Average Error: 29.2 → 22.1
Time: 19.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.277743339242776164382043874840818047014 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 8.740688882085781214297279027767987281218 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \left(\frac{1}{3} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -9.277743339242776164382043874840818047014 \cdot 10^{-13}:\\
\;\;\;\;\left(2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 8.740688882085781214297279027767987281218 \cdot 10^{-15}:\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \left(\frac{1}{3} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\\

\end{array}
double f(double x, double n) {
        double r63597 = x;
        double r63598 = 1.0;
        double r63599 = r63597 + r63598;
        double r63600 = n;
        double r63601 = r63598 / r63600;
        double r63602 = pow(r63599, r63601);
        double r63603 = pow(r63597, r63601);
        double r63604 = r63602 - r63603;
        return r63604;
}
↓
double f(double x, double n) {
        double r63605 = 1.0;
        double r63606 = n;
        double r63607 = r63605 / r63606;
        double r63608 = -9.277743339242776e-13;
        bool r63609 = r63607 <= r63608;
        double r63610 = 2.0;
        double r63611 = x;
        double r63612 = r63611 + r63605;
        double r63613 = pow(r63612, r63607);
        double r63614 = exp(r63613);
        double r63615 = 0.3333333333333333;
        double r63616 = pow(r63614, r63615);
        double r63617 = log(r63616);
        double r63618 = r63610 * r63617;
        double r63619 = cbrt(r63614);
        double r63620 = log(r63619);
        double r63621 = r63618 + r63620;
        double r63622 = pow(r63611, r63607);
        double r63623 = r63621 - r63622;
        double r63624 = 8.740688882085781e-15;
        bool r63625 = r63607 <= r63624;
        double r63626 = r63605 / r63611;
        double r63627 = 1.0;
        double r63628 = r63627 / r63606;
        double r63629 = log(r63611);
        double r63630 = -r63629;
        double r63631 = pow(r63606, r63610);
        double r63632 = r63630 / r63631;
        double r63633 = r63628 - r63632;
        double r63634 = r63626 * r63633;
        double r63635 = 0.5;
        double r63636 = pow(r63611, r63610);
        double r63637 = r63636 * r63606;
        double r63638 = r63635 / r63637;
        double r63639 = r63634 - r63638;
        double r63640 = r63615 * r63613;
        double r63641 = r63640 - r63622;
        double r63642 = r63618 + r63641;
        double r63643 = r63625 ? r63639 : r63642;
        double r63644 = r63609 ? r63623 : r63643;
        return r63644;
}

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) < -9.277743339242776e-13

    1. Initial program 1.3

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

    3. Applied add-log-exp1.3

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

    5. Applied add-cube-cbrt1.4

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

    6. Applied log-prod1.4

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

    7. Simplified1.4

      \[\leadsto \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
    8. Using strategy rm

    9. Applied pow1/31.4

      \[\leadsto \left(2 \cdot \log \color{blue}{\left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]

    if -9.277743339242776e-13 < (/ 1.0 n) < 8.740688882085781e-15

    1. Initial program 45.4

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

    2. Taylor expanded around inf 33.2

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

    3. Simplified32.6

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

    if 8.740688882085781e-15 < (/ 1.0 n)

    1. Initial program 24.6

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

    3. Applied add-log-exp24.6

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

    5. Applied add-cube-cbrt26.0

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

    6. Applied log-prod26.0

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

    7. Simplified26.0

      \[\leadsto \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
    8. Using strategy rm

    9. Applied pow1/325.3

      \[\leadsto \left(2 \cdot \log \color{blue}{\left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
    10. Using strategy rm

    11. Applied associate--l+25.3

      \[\leadsto \color{blue}{2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \left(\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) - {x}^{\left(\frac{1}{n}\right)}\right)}\]

    12. Simplified24.8

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -9.277743339242776164382043874840818047014 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 8.740688882085781214297279027767987281218 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right) + \left(\frac{1}{3} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))