Average Error: 29.0 → 22.5
Time: 20.3s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le \frac{-3403751052409837}{8} \lor \neg \left(n \le 6797098439818426372325376\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le \frac{-3403751052409837}{8} \lor \neg \left(n \le 6797098439818426372325376\right):\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r177682 = x;
        double r177683 = 1.0;
        double r177684 = r177682 + r177683;
        double r177685 = n;
        double r177686 = r177683 / r177685;
        double r177687 = pow(r177684, r177686);
        double r177688 = pow(r177682, r177686);
        double r177689 = r177687 - r177688;
        return r177689;
}
↓
double f(double x, double n) {
        double r177690 = n;
        double r177691 = -3403751052409837.0;
        double r177692 = 8.0;
        double r177693 = r177691 / r177692;
        bool r177694 = r177690 <= r177693;
        double r177695 = 6.797098439818426e+24;
        bool r177696 = r177690 <= r177695;
        double r177697 = !r177696;
        bool r177698 = r177694 || r177697;
        double r177699 = 1.0;
        double r177700 = r177699 / r177690;
        double r177701 = x;
        double r177702 = r177700 / r177701;
        double r177703 = 2.0;
        double r177704 = r177699 / r177703;
        double r177705 = r177704 / r177690;
        double r177706 = 2.0;
        double r177707 = pow(r177701, r177706);
        double r177708 = r177705 / r177707;
        double r177709 = log(r177701);
        double r177710 = r177709 * r177699;
        double r177711 = pow(r177690, r177706);
        double r177712 = r177701 * r177711;
        double r177713 = r177710 / r177712;
        double r177714 = r177708 - r177713;
        double r177715 = r177702 - r177714;
        double r177716 = r177701 + r177699;
        double r177717 = cbrt(r177716);
        double r177718 = r177717 * r177717;
        double r177719 = pow(r177718, r177700);
        double r177720 = pow(r177717, r177700);
        double r177721 = sqrt(r177720);
        double r177722 = r177721 * r177721;
        double r177723 = r177719 * r177722;
        double r177724 = pow(r177701, r177700);
        double r177725 = r177723 - r177724;
        double r177726 = 3.0;
        double r177727 = pow(r177725, r177726);
        double r177728 = cbrt(r177727);
        double r177729 = r177698 ? r177715 : r177728;
        return r177729;
}

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 < -425468881551229.6 or 6.797098439818426e+24 < n

    1. Initial program 44.5

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

    2. Taylor expanded around inf 33.1

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

    3. Simplified32.6

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

    if -425468881551229.6 < n < 6.797098439818426e+24

    1. Initial program 10.0

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

    3. Applied add-cube-cbrt10.1

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

    4. Applied unpow-prod-down10.1

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

    6. Applied add-sqr-sqrt10.1

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

    8. Applied add-cbrt-cube10.1

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

    9. Simplified10.1

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le \frac{-3403751052409837}{8} \lor \neg \left(n \le 6797098439818426372325376\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 350497007 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))