Average Error: 29.1 → 22.0
Time: 32.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \left(\frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)} - \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\
\;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\
\;\;\;\;\frac{\frac{1}{x}}{n} - \left(\frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)} - \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\

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

\end{array}
double f(double x, double n) {
        double r1662112 = x;
        double r1662113 = 1.0;
        double r1662114 = r1662112 + r1662113;
        double r1662115 = n;
        double r1662116 = r1662113 / r1662115;
        double r1662117 = pow(r1662114, r1662116);
        double r1662118 = pow(r1662112, r1662116);
        double r1662119 = r1662117 - r1662118;
        return r1662119;
}
↓
double f(double x, double n) {
        double r1662120 = 1.0;
        double r1662121 = n;
        double r1662122 = r1662120 / r1662121;
        double r1662123 = -104130347.93584307;
        bool r1662124 = r1662122 <= r1662123;
        double r1662125 = x;
        double r1662126 = r1662125 + r1662120;
        double r1662127 = pow(r1662126, r1662122);
        double r1662128 = pow(r1662125, r1662122);
        double r1662129 = r1662127 - r1662128;
        double r1662130 = log(r1662129);
        double r1662131 = cbrt(r1662130);
        double r1662132 = r1662131 * r1662131;
        double r1662133 = exp(r1662132);
        double r1662134 = pow(r1662133, r1662131);
        double r1662135 = 2.8321253549703665e-07;
        bool r1662136 = r1662122 <= r1662135;
        double r1662137 = r1662120 / r1662125;
        double r1662138 = r1662137 / r1662121;
        double r1662139 = 0.5;
        double r1662140 = r1662125 * r1662121;
        double r1662141 = r1662125 * r1662140;
        double r1662142 = r1662139 / r1662141;
        double r1662143 = log(r1662125);
        double r1662144 = r1662121 * r1662140;
        double r1662145 = r1662143 / r1662144;
        double r1662146 = r1662142 - r1662145;
        double r1662147 = r1662138 - r1662146;
        double r1662148 = r1662136 ? r1662147 : r1662134;
        double r1662149 = r1662124 ? r1662134 : r1662148;
        return r1662149;
}

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 (/ 1 n) < -104130347.93584307 or 2.8321253549703665e-07 < (/ 1 n)

    1. Initial program 7.7

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

    3. Applied add-exp-log7.7

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

    5. Applied add-cube-cbrt7.7

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

    6. Applied exp-prod7.7

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

    if -104130347.93584307 < (/ 1 n) < 2.8321253549703665e-07

    1. Initial program 44.7

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

    3. Applied add-exp-log45.4

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

    4. Taylor expanded around inf 33.1

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

    5. Simplified32.4

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -104130347.93584307:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.8321253549703665 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \left(\frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)} - \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))