Average Error: 28.7 → 21.3
Time: 28.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1935346252.7263522:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{n \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 13344.856355641448:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{n}}{x \cdot n} + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1935346252.7263522:\\
\;\;\;\;\left(\frac{\frac{-1}{2}}{n \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\

\mathbf{elif}\;n \le 13344.856355641448:\\
\;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r2144460 = x;
        double r2144461 = 1.0;
        double r2144462 = r2144460 + r2144461;
        double r2144463 = n;
        double r2144464 = r2144461 / r2144463;
        double r2144465 = pow(r2144462, r2144464);
        double r2144466 = pow(r2144460, r2144464);
        double r2144467 = r2144465 - r2144466;
        return r2144467;
}
↓
double f(double x, double n) {
        double r2144468 = n;
        double r2144469 = -1935346252.7263522;
        bool r2144470 = r2144468 <= r2144469;
        double r2144471 = -0.5;
        double r2144472 = x;
        double r2144473 = r2144472 * r2144472;
        double r2144474 = r2144468 * r2144473;
        double r2144475 = r2144471 / r2144474;
        double r2144476 = 1.0;
        double r2144477 = r2144476 / r2144472;
        double r2144478 = r2144477 / r2144468;
        double r2144479 = r2144475 + r2144478;
        double r2144480 = log(r2144472);
        double r2144481 = r2144472 * r2144468;
        double r2144482 = r2144481 * r2144468;
        double r2144483 = r2144480 / r2144482;
        double r2144484 = r2144479 + r2144483;
        double r2144485 = 13344.856355641448;
        bool r2144486 = r2144468 <= r2144485;
        double r2144487 = r2144476 + r2144472;
        double r2144488 = r2144476 / r2144468;
        double r2144489 = pow(r2144487, r2144488);
        double r2144490 = pow(r2144472, r2144488);
        double r2144491 = r2144489 - r2144490;
        double r2144492 = cbrt(r2144491);
        double r2144493 = r2144492 * r2144492;
        double r2144494 = exp(r2144493);
        double r2144495 = log(r2144494);
        double r2144496 = r2144492 * r2144495;
        double r2144497 = r2144488 / r2144472;
        double r2144498 = r2144480 / r2144468;
        double r2144499 = r2144498 / r2144481;
        double r2144500 = r2144472 * r2144481;
        double r2144501 = r2144471 / r2144500;
        double r2144502 = r2144499 + r2144501;
        double r2144503 = r2144497 + r2144502;
        double r2144504 = r2144486 ? r2144496 : r2144503;
        double r2144505 = r2144470 ? r2144484 : r2144504;
        return r2144505;
}

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 < -1935346252.7263522

    1. Initial program 44.9

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

    3. Applied add-cube-cbrt44.9

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

    4. Taylor expanded around inf 32.4

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

    5. Simplified31.8

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

    if -1935346252.7263522 < n < 13344.856355641448

    1. Initial program 8.3

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

    3. Applied add-cube-cbrt8.3

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

    5. Applied add-log-exp8.6

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

    6. Applied add-log-exp8.5

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

    7. Applied diff-log8.6

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

    8. Simplified8.5

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

    10. Applied add-cube-cbrt8.5

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

    11. Applied exp-prod8.5

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

    12. Applied log-pow8.5

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

    if 13344.856355641448 < n

    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 31.6

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

    3. Simplified30.7

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

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1935346252.7263522:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{n \cdot \left(x \cdot x\right)} + \frac{\frac{1}{x}}{n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 13344.856355641448:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{n}}{x \cdot n} + \frac{\frac{-1}{2}}{x \cdot \left(x \cdot n\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))