Average Error: 29.3 → 22.0
Time: 2.0m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -24784678.5745056308805942535400390625:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 37.00630601379255324445693986490368843079:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(1, \frac{-\log x}{\left(x \cdot n\right) \cdot n}, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -24784678.5745056308805942535400390625:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1}{\left(x \cdot n\right) \cdot n}\\

\mathbf{elif}\;n \le 37.00630601379255324445693986490368843079:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(1, \frac{-\log x}{\left(x \cdot n\right) \cdot n}, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\

\end{array}
double f(double x, double n) {
        double r2801344 = x;
        double r2801345 = 1.0;
        double r2801346 = r2801344 + r2801345;
        double r2801347 = n;
        double r2801348 = r2801345 / r2801347;
        double r2801349 = pow(r2801346, r2801348);
        double r2801350 = pow(r2801344, r2801348);
        double r2801351 = r2801349 - r2801350;
        return r2801351;
}
↓
double f(double x, double n) {
        double r2801352 = n;
        double r2801353 = -24784678.57450563;
        bool r2801354 = r2801352 <= r2801353;
        double r2801355 = 1.0;
        double r2801356 = x;
        double r2801357 = r2801355 / r2801356;
        double r2801358 = r2801357 / r2801352;
        double r2801359 = 0.5;
        double r2801360 = r2801356 * r2801356;
        double r2801361 = r2801352 * r2801360;
        double r2801362 = r2801359 / r2801361;
        double r2801363 = r2801358 - r2801362;
        double r2801364 = log(r2801356);
        double r2801365 = r2801364 * r2801355;
        double r2801366 = r2801356 * r2801352;
        double r2801367 = r2801366 * r2801352;
        double r2801368 = r2801365 / r2801367;
        double r2801369 = r2801363 + r2801368;
        double r2801370 = 37.00630601379255;
        bool r2801371 = r2801352 <= r2801370;
        double r2801372 = r2801356 + r2801355;
        double r2801373 = r2801355 / r2801352;
        double r2801374 = pow(r2801372, r2801373);
        double r2801375 = cbrt(r2801374);
        double r2801376 = r2801375 * r2801375;
        double r2801377 = 2.0;
        double r2801378 = r2801373 / r2801377;
        double r2801379 = pow(r2801356, r2801378);
        double r2801380 = -r2801379;
        double r2801381 = r2801379 * r2801380;
        double r2801382 = fma(r2801376, r2801375, r2801381);
        double r2801383 = cbrt(r2801382);
        double r2801384 = r2801383 * r2801383;
        double r2801385 = r2801384 * r2801383;
        double r2801386 = r2801373 / r2801356;
        double r2801387 = -r2801364;
        double r2801388 = r2801387 / r2801367;
        double r2801389 = r2801359 / r2801352;
        double r2801390 = r2801389 / r2801360;
        double r2801391 = fma(r2801355, r2801388, r2801390);
        double r2801392 = r2801386 - r2801391;
        double r2801393 = r2801371 ? r2801385 : r2801392;
        double r2801394 = r2801354 ? r2801369 : r2801393;
        return r2801394;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -24784678.57450563

    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 33.0

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

    3. Simplified32.2

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

    if -24784678.57450563 < n < 37.00630601379255

    1. Initial program 8.2

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

    3. Applied sqr-pow8.2

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

    4. Applied add-cube-cbrt8.2

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

    5. Applied prod-diff8.2

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

    6. Simplified8.2

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \color{blue}{0}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt8.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} + 0\]

    if 37.00630601379255 < n

    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

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

    4. Applied pow-unpow44.9

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

    5. Taylor expanded around inf 32.9

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

    6. Simplified32.4

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(1, -\frac{\log x}{n \cdot \left(n \cdot x\right)}, \frac{\frac{0.5}{n}}{x \cdot x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -24784678.5745056308805942535400390625:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) + \frac{\log x \cdot 1}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 37.00630601379255324445693986490368843079:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot \left(-{x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(1, \frac{-\log x}{\left(x \cdot n\right) \cdot n}, \frac{\frac{0.5}{n}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))