Average Error: 29.4 → 21.9
Time: 24.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -594192892.16520202159881591796875:\\ \;\;\;\;1 \cdot \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;n \le 61241.10698845344450091943144798278808594:\\ \;\;\;\;\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\right) - \log \left(e^{\frac{\frac{0.5}{n}}{x \cdot x}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -594192892.16520202159881591796875:\\
\;\;\;\;1 \cdot \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\

\mathbf{elif}\;n \le 61241.10698845344450091943144798278808594:\\
\;\;\;\;\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\right) - \log \left(e^{\frac{\frac{0.5}{n}}{x \cdot x}}\right)\\

\end{array}
double f(double x, double n) {
        double r64694 = x;
        double r64695 = 1.0;
        double r64696 = r64694 + r64695;
        double r64697 = n;
        double r64698 = r64695 / r64697;
        double r64699 = pow(r64696, r64698);
        double r64700 = pow(r64694, r64698);
        double r64701 = r64699 - r64700;
        return r64701;
}
↓
double f(double x, double n) {
        double r64702 = n;
        double r64703 = -594192892.165202;
        bool r64704 = r64702 <= r64703;
        double r64705 = 1.0;
        double r64706 = 1.0;
        double r64707 = r64706 / r64702;
        double r64708 = x;
        double r64709 = r64707 / r64708;
        double r64710 = log(r64708);
        double r64711 = -r64710;
        double r64712 = r64702 * r64702;
        double r64713 = r64712 * r64708;
        double r64714 = r64711 / r64713;
        double r64715 = r64709 - r64714;
        double r64716 = r64705 * r64715;
        double r64717 = 0.5;
        double r64718 = r64717 / r64702;
        double r64719 = r64708 * r64708;
        double r64720 = r64718 / r64719;
        double r64721 = r64716 - r64720;
        double r64722 = 61241.106988453445;
        bool r64723 = r64702 <= r64722;
        double r64724 = r64705 + r64708;
        double r64725 = r64705 / r64702;
        double r64726 = pow(r64724, r64725);
        double r64727 = sqrt(r64726);
        double r64728 = r64727 * r64727;
        double r64729 = pow(r64708, r64725);
        double r64730 = r64728 - r64729;
        double r64731 = exp(r64720);
        double r64732 = log(r64731);
        double r64733 = r64716 - r64732;
        double r64734 = r64723 ? r64730 : r64733;
        double r64735 = r64704 ? r64721 : r64734;
        return r64735;
}

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 < -594192892.165202

    1. Initial program 44.8

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

    2. Taylor expanded around inf 32.0

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

    3. Simplified31.3

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

    4. Taylor expanded around inf 32.0

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

    5. Simplified31.3

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

    if -594192892.165202 < n < 61241.106988453445

    1. Initial program 8.8

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

    3. Applied add-sqr-sqrt8.8

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

    4. Simplified8.8

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

    5. Simplified8.8

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

    if 61241.106988453445 < n

    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

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

    3. Simplified31.9

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

    4. Taylor expanded around inf 32.6

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

    5. Simplified31.9

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{x \cdot \left(n \cdot n\right)}\right) \cdot 1} - \frac{\frac{0.5}{n}}{x \cdot x}\]
    6. Using strategy rm

    7. Applied add-log-exp32.0

      \[\leadsto \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{x \cdot \left(n \cdot n\right)}\right) \cdot 1 - \color{blue}{\log \left(e^{\frac{\frac{0.5}{n}}{x \cdot x}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -594192892.16520202159881591796875:\\ \;\;\;\;1 \cdot \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\right) - \frac{\frac{0.5}{n}}{x \cdot x}\\ \mathbf{elif}\;n \le 61241.10698845344450091943144798278808594:\\ \;\;\;\;\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(n \cdot n\right) \cdot x}\right) - \log \left(e^{\frac{\frac{0.5}{n}}{x \cdot x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))