Average Error: 29.1 → 22.3
Time: 10.0s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -210159165.5542463362216949462890625 \lor \neg \left(\frac{1}{n} \le 6.331169527565450284131583336320555667476 \cdot 10^{-11}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -210159165.5542463362216949462890625 \lor \neg \left(\frac{1}{n} \le 6.331169527565450284131583336320555667476 \cdot 10^{-11}\right):\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\end{array}
double f(double x, double n) {
        double r59591 = x;
        double r59592 = 1.0;
        double r59593 = r59591 + r59592;
        double r59594 = n;
        double r59595 = r59592 / r59594;
        double r59596 = pow(r59593, r59595);
        double r59597 = pow(r59591, r59595);
        double r59598 = r59596 - r59597;
        return r59598;
}
↓
double f(double x, double n) {
        double r59599 = 1.0;
        double r59600 = n;
        double r59601 = r59599 / r59600;
        double r59602 = -210159165.55424634;
        bool r59603 = r59601 <= r59602;
        double r59604 = 6.33116952756545e-11;
        bool r59605 = r59601 <= r59604;
        double r59606 = !r59605;
        bool r59607 = r59603 || r59606;
        double r59608 = x;
        double r59609 = r59608 + r59599;
        double r59610 = pow(r59609, r59601);
        double r59611 = pow(r59608, r59601);
        double r59612 = r59610 - r59611;
        double r59613 = 3.0;
        double r59614 = pow(r59612, r59613);
        double r59615 = cbrt(r59614);
        double r59616 = r59601 / r59608;
        double r59617 = 0.5;
        double r59618 = 2.0;
        double r59619 = pow(r59608, r59618);
        double r59620 = r59619 * r59600;
        double r59621 = exp(r59620);
        double r59622 = log(r59621);
        double r59623 = r59617 / r59622;
        double r59624 = exp(r59623);
        double r59625 = log(r59624);
        double r59626 = log(r59608);
        double r59627 = r59626 * r59599;
        double r59628 = pow(r59600, r59618);
        double r59629 = r59608 * r59628;
        double r59630 = r59627 / r59629;
        double r59631 = r59625 - r59630;
        double r59632 = r59616 - r59631;
        double r59633 = r59607 ? r59615 : r59632;
        return r59633;
}

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 2 regimes

  2. if (/ 1.0 n) < -210159165.55424634 or 6.33116952756545e-11 < (/ 1.0 n)

    1. Initial program 8.2

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

    3. Applied add-cbrt-cube8.2

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

    4. Simplified8.2

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

    if -210159165.55424634 < (/ 1.0 n) < 6.33116952756545e-11

    1. Initial program 43.7

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

    2. Taylor expanded around inf 32.6

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

    3. Simplified32.0

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

    5. Applied add-log-exp32.1

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

    6. Simplified32.1

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

    8. Applied add-log-exp32.1

      \[\leadsto \frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{\color{blue}{\log \left(e^{{x}^{2} \cdot n}\right)}}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -210159165.5542463362216949462890625 \lor \neg \left(\frac{1}{n} \le 6.331169527565450284131583336320555667476 \cdot 10^{-11}\right):\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))