Average Error: 29.8 → 22.8
Time: 30.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.1647883017738754 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \mathbf{elif}\;n \le 7547414330764314.0:\\ \;\;\;\;\sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \end{array}\]
{\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{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\

\mathbf{elif}\;n \le 7547414330764314.0:\\
\;\;\;\;\sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)} \cdot \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)\\

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

\end{array}
double f(double x, double n) {
        double r3106686 = x;
        double r3106687 = 1.0;
        double r3106688 = r3106686 + r3106687;
        double r3106689 = n;
        double r3106690 = r3106687 / r3106689;
        double r3106691 = pow(r3106688, r3106690);
        double r3106692 = pow(r3106686, r3106690);
        double r3106693 = r3106691 - r3106692;
        return r3106693;
}
↓
double f(double x, double n) {
        double r3106694 = n;
        double r3106695 = -3.1647883017738754e+21;
        bool r3106696 = r3106694 <= r3106695;
        double r3106697 = 1.0;
        double r3106698 = x;
        double r3106699 = r3106698 * r3106694;
        double r3106700 = r3106697 / r3106699;
        double r3106701 = 0.5;
        double r3106702 = r3106701 / r3106694;
        double r3106703 = r3106698 * r3106698;
        double r3106704 = r3106702 / r3106703;
        double r3106705 = log(r3106698);
        double r3106706 = r3106694 * r3106694;
        double r3106707 = r3106705 / r3106706;
        double r3106708 = r3106707 / r3106698;
        double r3106709 = r3106704 - r3106708;
        double r3106710 = r3106700 - r3106709;
        double r3106711 = 7547414330764314.0;
        bool r3106712 = r3106694 <= r3106711;
        double r3106713 = r3106697 / r3106694;
        double r3106714 = 2.0;
        double r3106715 = r3106713 / r3106714;
        double r3106716 = pow(r3106698, r3106715);
        double r3106717 = r3106697 + r3106698;
        double r3106718 = pow(r3106717, r3106713);
        double r3106719 = sqrt(r3106718);
        double r3106720 = r3106716 + r3106719;
        double r3106721 = r3106719 - r3106716;
        double r3106722 = exp(r3106721);
        double r3106723 = log(r3106722);
        double r3106724 = r3106720 * r3106723;
        double r3106725 = cbrt(r3106724);
        double r3106726 = pow(r3106698, r3106713);
        double r3106727 = r3106718 - r3106726;
        double r3106728 = cbrt(r3106727);
        double r3106729 = r3106728 * r3106728;
        double r3106730 = r3106725 * r3106729;
        double r3106731 = r3106712 ? r3106730 : r3106710;
        double r3106732 = r3106696 ? r3106710 : r3106731;
        return r3106732;
}

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. Simplified32.2

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

    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 sqr-pow10.7

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

    6. Applied add-sqr-sqrt10.7

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

    7. Applied difference-of-squares10.7

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\]
    8. Using strategy rm

    9. Applied add-log-exp10.8

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

  4. Final simplification22.8

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

Reproduce

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