Average Error: 29.6 → 22.5
Time: 9.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -20133141455.183243 \lor \neg \left(n \le 253373043857974.969\right):\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right)\right) - \left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -20133141455.183243 \lor \neg \left(n \le 253373043857974.969\right):\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} - \log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right)\right) - \left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r62413 = x;
        double r62414 = 1.0;
        double r62415 = r62413 + r62414;
        double r62416 = n;
        double r62417 = r62414 / r62416;
        double r62418 = pow(r62415, r62417);
        double r62419 = pow(r62413, r62417);
        double r62420 = r62418 - r62419;
        return r62420;
}
↓
double f(double x, double n) {
        double r62421 = n;
        double r62422 = -20133141455.183243;
        bool r62423 = r62421 <= r62422;
        double r62424 = 253373043857974.97;
        bool r62425 = r62421 <= r62424;
        double r62426 = !r62425;
        bool r62427 = r62423 || r62426;
        double r62428 = 1.0;
        double r62429 = r62428 / r62421;
        double r62430 = x;
        double r62431 = r62429 / r62430;
        double r62432 = 0.5;
        double r62433 = 2.0;
        double r62434 = pow(r62430, r62433);
        double r62435 = r62434 * r62421;
        double r62436 = r62432 / r62435;
        double r62437 = exp(r62436);
        double r62438 = cbrt(r62437);
        double r62439 = r62438 * r62438;
        double r62440 = log(r62439);
        double r62441 = r62431 - r62440;
        double r62442 = log(r62438);
        double r62443 = log(r62430);
        double r62444 = r62443 * r62428;
        double r62445 = pow(r62421, r62433);
        double r62446 = r62430 * r62445;
        double r62447 = r62444 / r62446;
        double r62448 = r62442 - r62447;
        double r62449 = r62441 - r62448;
        double r62450 = r62430 + r62428;
        double r62451 = pow(r62450, r62429);
        double r62452 = pow(r62430, r62429);
        double r62453 = r62451 - r62452;
        double r62454 = r62427 ? r62449 : r62453;
        return r62454;
}

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 n < -20133141455.183243 or 253373043857974.97 < n

    1. Initial program 45.2

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

    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

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

    5. Applied add-log-exp32.8

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

    6. Simplified32.8

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

    8. Applied add-cube-cbrt32.8

      \[\leadsto \frac{\frac{1}{n}}{x} - \left(\log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right)} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\]

    9. Applied log-prod32.8

      \[\leadsto \frac{\frac{1}{n}}{x} - \left(\color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) + \log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right)\right)} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\]

    10. Applied associate--l+32.8

      \[\leadsto \frac{\frac{1}{n}}{x} - \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) + \left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\right)}\]

    11. Applied associate--r+32.8

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} - \log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right)\right) - \left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\]

    if -20133141455.183243 < n < 253373043857974.97

    1. Initial program 8.8

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -20133141455.183243 \lor \neg \left(n \le 253373043857974.969\right):\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}} \cdot \sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right)\right) - \left(\log \left(\sqrt[3]{e^{\frac{0.5}{{x}^{2} \cdot n}}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))