Average Error: 29.1 → 22.1
Time: 30.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.062563002021386395830214673147437801342 \cdot 10^{-15}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.907831219059099599129334134830144599578 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) + \log \left(e^{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -4.062563002021386395830214673147437801342 \cdot 10^{-15}:\\
\;\;\;\;\sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 2.907831219059099599129334134830144599578 \cdot 10^{-26}:\\
\;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\

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

\end{array}
double f(double x, double n) {
        double r2898791 = x;
        double r2898792 = 1.0;
        double r2898793 = r2898791 + r2898792;
        double r2898794 = n;
        double r2898795 = r2898792 / r2898794;
        double r2898796 = pow(r2898793, r2898795);
        double r2898797 = pow(r2898791, r2898795);
        double r2898798 = r2898796 - r2898797;
        return r2898798;
}
↓
double f(double x, double n) {
        double r2898799 = 1.0;
        double r2898800 = n;
        double r2898801 = r2898799 / r2898800;
        double r2898802 = -4.0625630020213864e-15;
        bool r2898803 = r2898801 <= r2898802;
        double r2898804 = 1.0;
        double r2898805 = x;
        double r2898806 = r2898799 + r2898805;
        double r2898807 = pow(r2898806, r2898801);
        double r2898808 = 2.0;
        double r2898809 = r2898801 / r2898808;
        double r2898810 = pow(r2898805, r2898809);
        double r2898811 = r2898810 * r2898810;
        double r2898812 = -r2898811;
        double r2898813 = fma(r2898804, r2898807, r2898812);
        double r2898814 = cbrt(r2898813);
        double r2898815 = r2898814 * r2898814;
        double r2898816 = r2898814 * r2898815;
        double r2898817 = -r2898810;
        double r2898818 = fma(r2898817, r2898810, r2898811);
        double r2898819 = r2898816 + r2898818;
        double r2898820 = 2.9078312190590996e-26;
        bool r2898821 = r2898801 <= r2898820;
        double r2898822 = r2898799 / r2898805;
        double r2898823 = r2898822 / r2898800;
        double r2898824 = log(r2898805);
        double r2898825 = r2898800 * r2898800;
        double r2898826 = r2898824 / r2898825;
        double r2898827 = r2898826 / r2898805;
        double r2898828 = -r2898827;
        double r2898829 = 0.5;
        double r2898830 = r2898829 / r2898800;
        double r2898831 = r2898805 * r2898805;
        double r2898832 = r2898830 / r2898831;
        double r2898833 = fma(r2898828, r2898799, r2898832);
        double r2898834 = r2898823 - r2898833;
        double r2898835 = exp(r2898813);
        double r2898836 = log(r2898835);
        double r2898837 = exp(r2898818);
        double r2898838 = log(r2898837);
        double r2898839 = r2898836 + r2898838;
        double r2898840 = r2898821 ? r2898834 : r2898839;
        double r2898841 = r2898803 ? r2898819 : r2898840;
        return r2898841;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -4.0625630020213864e-15

    1. Initial program 1.3

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

    3. Applied sqr-pow1.3

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

    4. Applied *-un-lft-identity1.3

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

    5. Applied prod-diff1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt1.4

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

    if -4.0625630020213864e-15 < (/ 1.0 n) < 2.9078312190590996e-26

    1. Initial program 44.7

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

    2. Taylor expanded around inf 32.2

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

    3. Simplified31.7

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

    if 2.9078312190590996e-26 < (/ 1.0 n)

    1. Initial program 28.1

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

    3. Applied sqr-pow28.3

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

    4. Applied *-un-lft-identity28.3

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

    5. Applied prod-diff28.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\]
    6. Using strategy rm

    7. Applied add-log-exp28.3

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

    9. Applied add-log-exp28.4

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.062563002021386395830214673147437801342 \cdot 10^{-15}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) + \mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.907831219059099599129334134830144599578 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(-\frac{\frac{\log x}{n \cdot n}}{x}, 1, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) + \log \left(e^{\mathsf{fma}\left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))