Average Error: 29.3 → 22.7
Time: 31.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.234230860330125218:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.097464775265143 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) - \frac{-\log x}{x \cdot {n}^{2}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.234230860330125218:\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 3.097464775265143 \cdot 10^{-17}:\\
\;\;\;\;\left(\frac{1}{x \cdot n} - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) - \frac{-\log x}{x \cdot {n}^{2}} \cdot 1\\

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

\end{array}
double f(double x, double n) {
        double r45741 = x;
        double r45742 = 1.0;
        double r45743 = r45741 + r45742;
        double r45744 = n;
        double r45745 = r45742 / r45744;
        double r45746 = pow(r45743, r45745);
        double r45747 = pow(r45741, r45745);
        double r45748 = r45746 - r45747;
        return r45748;
}
↓
double f(double x, double n) {
        double r45749 = 1.0;
        double r45750 = n;
        double r45751 = r45749 / r45750;
        double r45752 = -0.23423086033012522;
        bool r45753 = r45751 <= r45752;
        double r45754 = x;
        double r45755 = r45754 + r45749;
        double r45756 = pow(r45755, r45751);
        double r45757 = 3.0;
        double r45758 = pow(r45756, r45757);
        double r45759 = cbrt(r45758);
        double r45760 = pow(r45754, r45751);
        double r45761 = r45759 - r45760;
        double r45762 = 3.097464775265143e-17;
        bool r45763 = r45751 <= r45762;
        double r45764 = r45754 * r45750;
        double r45765 = r45749 / r45764;
        double r45766 = 0.5;
        double r45767 = 2.0;
        double r45768 = pow(r45754, r45767);
        double r45769 = r45768 * r45750;
        double r45770 = exp(r45769);
        double r45771 = log(r45770);
        double r45772 = r45766 / r45771;
        double r45773 = r45765 - r45772;
        double r45774 = log(r45754);
        double r45775 = -r45774;
        double r45776 = pow(r45750, r45767);
        double r45777 = r45754 * r45776;
        double r45778 = r45775 / r45777;
        double r45779 = r45778 * r45749;
        double r45780 = r45773 - r45779;
        double r45781 = r45767 * r45751;
        double r45782 = pow(r45755, r45781);
        double r45783 = pow(r45754, r45781);
        double r45784 = r45782 - r45783;
        double r45785 = r45756 + r45760;
        double r45786 = r45784 / r45785;
        double r45787 = r45763 ? r45780 : r45786;
        double r45788 = r45753 ? r45761 : r45787;
        return r45788;
}

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) < -0.23423086033012522

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    if -0.23423086033012522 < (/ 1.0 n) < 3.097464775265143e-17

    1. Initial program 44.8

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

    3. Applied add-cbrt-cube44.9

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

    4. Simplified44.9

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

    5. Taylor expanded around inf 32.5

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

    6. Simplified32.5

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

    8. Applied add-log-exp32.5

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

    if 3.097464775265143e-17 < (/ 1.0 n)

    1. Initial program 25.5

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

    3. Applied flip--28.5

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

    4. Simplified28.4

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.234230860330125218:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.097464775265143 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) - \frac{-\log x}{x \cdot {n}^{2}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))