Average Error: 29.3 → 22.6
Time: 10.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.62886214871615791 \cdot 10^{-17}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.0466841968524116 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.62886214871615791 \cdot 10^{-17}:\\
\;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 4.0466841968524116 \cdot 10^{-15}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\

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

\end{array}
double f(double x, double n) {
        double r52850 = x;
        double r52851 = 1.0;
        double r52852 = r52850 + r52851;
        double r52853 = n;
        double r52854 = r52851 / r52853;
        double r52855 = pow(r52852, r52854);
        double r52856 = pow(r52850, r52854);
        double r52857 = r52855 - r52856;
        return r52857;
}
↓
double f(double x, double n) {
        double r52858 = 1.0;
        double r52859 = n;
        double r52860 = r52858 / r52859;
        double r52861 = -3.628862148716158e-17;
        bool r52862 = r52860 <= r52861;
        double r52863 = x;
        double r52864 = r52863 + r52858;
        double r52865 = pow(r52864, r52860);
        double r52866 = pow(r52863, r52860);
        double r52867 = r52865 - r52866;
        double r52868 = exp(r52867);
        double r52869 = log(r52868);
        double r52870 = 4.046684196852412e-15;
        bool r52871 = r52860 <= r52870;
        double r52872 = 1.0;
        double r52873 = r52863 * r52859;
        double r52874 = r52872 / r52873;
        double r52875 = 0.5;
        double r52876 = 2.0;
        double r52877 = pow(r52863, r52876);
        double r52878 = r52877 * r52859;
        double r52879 = r52872 / r52878;
        double r52880 = r52872 / r52863;
        double r52881 = log(r52880);
        double r52882 = pow(r52859, r52876);
        double r52883 = r52863 * r52882;
        double r52884 = r52881 / r52883;
        double r52885 = r52858 * r52884;
        double r52886 = fma(r52875, r52879, r52885);
        double r52887 = -r52886;
        double r52888 = fma(r52858, r52874, r52887);
        double r52889 = sqrt(r52869);
        double r52890 = sqrt(r52867);
        double r52891 = r52889 * r52890;
        double r52892 = r52871 ? r52888 : r52891;
        double r52893 = r52862 ? r52869 : r52892;
        return r52893;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -3.628862148716158e-17

    1. Initial program 2.1

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

    3. Applied add-log-exp2.5

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

    4. Applied add-log-exp2.4

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

    5. Applied diff-log2.4

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

    6. Simplified2.4

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

    if -3.628862148716158e-17 < (/ 1.0 n) < 4.046684196852412e-15

    1. Initial program 44.9

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

    2. Taylor expanded around inf 32.7

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

    3. Simplified32.7

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

    if 4.046684196852412e-15 < (/ 1.0 n)

    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-sqr-sqrt25.1

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

    5. Applied add-log-exp25.1

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

    6. Applied add-log-exp25.1

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

    7. Applied diff-log25.1

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

    8. Simplified25.1

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

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.62886214871615791 \cdot 10^{-17}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.0466841968524116 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))