Average Error: 29.3 → 19.4
Time: 5.5m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3253307.1591160325:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{1}{x \cdot \left(x \cdot n\right)}\right), \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right)\right)\\ \mathbf{elif}\;n \le -3.274682954984462 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\left(\log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}} \cdot \sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)\right)\\ \mathbf{elif}\;n \le 1.3579770284385667 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{1}{x \cdot \left(x \cdot n\right)}\right), \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right)\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -3253307.1591160325:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{1}{x \cdot \left(x \cdot n\right)}\right), \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right)\right)\\

\mathbf{elif}\;n \le -3.274682954984462 \cdot 10^{-305}:\\
\;\;\;\;\mathsf{log1p}\left(\left(\left(\log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}} \cdot \sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)\right)\\

\mathbf{elif}\;n \le 1.3579770284385667 \cdot 10^{+23}:\\
\;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)\\

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

\end{array}
double f(double x, double n) {
        double r30905922 = x;
        double r30905923 = 1.0;
        double r30905924 = r30905922 + r30905923;
        double r30905925 = n;
        double r30905926 = r30905923 / r30905925;
        double r30905927 = pow(r30905924, r30905926);
        double r30905928 = pow(r30905922, r30905926);
        double r30905929 = r30905927 - r30905928;
        return r30905929;
}
↓
double f(double x, double n) {
        double r30905930 = n;
        double r30905931 = -3253307.1591160325;
        bool r30905932 = r30905930 <= r30905931;
        double r30905933 = -0.5;
        double r30905934 = 1.0;
        double r30905935 = x;
        double r30905936 = r30905935 * r30905930;
        double r30905937 = r30905935 * r30905936;
        double r30905938 = r30905934 / r30905937;
        double r30905939 = r30905934 / r30905936;
        double r30905940 = log(r30905935);
        double r30905941 = -r30905940;
        double r30905942 = r30905930 * r30905936;
        double r30905943 = r30905941 / r30905942;
        double r30905944 = r30905939 - r30905943;
        double r30905945 = fma(r30905933, r30905938, r30905944);
        double r30905946 = -3.274682954984462e-305;
        bool r30905947 = r30905930 <= r30905946;
        double r30905948 = r30905935 + r30905934;
        double r30905949 = r30905934 / r30905930;
        double r30905950 = pow(r30905948, r30905949);
        double r30905951 = pow(r30905935, r30905949);
        double r30905952 = r30905950 - r30905951;
        double r30905953 = expm1(r30905952);
        double r30905954 = cbrt(r30905953);
        double r30905955 = r30905954 * r30905954;
        double r30905956 = exp(r30905955);
        double r30905957 = cbrt(r30905956);
        double r30905958 = r30905957 * r30905957;
        double r30905959 = log(r30905958);
        double r30905960 = log(r30905957);
        double r30905961 = r30905959 + r30905960;
        double r30905962 = r30905961 * r30905954;
        double r30905963 = log1p(r30905962);
        double r30905964 = 1.3579770284385667e+23;
        bool r30905965 = r30905930 <= r30905964;
        double r30905966 = log1p(r30905935);
        double r30905967 = r30905966 / r30905930;
        double r30905968 = exp(r30905967);
        double r30905969 = r30905968 - r30905951;
        double r30905970 = expm1(r30905969);
        double r30905971 = log1p(r30905970);
        double r30905972 = r30905965 ? r30905971 : r30905945;
        double r30905973 = r30905947 ? r30905963 : r30905972;
        double r30905974 = r30905932 ? r30905945 : r30905973;
        return r30905974;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -3253307.1591160325 or 1.3579770284385667e+23 < n

    1. Initial program 44.0

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

    2. Taylor expanded around inf 31.7

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

    3. Simplified31.7

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

    if -3253307.1591160325 < n < -3.274682954984462e-305

    1. Initial program 0.6

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

    3. Applied log1p-expm1-u0.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp0.7

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\log \left(e^{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)\right)}\right)\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.7

      \[\leadsto \mathsf{log1p}\left(\left(\log \left(e^{\color{blue}{\left(\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right)\right)\right)\]

    8. Applied exp-prod0.7

      \[\leadsto \mathsf{log1p}\left(\left(\log \color{blue}{\left({\left(e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}\right)}^{\left(\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)}\right)}\right)\right)\]

    9. Applied log-pow0.7

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \log \left(e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}\right)\right)}\right)\]
    10. Using strategy rm

    11. Applied add-cube-cbrt0.7

      \[\leadsto \mathsf{log1p}\left(\left(\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}} \cdot \sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right) \cdot \sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right)}\right)\right)\]

    12. Applied log-prod0.7

      \[\leadsto \mathsf{log1p}\left(\left(\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}} \cdot \sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right)\right)}\right)\right)\]

    if -3.274682954984462e-305 < n < 1.3579770284385667e+23

    1. Initial program 26.9

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

    3. Applied log1p-expm1-u27.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied add-exp-log27.0

      \[\leadsto \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)\]

    6. Simplified8.2

      \[\leadsto \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification19.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3253307.1591160325:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{1}{x \cdot \left(x \cdot n\right)}\right), \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right)\right)\\ \mathbf{elif}\;n \le -3.274682954984462 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\left(\log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}} \cdot \sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}}}\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\right)\right)\\ \mathbf{elif}\;n \le 1.3579770284385667 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{1}{x \cdot \left(x \cdot n\right)}\right), \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))