Average Error: 29.6 → 22.0
Time: 36.0s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -47651231.911831676959991455078125:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{1}{x}, -\frac{\log x}{n \cdot n}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \mathbf{elif}\;n \le 1096.357908857971551697119139134883880615:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{1}{x}, -\frac{\log x}{n \cdot n}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -47651231.911831676959991455078125:\\
\;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{1}{x}, -\frac{\log x}{n \cdot n}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\

\mathbf{elif}\;n \le 1096.357908857971551697119139134883880615:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{1}{x}, -\frac{\log x}{n \cdot n}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\

\end{array}
double f(double x, double n) {
        double r2524859 = x;
        double r2524860 = 1.0;
        double r2524861 = r2524859 + r2524860;
        double r2524862 = n;
        double r2524863 = r2524860 / r2524862;
        double r2524864 = pow(r2524861, r2524863);
        double r2524865 = pow(r2524859, r2524863);
        double r2524866 = r2524864 - r2524865;
        return r2524866;
}
↓
double f(double x, double n) {
        double r2524867 = n;
        double r2524868 = -47651231.91183168;
        bool r2524869 = r2524867 <= r2524868;
        double r2524870 = 1.0;
        double r2524871 = x;
        double r2524872 = r2524870 / r2524871;
        double r2524873 = r2524872 / r2524867;
        double r2524874 = log(r2524871);
        double r2524875 = r2524867 * r2524867;
        double r2524876 = r2524874 / r2524875;
        double r2524877 = -r2524876;
        double r2524878 = 0.5;
        double r2524879 = r2524871 * r2524871;
        double r2524880 = r2524879 * r2524867;
        double r2524881 = r2524878 / r2524880;
        double r2524882 = fma(r2524872, r2524877, r2524881);
        double r2524883 = r2524873 - r2524882;
        double r2524884 = 1096.3579088579716;
        bool r2524885 = r2524867 <= r2524884;
        double r2524886 = r2524870 + r2524871;
        double r2524887 = r2524870 / r2524867;
        double r2524888 = pow(r2524886, r2524887);
        double r2524889 = pow(r2524871, r2524887);
        double r2524890 = r2524888 - r2524889;
        double r2524891 = exp(r2524890);
        double r2524892 = log(r2524891);
        double r2524893 = cbrt(r2524892);
        double r2524894 = r2524893 * r2524893;
        double r2524895 = r2524894 * r2524893;
        double r2524896 = r2524885 ? r2524895 : r2524883;
        double r2524897 = r2524869 ? r2524883 : r2524896;
        return r2524897;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -47651231.91183168 or 1096.3579088579716 < 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

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

    7. Taylor expanded around inf 32.2

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

    8. Simplified31.5

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

    if -47651231.91183168 < n < 1096.3579088579716

    1. Initial program 8.6

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

    3. Applied add-log-exp8.8

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

    4. Applied add-log-exp8.7

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

    5. Applied diff-log8.7

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

    6. Simplified8.7

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

    8. Applied add-cube-cbrt8.7

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -47651231.911831676959991455078125:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{1}{x}, -\frac{\log x}{n \cdot n}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \mathbf{elif}\;n \le 1096.357908857971551697119139134883880615:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(\frac{1}{x}, -\frac{\log x}{n \cdot n}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))