Average Error: 29.4 → 18.9
Time: 3.4m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1650.055784837749:\\ \;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\\ \mathbf{elif}\;n \le 9.8058607172592 \cdot 10^{-311}:\\ \;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)}\\ \mathbf{elif}\;n \le 271.3240147216949:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\\ \end{array}\]
double f(double x, double n) {
        double r14908862 = x;
        double r14908863 = 1.0;
        double r14908864 = r14908862 + r14908863;
        double r14908865 = n;
        double r14908866 = r14908863 / r14908865;
        double r14908867 = pow(r14908864, r14908866);
        double r14908868 = pow(r14908862, r14908866);
        double r14908869 = r14908867 - r14908868;
        return r14908869;
}
↓
double f(double x, double n) {
        double r14908870 = n;
        double r14908871 = -1650.055784837749;
        bool r14908872 = r14908870 <= r14908871;
        double r14908873 = x;
        double r14908874 = 1.0;
        double r14908875 = r14908874 / r14908870;
        double r14908876 = 2.0;
        double r14908877 = r14908875 / r14908876;
        double r14908878 = pow(r14908873, r14908877);
        double r14908879 = log(r14908878);
        double r14908880 = exp(r14908879);
        double r14908881 = r14908873 + r14908874;
        double r14908882 = pow(r14908881, r14908877);
        double r14908883 = r14908880 + r14908882;
        double r14908884 = 0.25;
        double r14908885 = r14908884 / r14908870;
        double r14908886 = r14908885 / r14908870;
        double r14908887 = log(r14908873);
        double r14908888 = r14908887 / r14908873;
        double r14908889 = 0.5;
        double r14908890 = r14908873 * r14908870;
        double r14908891 = r14908889 / r14908890;
        double r14908892 = fma(r14908886, r14908888, r14908891);
        double r14908893 = r14908873 * r14908873;
        double r14908894 = r14908885 / r14908893;
        double r14908895 = r14908892 - r14908894;
        double r14908896 = r14908883 * r14908895;
        double r14908897 = 9.8058607172592e-311;
        bool r14908898 = r14908870 <= r14908897;
        double r14908899 = r14908882 - r14908880;
        double r14908900 = r14908899 * r14908899;
        double r14908901 = r14908900 * r14908899;
        double r14908902 = cbrt(r14908901);
        double r14908903 = r14908883 * r14908902;
        double r14908904 = 271.3240147216949;
        bool r14908905 = r14908870 <= r14908904;
        double r14908906 = log1p(r14908873);
        double r14908907 = r14908906 / r14908870;
        double r14908908 = exp(r14908907);
        double r14908909 = pow(r14908873, r14908875);
        double r14908910 = r14908908 - r14908909;
        double r14908911 = r14908905 ? r14908910 : r14908896;
        double r14908912 = r14908898 ? r14908903 : r14908911;
        double r14908913 = r14908872 ? r14908896 : r14908912;
        return r14908913;
}


{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1650.055784837749:\\
\;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\\

\mathbf{elif}\;n \le 9.8058607172592 \cdot 10^{-311}:\\
\;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)}\\

\mathbf{elif}\;n \le 271.3240147216949:\\
\;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\\

\end{array}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -1650.055784837749 or 271.3240147216949 < n

    1. Initial program 44.6

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

    3. Applied add-exp-log44.6

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

    5. Applied sqr-pow44.7

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

    6. Applied log-prod44.7

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

    7. Applied exp-sum44.7

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

    8. Applied sqr-pow44.7

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

    9. Applied difference-of-squares44.7

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

    10. Taylor expanded around inf 32.4

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

    11. Simplified32.4

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

    if -1650.055784837749 < n < 9.8058607172592e-311

    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 add-exp-log0.2

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

    5. Applied sqr-pow0.2

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

    6. Applied log-prod0.2

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

    7. Applied exp-sum0.2

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

    8. Applied sqr-pow0.2

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

    9. Applied difference-of-squares0.2

      \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)}\]
    10. Using strategy rm

    11. Applied add-cbrt-cube0.2

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

    if 9.8058607172592e-311 < n < 271.3240147216949

    1. Initial program 25.1

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

    3. Applied add-exp-log25.1

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

    4. Applied pow-exp25.1

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

    5. Simplified0.5

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

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1650.055784837749:\\ \;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\\ \mathbf{elif}\;n \le 9.8058607172592 \cdot 10^{-311}:\\ \;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)}\\ \mathbf{elif}\;n \le 271.3240147216949:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))