Average Error: 29.4 → 22.4
Time: 58.0s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -27159841.041252471506595611572265625:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot \left(n \cdot n\right)}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \mathbf{elif}\;n \le 353.8529854336741777842689771205186843872:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \log \left(e^{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot \left(n \cdot n\right)}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -27159841.041252471506595611572265625:\\
\;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot \left(n \cdot n\right)}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\

\mathbf{elif}\;n \le 353.8529854336741777842689771205186843872:\\
\;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \log \left(e^{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r2791271 = x;
        double r2791272 = 1.0;
        double r2791273 = r2791271 + r2791272;
        double r2791274 = n;
        double r2791275 = r2791272 / r2791274;
        double r2791276 = pow(r2791273, r2791275);
        double r2791277 = pow(r2791271, r2791275);
        double r2791278 = r2791276 - r2791277;
        return r2791278;
}
↓
double f(double x, double n) {
        double r2791279 = n;
        double r2791280 = -27159841.04125247;
        bool r2791281 = r2791279 <= r2791280;
        double r2791282 = 1.0;
        double r2791283 = x;
        double r2791284 = r2791282 / r2791283;
        double r2791285 = r2791284 / r2791279;
        double r2791286 = log(r2791283);
        double r2791287 = -r2791286;
        double r2791288 = r2791279 * r2791279;
        double r2791289 = r2791283 * r2791288;
        double r2791290 = r2791287 / r2791289;
        double r2791291 = 0.5;
        double r2791292 = r2791283 * r2791283;
        double r2791293 = r2791292 * r2791279;
        double r2791294 = r2791291 / r2791293;
        double r2791295 = fma(r2791282, r2791290, r2791294);
        double r2791296 = r2791285 - r2791295;
        double r2791297 = 353.8529854336742;
        bool r2791298 = r2791279 <= r2791297;
        double r2791299 = r2791282 + r2791283;
        double r2791300 = r2791282 / r2791279;
        double r2791301 = pow(r2791299, r2791300);
        double r2791302 = cbrt(r2791283);
        double r2791303 = exp(r2791302);
        double r2791304 = log(r2791303);
        double r2791305 = r2791302 * r2791304;
        double r2791306 = pow(r2791305, r2791300);
        double r2791307 = pow(r2791302, r2791300);
        double r2791308 = r2791306 * r2791307;
        double r2791309 = r2791301 - r2791308;
        double r2791310 = r2791298 ? r2791309 : r2791296;
        double r2791311 = r2791281 ? r2791296 : r2791310;
        return r2791311;
}

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -27159841.04125247 or 353.8529854336742 < n

    1. Initial program 44.7

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

    3. Applied sqr-pow44.7

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

    4. Taylor expanded around inf 32.7

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

    5. Simplified32.2

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

    if -27159841.04125247 < n < 353.8529854336742

    1. Initial program 8.4

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

    3. Applied sqr-pow8.4

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

    5. Applied add-cube-cbrt8.4

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

    6. Applied unpow-prod-down8.4

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

    7. Applied prod-diff8.4

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

    8. Simplified8.4

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

    9. Taylor expanded around 0 8.4

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

    11. Applied add-log-exp9.0

      \[\leadsto \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \color{blue}{\log \left(e^{\sqrt[3]{x}}\right)}\right)}^{\left(\frac{1}{n}\right)}\right) + 0\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -27159841.041252471506595611572265625:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot \left(n \cdot n\right)}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \mathbf{elif}\;n \le 353.8529854336741777842689771205186843872:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \log \left(e^{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot \left(n \cdot n\right)}, \frac{0.5}{\left(x \cdot x\right) \cdot n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))