        double r2128590 = r2128588 * r2128589;
        double r2128591 = 4.948562593498665e-24;
        bool r2128592 = r2128577 <= r2128591;
        double r2128593 = log(r2128580);
        double r2128594 = r2128580 * r2128576;
        double r2128595 = r2128576 * r2128594;
        double r2128596 = r2128593 / r2128595;
        double r2128597 = 0.5;
        double r2128598 = r2128580 * r2128580;
        double r2128599 = r2128597 / r2128598;
        double r2128600 = r2128599 / r2128576;
        double r2128601 = r2128596 - r2128600;
        double r2128602 = r2128577 / r2128580;
        double r2128603 = r2128601 + r2128602;
        double r2128604 = r2128592 ? r2128603 : r2128590;
        double r2128605 = r2128579 ? r2128590 : r2128604;
        return r2128605;
}

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 (/ 1 n) < -550033767897.8672 or 4.948562593498665e-24 < (/ 1 n)

    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 sqr-pow9.6

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

    4. Applied sqr-pow9.6

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

    5. Applied difference-of-squares9.6

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

    7. Applied add-exp-log9.6

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

    if -550033767897.8672 < (/ 1 n) < 4.948562593498665e-24

    1. Initial program 44.3

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

    2. Taylor expanded around inf 32.3

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

    3. Simplified31.8

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

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -550033767897.8672:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.948562593498665 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))