Average Error: 28.9 → 22.4
Time: 14.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1469491090180712 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.2787797536218279 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot {\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -1.1469491090180712 \cdot 10^{-6}:\\
\;\;\;\;{\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(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\\

\mathbf{elif}\;\frac{1}{n} \le 1.2787797536218279 \cdot 10^{-20}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot {\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\

\end{array}
double f(double x, double n) {
        double r77373 = x;
        double r77374 = 1.0;
        double r77375 = r77373 + r77374;
        double r77376 = n;
        double r77377 = r77374 / r77376;
        double r77378 = pow(r77375, r77377);
        double r77379 = pow(r77373, r77377);
        double r77380 = r77378 - r77379;
        return r77380;
}
↓
double f(double x, double n) {
        double r77381 = 1.0;
        double r77382 = n;
        double r77383 = r77381 / r77382;
        double r77384 = -1.1469491090180712e-06;
        bool r77385 = r77383 <= r77384;
        double r77386 = x;
        double r77387 = r77386 + r77381;
        double r77388 = cbrt(r77387);
        double r77389 = r77388 * r77388;
        double r77390 = pow(r77389, r77383);
        double r77391 = pow(r77388, r77383);
        double r77392 = r77390 * r77391;
        double r77393 = pow(r77386, r77383);
        double r77394 = cbrt(r77393);
        double r77395 = r77394 * r77394;
        double r77396 = r77395 * r77394;
        double r77397 = r77392 - r77396;
        double r77398 = 1.2787797536218279e-20;
        bool r77399 = r77383 <= r77398;
        double r77400 = 1.0;
        double r77401 = r77386 * r77382;
        double r77402 = r77400 / r77401;
        double r77403 = 0.5;
        double r77404 = 2.0;
        double r77405 = pow(r77386, r77404);
        double r77406 = r77405 * r77382;
        double r77407 = r77400 / r77406;
        double r77408 = r77400 / r77386;
        double r77409 = log(r77408);
        double r77410 = pow(r77382, r77404);
        double r77411 = r77386 * r77410;
        double r77412 = r77409 / r77411;
        double r77413 = r77381 * r77412;
        double r77414 = fma(r77403, r77407, r77413);
        double r77415 = -r77414;
        double r77416 = fma(r77381, r77402, r77415);
        double r77417 = 0.6666666666666666;
        double r77418 = r77417 * r77383;
        double r77419 = pow(r77387, r77418);
        double r77420 = cbrt(r77419);
        double r77421 = r77420 * r77420;
        double r77422 = r77421 * r77420;
        double r77423 = cbrt(r77383);
        double r77424 = r77423 * r77423;
        double r77425 = pow(r77388, r77424);
        double r77426 = pow(r77425, r77423);
        double r77427 = r77422 * r77426;
        double r77428 = r77427 - r77393;
        double r77429 = r77399 ? r77416 : r77428;
        double r77430 = r77385 ? r77397 : r77429;
        return r77430;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (/ 1.0 n) < -1.1469491090180712e-06

    1. Initial program 0.6

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

    3. Applied add-cube-cbrt0.6

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

    4. Applied unpow-prod-down0.6

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

    6. Applied add-cube-cbrt0.6

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

    if -1.1469491090180712e-06 < (/ 1.0 n) < 1.2787797536218279e-20

    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 32.7

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

    3. Simplified32.7

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

    if 1.2787797536218279e-20 < (/ 1.0 n)

    1. Initial program 27.1

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

    3. Applied add-cube-cbrt27.1

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

    4. Applied unpow-prod-down27.2

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

    6. Applied pow1/327.2

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

    7. Applied pow1/327.2

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

    8. Applied pow-prod-up27.2

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

    9. Applied pow-pow27.1

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

    10. Simplified27.1

      \[\leadsto {\left(x + 1\right)}^{\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt27.2

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

    13. Applied pow-unpow27.2

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)} \cdot \color{blue}{{\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)}} - {x}^{\left(\frac{1}{n}\right)}\]
    14. Using strategy rm

    15. Applied add-cube-cbrt27.2

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1469491090180712 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.2787797536218279 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{2}{3} \cdot \frac{1}{n}\right)}}\right) \cdot {\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))