Average Error: 29.5 → 22.0
Time: 25.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.724877157437157453427224965446109160238 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.130157691600053332981091905665160801675 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.724877157437157453427224965446109160238 \cdot 10^{-20}:\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\\

\mathbf{elif}\;\frac{1}{n} \le 2.130157691600053332981091905665160801675 \cdot 10^{-7}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r41710 = x;
        double r41711 = 1.0;
        double r41712 = r41710 + r41711;
        double r41713 = n;
        double r41714 = r41711 / r41713;
        double r41715 = pow(r41712, r41714);
        double r41716 = pow(r41710, r41714);
        double r41717 = r41715 - r41716;
        return r41717;
}
↓
double f(double x, double n) {
        double r41718 = 1.0;
        double r41719 = n;
        double r41720 = r41718 / r41719;
        double r41721 = -1.7248771574371575e-20;
        bool r41722 = r41720 <= r41721;
        double r41723 = x;
        double r41724 = r41723 + r41718;
        double r41725 = pow(r41724, r41720);
        double r41726 = pow(r41723, r41720);
        double r41727 = 3.0;
        double r41728 = pow(r41726, r41727);
        double r41729 = cbrt(r41728);
        double r41730 = r41725 - r41729;
        double r41731 = pow(r41730, r41727);
        double r41732 = cbrt(r41731);
        double r41733 = 2.1301576916000533e-07;
        bool r41734 = r41720 <= r41733;
        double r41735 = r41720 / r41723;
        double r41736 = 0.5;
        double r41737 = r41736 / r41719;
        double r41738 = 2.0;
        double r41739 = pow(r41723, r41738);
        double r41740 = r41737 / r41739;
        double r41741 = log(r41723);
        double r41742 = r41741 * r41718;
        double r41743 = pow(r41719, r41738);
        double r41744 = r41723 * r41743;
        double r41745 = r41742 / r41744;
        double r41746 = r41740 - r41745;
        double r41747 = r41735 - r41746;
        double r41748 = cbrt(r41723);
        double r41749 = r41748 * r41748;
        double r41750 = pow(r41749, r41720);
        double r41751 = pow(r41748, r41720);
        double r41752 = r41750 * r41751;
        double r41753 = r41725 - r41752;
        double r41754 = log(r41753);
        double r41755 = exp(r41754);
        double r41756 = r41734 ? r41747 : r41755;
        double r41757 = r41722 ? r41732 : r41756;
        return r41757;
}

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) < -1.7248771574371575e-20

    1. Initial program 2.4

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

    3. Applied add-cbrt-cube2.7

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

    4. Simplified2.7

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

    6. Applied add-cbrt-cube2.7

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

    7. Simplified2.7

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

    if -1.7248771574371575e-20 < (/ 1.0 n) < 2.1301576916000533e-07

    1. Initial program 45.1

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

    2. Taylor expanded around inf 32.3

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

    3. Simplified31.7

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

    if 2.1301576916000533e-07 < (/ 1.0 n)

    1. Initial program 23.9

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

    3. Applied add-cube-cbrt23.9

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

    4. Applied unpow-prod-down23.9

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

    6. Applied add-exp-log23.9

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.724877157437157453427224965446109160238 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 2.130157691600053332981091905665160801675 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))