Average Error: 29.4 → 22.5
Time: 1.0m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6458580981343804 \cdot 10^{-16}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 6.469269257276286 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.6458580981343804 \cdot 10^{-16}:\\
\;\;\;\;\log \left(\sqrt{e^{{\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) + \log \left(\sqrt{e^{{\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)\\

\mathbf{elif}\;\frac{1}{n} \le 6.469269257276286 \cdot 10^{-11}:\\
\;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\

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

\end{array}
double f(double x, double n) {
        double r6248783 = x;
        double r6248784 = 1.0;
        double r6248785 = r6248783 + r6248784;
        double r6248786 = n;
        double r6248787 = r6248784 / r6248786;
        double r6248788 = pow(r6248785, r6248787);
        double r6248789 = pow(r6248783, r6248787);
        double r6248790 = r6248788 - r6248789;
        return r6248790;
}
↓
double f(double x, double n) {
        double r6248791 = 1.0;
        double r6248792 = n;
        double r6248793 = r6248791 / r6248792;
        double r6248794 = -1.6458580981343804e-16;
        bool r6248795 = r6248793 <= r6248794;
        double r6248796 = x;
        double r6248797 = r6248796 + r6248791;
        double r6248798 = pow(r6248797, r6248793);
        double r6248799 = cbrt(r6248796);
        double r6248800 = r6248799 * r6248799;
        double r6248801 = pow(r6248800, r6248793);
        double r6248802 = pow(r6248799, r6248793);
        double r6248803 = r6248801 * r6248802;
        double r6248804 = r6248798 - r6248803;
        double r6248805 = exp(r6248804);
        double r6248806 = sqrt(r6248805);
        double r6248807 = log(r6248806);
        double r6248808 = r6248807 + r6248807;
        double r6248809 = 6.469269257276286e-11;
        bool r6248810 = r6248793 <= r6248809;
        double r6248811 = r6248796 * r6248792;
        double r6248812 = r6248791 / r6248811;
        double r6248813 = log(r6248796);
        double r6248814 = -r6248813;
        double r6248815 = r6248811 * r6248792;
        double r6248816 = r6248814 / r6248815;
        double r6248817 = r6248812 - r6248816;
        double r6248818 = -0.5;
        double r6248819 = r6248796 * r6248811;
        double r6248820 = r6248818 / r6248819;
        double r6248821 = r6248817 + r6248820;
        double r6248822 = r6248798 * r6248798;
        double r6248823 = r6248803 * r6248803;
        double r6248824 = r6248822 - r6248823;
        double r6248825 = r6248798 + r6248803;
        double r6248826 = r6248824 / r6248825;
        double r6248827 = r6248810 ? r6248821 : r6248826;
        double r6248828 = r6248795 ? r6248808 : r6248827;
        return r6248828;
}

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 n) < -1.6458580981343804e-16

    1. Initial program 2.3

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

    3. Applied add-cube-cbrt2.3

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

    4. Applied unpow-prod-down2.4

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

    6. Applied add-log-exp2.6

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

    8. Applied add-sqr-sqrt2.6

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

    9. Applied log-prod2.6

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

    if -1.6458580981343804e-16 < (/ 1 n) < 6.469269257276286e-11

    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 31.8

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

    3. Simplified31.8

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

    if 6.469269257276286e-11 < (/ 1 n)

    1. Initial program 23.5

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

    3. Applied add-cube-cbrt23.5

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

    4. Applied unpow-prod-down23.5

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

    6. Applied flip--27.0

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.6458580981343804 \cdot 10^{-16}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 6.469269257276286 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right) + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))