Average Error: 29.4 → 22.3
Time: 33.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -155965176.56480727:\\ \;\;\;\;\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{x \cdot n} - \frac{\frac{1}{4}}{x \cdot \left(x \cdot n\right)}\right) - \frac{\log x}{\left(x \cdot n\right) \cdot n} \cdot \frac{-1}{4}\right)\\ \mathbf{elif}\;n \le 113915774.22778098:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x \cdot n}}{n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -155965176.56480727:\\
\;\;\;\;\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{x \cdot n} - \frac{\frac{1}{4}}{x \cdot \left(x \cdot n\right)}\right) - \frac{\log x}{\left(x \cdot n\right) \cdot n} \cdot \frac{-1}{4}\right)\\

\mathbf{elif}\;n \le 113915774.22778098:\\
\;\;\;\;\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{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x \cdot n}}{n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\\

\end{array}
double f(double x, double n) {
        double r1809659 = x;
        double r1809660 = 1.0;
        double r1809661 = r1809659 + r1809660;
        double r1809662 = n;
        double r1809663 = r1809660 / r1809662;
        double r1809664 = pow(r1809661, r1809663);
        double r1809665 = pow(r1809659, r1809663);
        double r1809666 = r1809664 - r1809665;
        return r1809666;
}
↓
double f(double x, double n) {
        double r1809667 = n;
        double r1809668 = -155965176.56480727;
        bool r1809669 = r1809667 <= r1809668;
        double r1809670 = x;
        double r1809671 = 1.0;
        double r1809672 = r1809671 / r1809667;
        double r1809673 = 2.0;
        double r1809674 = r1809672 / r1809673;
        double r1809675 = pow(r1809670, r1809674);
        double r1809676 = r1809670 + r1809671;
        double r1809677 = pow(r1809676, r1809674);
        double r1809678 = r1809675 + r1809677;
        double r1809679 = 0.5;
        double r1809680 = r1809670 * r1809667;
        double r1809681 = r1809679 / r1809680;
        double r1809682 = 0.25;
        double r1809683 = r1809670 * r1809680;
        double r1809684 = r1809682 / r1809683;
        double r1809685 = r1809681 - r1809684;
        double r1809686 = log(r1809670);
        double r1809687 = r1809680 * r1809667;
        double r1809688 = r1809686 / r1809687;
        double r1809689 = -0.25;
        double r1809690 = r1809688 * r1809689;
        double r1809691 = r1809685 - r1809690;
        double r1809692 = r1809678 * r1809691;
        double r1809693 = 113915774.22778098;
        bool r1809694 = r1809667 <= r1809693;
        double r1809695 = r1809677 - r1809675;
        double r1809696 = r1809695 * r1809678;
        double r1809697 = r1809672 / r1809670;
        double r1809698 = r1809686 / r1809680;
        double r1809699 = r1809698 / r1809667;
        double r1809700 = r1809679 / r1809670;
        double r1809701 = r1809700 / r1809680;
        double r1809702 = r1809699 - r1809701;
        double r1809703 = r1809697 + r1809702;
        double r1809704 = r1809694 ? r1809696 : r1809703;
        double r1809705 = r1809669 ? r1809692 : r1809704;
        return r1809705;
}

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 n < -155965176.56480727

    1. Initial program 44.5

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

    3. Applied sqr-pow44.6

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

    4. Applied sqr-pow44.5

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

    5. Applied difference-of-squares44.5

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

    6. Taylor expanded around inf 32.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{4} \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot n}\right)\right)}\]

    7. Simplified32.7

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\frac{\frac{1}{2}}{n \cdot x} - \frac{\frac{1}{4}}{x \cdot \left(n \cdot x\right)}\right) - \frac{-\log x}{n \cdot \left(n \cdot x\right)} \cdot \frac{1}{4}\right)}\]

    if -155965176.56480727 < n < 113915774.22778098

    1. Initial program 8.3

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

    3. Applied sqr-pow8.3

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

    4. Applied sqr-pow8.3

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

    5. Applied difference-of-squares8.3

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

    if 113915774.22778098 < n

    1. Initial program 45.1

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

    2. Taylor expanded around inf 32.6

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

    3. Simplified32.1

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

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -155965176.56480727:\\ \;\;\;\;\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{x \cdot n} - \frac{\frac{1}{4}}{x \cdot \left(x \cdot n\right)}\right) - \frac{\log x}{\left(x \cdot n\right) \cdot n} \cdot \frac{-1}{4}\right)\\ \mathbf{elif}\;n \le 113915774.22778098:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x \cdot n}}{n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))