Average Error: 29.3 → 22.5
Time: 19.2s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le \frac{-4175041654096867}{2251799813685248} \lor \neg \left(n \le \frac{523458166673859}{562949953421312}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\log \left(e^{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le \frac{-4175041654096867}{2251799813685248} \lor \neg \left(n \le \frac{523458166673859}{562949953421312}\right):\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\log \left(e^{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\

\end{array}
double f(double x, double n) {
        double r75459 = x;
        double r75460 = 1.0;
        double r75461 = r75459 + r75460;
        double r75462 = n;
        double r75463 = r75460 / r75462;
        double r75464 = pow(r75461, r75463);
        double r75465 = pow(r75459, r75463);
        double r75466 = r75464 - r75465;
        return r75466;
}
↓
double f(double x, double n) {
        double r75467 = n;
        double r75468 = -4175041654096867.0;
        double r75469 = 2251799813685248.0;
        double r75470 = r75468 / r75469;
        bool r75471 = r75467 <= r75470;
        double r75472 = 523458166673859.0;
        double r75473 = 562949953421312.0;
        double r75474 = r75472 / r75473;
        bool r75475 = r75467 <= r75474;
        double r75476 = !r75475;
        bool r75477 = r75471 || r75476;
        double r75478 = 1.0;
        double r75479 = r75478 / r75467;
        double r75480 = x;
        double r75481 = r75479 / r75480;
        double r75482 = 2.0;
        double r75483 = r75478 / r75482;
        double r75484 = r75483 / r75467;
        double r75485 = 2.0;
        double r75486 = pow(r75480, r75485);
        double r75487 = r75484 / r75486;
        double r75488 = log(r75480);
        double r75489 = r75488 * r75478;
        double r75490 = pow(r75467, r75485);
        double r75491 = r75480 * r75490;
        double r75492 = r75489 / r75491;
        double r75493 = r75487 - r75492;
        double r75494 = r75481 - r75493;
        double r75495 = r75480 + r75478;
        double r75496 = pow(r75495, r75479);
        double r75497 = cbrt(r75480);
        double r75498 = cbrt(r75497);
        double r75499 = 4.0;
        double r75500 = pow(r75498, r75499);
        double r75501 = exp(r75500);
        double r75502 = log(r75501);
        double r75503 = r75498 * r75498;
        double r75504 = r75502 * r75503;
        double r75505 = pow(r75504, r75479);
        double r75506 = pow(r75497, r75479);
        double r75507 = r75505 * r75506;
        double r75508 = r75496 - r75507;
        double r75509 = r75477 ? r75494 : r75508;
        return r75509;
}

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 < -1.854090949258976 or 0.9298484945110257 < n

    1. Initial program 44.7

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

    2. Taylor expanded around inf 33.5

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

    3. Simplified33.0

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

    if -1.854090949258976 < n < 0.9298484945110257

    1. Initial program 7.9

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

    3. Applied add-cube-cbrt7.9

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

    4. Applied unpow-prod-down7.9

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

    6. Applied add-cube-cbrt7.9

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

    7. Applied add-cube-cbrt7.9

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

    8. Applied swap-sqr7.9

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

    9. Simplified7.9

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

    11. Applied add-log-exp7.9

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\color{blue}{\log \left(e^{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le \frac{-4175041654096867}{2251799813685248} \lor \neg \left(n \le \frac{523458166673859}{562949953421312}\right):\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\log \left(e^{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))