Average Error: 29.4 → 22.1
Time: 30.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le \frac{-5927677248097903}{268435456} \lor \neg \left(n \le \frac{6854405603415341}{65536}\right):\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^{2}}\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le \frac{-5927677248097903}{268435456} \lor \neg \left(n \le \frac{6854405603415341}{65536}\right):\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^{2}}\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\\

\mathbf{else}:\\
\;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)\\

\end{array}
double f(double x, double n) {
        double r84592 = x;
        double r84593 = 1.0;
        double r84594 = r84592 + r84593;
        double r84595 = n;
        double r84596 = r84593 / r84595;
        double r84597 = pow(r84594, r84596);
        double r84598 = pow(r84592, r84596);
        double r84599 = r84597 - r84598;
        return r84599;
}
↓
double f(double x, double n) {
        double r84600 = n;
        double r84601 = -5927677248097903.0;
        double r84602 = 268435456.0;
        double r84603 = r84601 / r84602;
        bool r84604 = r84600 <= r84603;
        double r84605 = 6854405603415341.0;
        double r84606 = 65536.0;
        double r84607 = r84605 / r84606;
        bool r84608 = r84600 <= r84607;
        double r84609 = !r84608;
        bool r84610 = r84604 || r84609;
        double r84611 = 1.0;
        double r84612 = r84611 / r84600;
        double r84613 = x;
        double r84614 = r84612 / r84613;
        double r84615 = 2.0;
        double r84616 = r84611 / r84615;
        double r84617 = r84616 / r84600;
        double r84618 = 2.0;
        double r84619 = pow(r84613, r84618);
        double r84620 = r84617 / r84619;
        double r84621 = r84614 - r84620;
        double r84622 = log(r84613);
        double r84623 = r84622 * r84611;
        double r84624 = pow(r84600, r84618);
        double r84625 = r84613 * r84624;
        double r84626 = r84623 / r84625;
        double r84627 = r84621 + r84626;
        double r84628 = r84613 + r84611;
        double r84629 = cbrt(r84628);
        double r84630 = r84629 * r84629;
        double r84631 = pow(r84630, r84612);
        double r84632 = pow(r84629, r84612);
        double r84633 = r84631 * r84632;
        double r84634 = pow(r84613, r84612);
        double r84635 = -r84634;
        double r84636 = r84633 + r84635;
        double r84637 = r84610 ? r84627 : r84636;
        return r84637;
}

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 n < -22082318.544752534

    1. Initial program 45.0

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

    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

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

    5. Applied add-log-exp32.7

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

    6. Simplified32.7

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

    if -22082318.544752534 < n < 104589929251.33272

    1. Initial program 8.8

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

    3. Applied add-cube-cbrt8.8

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

    4. Applied unpow-prod-down8.9

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

    6. Applied add-cbrt-cube8.9

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

    7. Simplified8.9

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

    if 104589929251.33272 < n

    1. Initial program 44.8

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

    2. Taylor expanded around inf 32.1

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

    3. Simplified31.5

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

    5. Applied add-log-exp31.6

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le \frac{-5927677248097903}{268435456} \lor \neg \left(n \le \frac{6854405603415341}{65536}\right):\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^{2}}\right) + \frac{\log x \cdot 1}{x \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))