Average Error: 32.1 → 24.2
Time: 17.7s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -10036768523789454:\\ \;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)\right)}^{3}}\right)}^{3}}}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.86323975548902723 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\log \left({\left(\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\right)\right)}^{3}}}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -10036768523789454:\\
\;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)\right)}^{3}}\right)}^{3}}}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.86323975548902723 \cdot 10^{-9}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r82998 = x;
        double r82999 = 1.0;
        double r83000 = r82998 + r82999;
        double r83001 = n;
        double r83002 = r82999 / r83001;
        double r83003 = pow(r83000, r83002);
        double r83004 = pow(r82998, r83002);
        double r83005 = r83003 - r83004;
        return r83005;
}
↓
double f(double x, double n) {
        double r83006 = 1.0;
        double r83007 = n;
        double r83008 = r83006 / r83007;
        double r83009 = -10036768523789454.0;
        bool r83010 = r83008 <= r83009;
        double r83011 = x;
        double r83012 = r83011 + r83006;
        double r83013 = pow(r83012, r83008);
        double r83014 = pow(r83011, r83008);
        double r83015 = r83013 - r83014;
        double r83016 = 3.0;
        double r83017 = pow(r83015, r83016);
        double r83018 = log(r83017);
        double r83019 = pow(r83018, r83016);
        double r83020 = cbrt(r83019);
        double r83021 = pow(r83020, r83016);
        double r83022 = cbrt(r83021);
        double r83023 = exp(r83022);
        double r83024 = cbrt(r83023);
        double r83025 = 1.8632397554890272e-09;
        bool r83026 = r83008 <= r83025;
        double r83027 = r83008 / r83011;
        double r83028 = 0.5;
        double r83029 = r83028 / r83007;
        double r83030 = 2.0;
        double r83031 = pow(r83011, r83030);
        double r83032 = r83029 / r83031;
        double r83033 = log(r83011);
        double r83034 = r83006 * r83033;
        double r83035 = pow(r83007, r83030);
        double r83036 = r83011 * r83035;
        double r83037 = r83034 / r83036;
        double r83038 = r83032 - r83037;
        double r83039 = r83027 - r83038;
        double r83040 = r83030 * r83008;
        double r83041 = pow(r83012, r83040);
        double r83042 = pow(r83011, r83040);
        double r83043 = -r83042;
        double r83044 = r83041 + r83043;
        double r83045 = r83013 + r83014;
        double r83046 = r83044 / r83045;
        double r83047 = pow(r83046, r83016);
        double r83048 = log(r83047);
        double r83049 = pow(r83048, r83016);
        double r83050 = cbrt(r83049);
        double r83051 = exp(r83050);
        double r83052 = cbrt(r83051);
        double r83053 = r83026 ? r83039 : r83052;
        double r83054 = r83010 ? r83024 : r83053;
        return r83054;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -10036768523789454.0

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Simplified0

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied add-exp-log0

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

    7. Applied pow-exp0

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

    8. Simplified0

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

    10. Applied add-cbrt-cube0

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

    11. Simplified0

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

    13. Applied add-cbrt-cube0

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

    14. Simplified0

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

    if -10036768523789454.0 < (/ 1.0 n) < 1.8632397554890272e-09

    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 33.6

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

    3. Simplified33.0

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

    if 1.8632397554890272e-09 < (/ 1.0 n)

    1. Initial program 5.9

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

    3. Applied add-cbrt-cube5.9

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

    4. Simplified5.9

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied add-exp-log5.9

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

    7. Applied pow-exp5.9

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

    8. Simplified5.9

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

    10. Applied add-cbrt-cube5.9

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

    11. Simplified5.9

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

    13. Applied flip--5.9

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

    14. Simplified5.9

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

  4. Final simplification24.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -10036768523789454:\\ \;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)\right)}^{3}}\right)}^{3}}}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.86323975548902723 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{1 \cdot \log x}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\sqrt[3]{{\left(\log \left({\left(\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\right)\right)}^{3}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))