Average Error: 29.7 → 22.2
Time: 1.0m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.011658222809468368:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 6.044282672011111 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\left(x \cdot n\right) \cdot x} + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{x \cdot \left(n \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.011658222809468368:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 6.044282672011111 \cdot 10^{-10}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\left(x \cdot n\right) \cdot x} + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{x \cdot \left(n \cdot n\right)}\right)\\

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

\end{array}
double f(double x, double n) {
        double r5060699 = x;
        double r5060700 = 1.0;
        double r5060701 = r5060699 + r5060700;
        double r5060702 = n;
        double r5060703 = r5060700 / r5060702;
        double r5060704 = pow(r5060701, r5060703);
        double r5060705 = pow(r5060699, r5060703);
        double r5060706 = r5060704 - r5060705;
        return r5060706;
}
↓
double f(double x, double n) {
        double r5060707 = 1.0;
        double r5060708 = n;
        double r5060709 = r5060707 / r5060708;
        double r5060710 = -0.011658222809468368;
        bool r5060711 = r5060709 <= r5060710;
        double r5060712 = x;
        double r5060713 = r5060712 + r5060707;
        double r5060714 = pow(r5060713, r5060709);
        double r5060715 = 2.0;
        double r5060716 = r5060709 / r5060715;
        double r5060717 = pow(r5060712, r5060716);
        double r5060718 = r5060717 * r5060717;
        double r5060719 = r5060714 - r5060718;
        double r5060720 = 6.044282672011111e-10;
        bool r5060721 = r5060709 <= r5060720;
        double r5060722 = -0.5;
        double r5060723 = r5060712 * r5060708;
        double r5060724 = r5060723 * r5060712;
        double r5060725 = r5060722 / r5060724;
        double r5060726 = r5060707 / r5060712;
        double r5060727 = r5060726 / r5060708;
        double r5060728 = log(r5060712);
        double r5060729 = r5060708 * r5060708;
        double r5060730 = r5060712 * r5060729;
        double r5060731 = r5060728 / r5060730;
        double r5060732 = r5060727 + r5060731;
        double r5060733 = r5060725 + r5060732;
        double r5060734 = pow(r5060712, r5060709);
        double r5060735 = r5060714 - r5060734;
        double r5060736 = log(r5060735);
        double r5060737 = exp(r5060736);
        double r5060738 = r5060721 ? r5060733 : r5060737;
        double r5060739 = r5060711 ? r5060719 : r5060738;
        return r5060739;
}

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) < -0.011658222809468368

    1. Initial program 0.2

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

    3. Applied sqr-pow0.2

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

    if -0.011658222809468368 < (/ 1 n) < 6.044282672011111e-10

    1. Initial program 44.9

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

    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. Simplified32.6

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

    4. Taylor expanded around 0 32.6

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

    5. Simplified32.0

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

    if 6.044282672011111e-10 < (/ 1 n)

    1. Initial program 25.9

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

    3. Applied add-exp-log25.9

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.011658222809468368:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 6.044282672011111 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\left(x \cdot n\right) \cdot x} + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{x \cdot \left(n \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))