Average Error: 29.0 → 22.2
Time: 40.7s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.858253988214169272711275460348290256007 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.863364080616155825796168983867868088999 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -3.858253988214169272711275460348290256007 \cdot 10^{-8}:\\
\;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 1.863364080616155825796168983867868088999 \cdot 10^{-31}:\\
\;\;\;\;\left(\frac{1}{x \cdot n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1}{x \cdot \left(n \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right) + \log \left(\sqrt{e^{{\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 r11104652 = x;
        double r11104653 = 1.0;
        double r11104654 = r11104652 + r11104653;
        double r11104655 = n;
        double r11104656 = r11104653 / r11104655;
        double r11104657 = pow(r11104654, r11104656);
        double r11104658 = pow(r11104652, r11104656);
        double r11104659 = r11104657 - r11104658;
        return r11104659;
}
↓
double f(double x, double n) {
        double r11104660 = 1.0;
        double r11104661 = n;
        double r11104662 = r11104660 / r11104661;
        double r11104663 = -3.858253988214169e-08;
        bool r11104664 = r11104662 <= r11104663;
        double r11104665 = x;
        double r11104666 = r11104665 + r11104660;
        double r11104667 = pow(r11104666, r11104662);
        double r11104668 = pow(r11104665, r11104662);
        double r11104669 = r11104667 - r11104668;
        double r11104670 = cbrt(r11104669);
        double r11104671 = sqrt(r11104667);
        double r11104672 = 2.0;
        double r11104673 = r11104662 / r11104672;
        double r11104674 = pow(r11104665, r11104673);
        double r11104675 = r11104671 + r11104674;
        double r11104676 = r11104671 - r11104674;
        double r11104677 = r11104675 * r11104676;
        double r11104678 = cbrt(r11104677);
        double r11104679 = r11104670 * r11104678;
        double r11104680 = exp(r11104679);
        double r11104681 = log(r11104680);
        double r11104682 = r11104670 * r11104681;
        double r11104683 = 1.8633640806161558e-31;
        bool r11104684 = r11104662 <= r11104683;
        double r11104685 = r11104665 * r11104661;
        double r11104686 = r11104660 / r11104685;
        double r11104687 = 0.5;
        double r11104688 = r11104665 * r11104665;
        double r11104689 = r11104661 * r11104688;
        double r11104690 = r11104687 / r11104689;
        double r11104691 = r11104686 - r11104690;
        double r11104692 = log(r11104665);
        double r11104693 = r11104692 * r11104660;
        double r11104694 = r11104661 * r11104661;
        double r11104695 = r11104665 * r11104694;
        double r11104696 = r11104693 / r11104695;
        double r11104697 = r11104691 + r11104696;
        double r11104698 = exp(r11104667);
        double r11104699 = sqrt(r11104698);
        double r11104700 = log(r11104699);
        double r11104701 = -r11104668;
        double r11104702 = exp(r11104701);
        double r11104703 = sqrt(r11104702);
        double r11104704 = log(r11104703);
        double r11104705 = r11104700 + r11104704;
        double r11104706 = exp(r11104669);
        double r11104707 = sqrt(r11104706);
        double r11104708 = log(r11104707);
        double r11104709 = r11104705 + r11104708;
        double r11104710 = r11104684 ? r11104697 : r11104709;
        double r11104711 = r11104664 ? r11104682 : r11104710;
        return r11104711;
}

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) < -3.858253988214169e-08

    1. Initial program 0.9

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

    3. Applied add-log-exp1.3

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

    4. Applied add-log-exp1.1

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

    5. Applied diff-log1.1

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

    6. Simplified1.1

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

    8. Applied add-cube-cbrt1.1

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

    9. Applied exp-prod1.1

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

    10. Applied log-pow1.1

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

    12. Applied sqr-pow1.1

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

    13. Applied add-sqr-sqrt1.1

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

    14. Applied difference-of-squares1.1

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

    if -3.858253988214169e-08 < (/ 1.0 n) < 1.8633640806161558e-31

    1. Initial program 44.4

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

    2. Taylor expanded around inf 31.9

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

    3. Simplified31.9

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

    if 1.8633640806161558e-31 < (/ 1.0 n)

    1. Initial program 28.5

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

    3. Applied add-log-exp28.6

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

    4. Applied add-log-exp28.6

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

    5. Applied diff-log28.7

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

    6. Simplified28.6

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

    8. Applied add-sqr-sqrt28.7

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

    9. Applied log-prod28.7

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

    11. Applied sub-neg28.7

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

    12. Applied exp-sum28.7

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

    13. Applied sqrt-prod28.7

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

    14. Applied log-prod28.7

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.858253988214169272711275460348290256007 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.863364080616155825796168983867868088999 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1}{x \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))