Average Error: 29.7 → 22.4
Time: 32.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1303502725104472424448:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{0.5}{x \cdot n}}{x}\right) + \frac{1 \cdot \log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 9526334.6405404396355152130126953125:\\ \;\;\;\;\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 \cdot \log x}{\left(x \cdot n\right) \cdot n} + \frac{\frac{1}{x}}{n}\right) - \frac{\frac{0.5}{x \cdot x}}{n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1303502725104472424448:\\
\;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{0.5}{x \cdot n}}{x}\right) + \frac{1 \cdot \log x}{\left(x \cdot n\right) \cdot n}\\

\mathbf{elif}\;n \le 9526334.6405404396355152130126953125:\\
\;\;\;\;\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}:\\
\;\;\;\;\left(\frac{1 \cdot \log x}{\left(x \cdot n\right) \cdot n} + \frac{\frac{1}{x}}{n}\right) - \frac{\frac{0.5}{x \cdot x}}{n}\\

\end{array}
double f(double x, double n) {
        double r3189816 = x;
        double r3189817 = 1.0;
        double r3189818 = r3189816 + r3189817;
        double r3189819 = n;
        double r3189820 = r3189817 / r3189819;
        double r3189821 = pow(r3189818, r3189820);
        double r3189822 = pow(r3189816, r3189820);
        double r3189823 = r3189821 - r3189822;
        return r3189823;
}
↓
double f(double x, double n) {
        double r3189824 = n;
        double r3189825 = -1.3035027251044724e+21;
        bool r3189826 = r3189824 <= r3189825;
        double r3189827 = 1.0;
        double r3189828 = x;
        double r3189829 = r3189828 * r3189824;
        double r3189830 = r3189827 / r3189829;
        double r3189831 = 0.5;
        double r3189832 = r3189831 / r3189829;
        double r3189833 = r3189832 / r3189828;
        double r3189834 = r3189830 - r3189833;
        double r3189835 = log(r3189828);
        double r3189836 = r3189827 * r3189835;
        double r3189837 = r3189829 * r3189824;
        double r3189838 = r3189836 / r3189837;
        double r3189839 = r3189834 + r3189838;
        double r3189840 = 9526334.64054044;
        bool r3189841 = r3189824 <= r3189840;
        double r3189842 = r3189827 + r3189828;
        double r3189843 = r3189827 / r3189824;
        double r3189844 = pow(r3189842, r3189843);
        double r3189845 = sqrt(r3189844);
        double r3189846 = r3189845 * r3189845;
        double r3189847 = pow(r3189828, r3189843);
        double r3189848 = r3189846 - r3189847;
        double r3189849 = r3189827 / r3189828;
        double r3189850 = r3189849 / r3189824;
        double r3189851 = r3189838 + r3189850;
        double r3189852 = r3189828 * r3189828;
        double r3189853 = r3189831 / r3189852;
        double r3189854 = r3189853 / r3189824;
        double r3189855 = r3189851 - r3189854;
        double r3189856 = r3189841 ? r3189848 : r3189855;
        double r3189857 = r3189826 ? r3189839 : r3189856;
        return r3189857;
}

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 < -1.3035027251044724e+21

    1. Initial program 45.9

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

    3. Applied add-sqr-sqrt45.9

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

    4. Taylor expanded around inf 32.6

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

    5. Simplified32.4

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

    if -1.3035027251044724e+21 < n < 9526334.64054044

    1. Initial program 9.4

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

    3. Applied add-sqr-sqrt9.5

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

    if 9526334.64054044 < n

    1. Initial program 44.7

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

    3. Applied add-log-exp44.7

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

    4. Applied add-log-exp44.7

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

    5. Applied diff-log44.7

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

    6. Simplified44.7

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

    7. Taylor expanded around inf 32.7

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

    8. Simplified32.0

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1303502725104472424448:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{0.5}{x \cdot n}}{x}\right) + \frac{1 \cdot \log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 9526334.6405404396355152130126953125:\\ \;\;\;\;\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 \cdot \log x}{\left(x \cdot n\right) \cdot n} + \frac{\frac{1}{x}}{n}\right) - \frac{\frac{0.5}{x \cdot x}}{n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))