Average Error: 29.8 → 22.5
Time: 30.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.1647883017738754 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{\left(n \cdot n\right) \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{elif}\;n \le 7547414330764314.0:\\ \;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\log \left(\frac{1}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \left(1 - \log \left(\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{\left(n \cdot n\right) \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -3.1647883017738754 \cdot 10^{+21}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{\left(n \cdot n\right) \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\

\mathbf{elif}\;n \le 7547414330764314.0:\\
\;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\log \left(\frac{1}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \left(1 - \log \left(\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)}\\

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

\end{array}
double f(double x, double n) {
        double r3059106 = x;
        double r3059107 = 1.0;
        double r3059108 = r3059106 + r3059107;
        double r3059109 = n;
        double r3059110 = r3059107 / r3059109;
        double r3059111 = pow(r3059108, r3059110);
        double r3059112 = pow(r3059106, r3059110);
        double r3059113 = r3059111 - r3059112;
        return r3059113;
}
↓
double f(double x, double n) {
        double r3059114 = n;
        double r3059115 = -3.1647883017738754e+21;
        bool r3059116 = r3059114 <= r3059115;
        double r3059117 = 1.0;
        double r3059118 = r3059117 / r3059114;
        double r3059119 = x;
        double r3059120 = r3059118 / r3059119;
        double r3059121 = log(r3059119);
        double r3059122 = r3059114 * r3059114;
        double r3059123 = r3059122 * r3059119;
        double r3059124 = r3059121 / r3059123;
        double r3059125 = 0.5;
        double r3059126 = r3059125 / r3059114;
        double r3059127 = r3059119 * r3059119;
        double r3059128 = r3059126 / r3059127;
        double r3059129 = r3059124 - r3059128;
        double r3059130 = r3059120 + r3059129;
        double r3059131 = 7547414330764314.0;
        bool r3059132 = r3059114 <= r3059131;
        double r3059133 = r3059117 + r3059119;
        double r3059134 = pow(r3059133, r3059118);
        double r3059135 = pow(r3059119, r3059118);
        double r3059136 = r3059134 - r3059135;
        double r3059137 = cbrt(r3059136);
        double r3059138 = r3059137 * r3059137;
        double r3059139 = exp(r3059135);
        double r3059140 = sqrt(r3059139);
        double r3059141 = r3059117 / r3059140;
        double r3059142 = log(r3059141);
        double r3059143 = log(r3059140);
        double r3059144 = r3059117 - r3059143;
        double r3059145 = r3059142 + r3059144;
        double r3059146 = cbrt(r3059145);
        double r3059147 = r3059138 * r3059146;
        double r3059148 = r3059132 ? r3059147 : r3059130;
        double r3059149 = r3059116 ? r3059130 : r3059148;
        return r3059149;
}

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 2 regimes

  2. if n < -3.1647883017738754e+21 or 7547414330764314.0 < n

    1. Initial program 44.8

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

    2. Taylor expanded around inf 32.2

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

    3. Simplified31.5

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

    if -3.1647883017738754e+21 < n < 7547414330764314.0

    1. Initial program 10.8

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

    3. Applied add-cube-cbrt10.8

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

    5. Applied add-log-exp10.9

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

    6. Applied add-log-exp10.9

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

    7. Applied diff-log10.9

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

    9. Applied add-sqr-sqrt10.9

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

    10. Applied *-un-lft-identity10.9

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

    11. Applied times-frac10.9

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

    12. Applied log-prod10.9

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

    13. Simplified10.9

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

    14. Taylor expanded around 0 10.9

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.1647883017738754 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{\left(n \cdot n\right) \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{elif}\;n \le 7547414330764314.0:\\ \;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\log \left(\frac{1}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \left(1 - \log \left(\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\log x}{\left(n \cdot n\right) \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))