Average Error: 29.4 → 22.0
Time: 31.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -48075474.504670314:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \mathbf{elif}\;n \le 55908.36668550803:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -48075474.504670314:\\
\;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\

\mathbf{elif}\;n \le 55908.36668550803:\\
\;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r3533962 = x;
        double r3533963 = 1.0;
        double r3533964 = r3533962 + r3533963;
        double r3533965 = n;
        double r3533966 = r3533963 / r3533965;
        double r3533967 = pow(r3533964, r3533966);
        double r3533968 = pow(r3533962, r3533966);
        double r3533969 = r3533967 - r3533968;
        return r3533969;
}
↓
double f(double x, double n) {
        double r3533970 = n;
        double r3533971 = -48075474.504670314;
        bool r3533972 = r3533970 <= r3533971;
        double r3533973 = 1.0;
        double r3533974 = r3533973 / r3533970;
        double r3533975 = x;
        double r3533976 = r3533974 / r3533975;
        double r3533977 = log(r3533975);
        double r3533978 = r3533977 / r3533975;
        double r3533979 = r3533970 * r3533970;
        double r3533980 = r3533978 / r3533979;
        double r3533981 = 0.5;
        double r3533982 = r3533975 * r3533975;
        double r3533983 = r3533981 / r3533982;
        double r3533984 = r3533983 / r3533970;
        double r3533985 = r3533980 - r3533984;
        double r3533986 = r3533976 + r3533985;
        double r3533987 = 55908.36668550803;
        bool r3533988 = r3533970 <= r3533987;
        double r3533989 = r3533973 + r3533975;
        double r3533990 = pow(r3533989, r3533974);
        double r3533991 = pow(r3533975, r3533974);
        double r3533992 = r3533990 - r3533991;
        double r3533993 = cbrt(r3533992);
        double r3533994 = sqrt(r3533990);
        double r3533995 = sqrt(r3533991);
        double r3533996 = r3533994 - r3533995;
        double r3533997 = r3533995 + r3533994;
        double r3533998 = r3533996 * r3533997;
        double r3533999 = cbrt(r3533998);
        double r3534000 = r3533999 * r3533993;
        double r3534001 = exp(r3534000);
        double r3534002 = log(r3534001);
        double r3534003 = r3533993 * r3534002;
        double r3534004 = r3533988 ? r3534003 : r3533986;
        double r3534005 = r3533972 ? r3533986 : r3534004;
        return r3534005;
}

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 n < -48075474.504670314 or 55908.36668550803 < n

    1. Initial program 45.6

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

    2. Taylor expanded around inf 33.2

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

    3. Simplified32.4

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

    if -48075474.504670314 < n < 55908.36668550803

    1. Initial program 8.2

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

    3. Applied add-log-exp8.3

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

    4. Applied add-log-exp8.3

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

    5. Applied diff-log8.3

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

    6. Simplified8.3

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

    8. Applied add-cube-cbrt8.3

      \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\]

    9. Applied exp-prod8.3

      \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right)}\]

    10. Applied log-pow8.3

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt8.3

      \[\leadsto \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}}\right)\]

    13. Applied add-sqr-sqrt8.3

      \[\leadsto \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\]

    14. Applied difference-of-squares8.3

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -48075474.504670314:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \mathbf{elif}\;n \le 55908.36668550803:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))