Average Error: 29.7 → 22.6
Time: 42.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.830689015669747194744665829427838275581 \cdot 10^{-22}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \left(\log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 8.232517658085414805800372013352972366818 \cdot 10^{-8}:\\ \;\;\;\;\frac{-0.5}{x \cdot \left(x \cdot n\right)} + \left(\frac{1}{x \cdot n} + \frac{\log x \cdot 1}{\left(x \cdot n\right) \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -7.830689015669747194744665829427838275581 \cdot 10^{-22}:\\
\;\;\;\;\left(\log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \left(\log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\

\mathbf{elif}\;\frac{1}{n} \le 8.232517658085414805800372013352972366818 \cdot 10^{-8}:\\
\;\;\;\;\frac{-0.5}{x \cdot \left(x \cdot n\right)} + \left(\frac{1}{x \cdot n} + \frac{\log x \cdot 1}{\left(x \cdot n\right) \cdot n}\right)\\

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

\end{array}
double f(double x, double n) {
        double r3045246 = x;
        double r3045247 = 1.0;
        double r3045248 = r3045246 + r3045247;
        double r3045249 = n;
        double r3045250 = r3045247 / r3045249;
        double r3045251 = pow(r3045248, r3045250);
        double r3045252 = pow(r3045246, r3045250);
        double r3045253 = r3045251 - r3045252;
        return r3045253;
}
↓
double f(double x, double n) {
        double r3045254 = 1.0;
        double r3045255 = n;
        double r3045256 = r3045254 / r3045255;
        double r3045257 = -7.830689015669747e-22;
        bool r3045258 = r3045256 <= r3045257;
        double r3045259 = x;
        double r3045260 = r3045259 + r3045254;
        double r3045261 = pow(r3045260, r3045256);
        double r3045262 = pow(r3045259, r3045256);
        double r3045263 = r3045261 - r3045262;
        double r3045264 = exp(r3045263);
        double r3045265 = cbrt(r3045264);
        double r3045266 = sqrt(r3045265);
        double r3045267 = log(r3045266);
        double r3045268 = r3045265 * r3045265;
        double r3045269 = sqrt(r3045268);
        double r3045270 = log(r3045269);
        double r3045271 = r3045267 + r3045270;
        double r3045272 = r3045271 + r3045271;
        double r3045273 = 8.232517658085415e-08;
        bool r3045274 = r3045256 <= r3045273;
        double r3045275 = 0.5;
        double r3045276 = -r3045275;
        double r3045277 = r3045259 * r3045255;
        double r3045278 = r3045259 * r3045277;
        double r3045279 = r3045276 / r3045278;
        double r3045280 = r3045254 / r3045277;
        double r3045281 = log(r3045259);
        double r3045282 = r3045281 * r3045254;
        double r3045283 = r3045277 * r3045255;
        double r3045284 = r3045282 / r3045283;
        double r3045285 = r3045280 + r3045284;
        double r3045286 = r3045279 + r3045285;
        double r3045287 = log(r3045264);
        double r3045288 = r3045274 ? r3045286 : r3045287;
        double r3045289 = r3045258 ? r3045272 : r3045288;
        return r3045289;
}

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 (/ 1.0 n) < -7.830689015669747e-22

    1. Initial program 3.0

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

    3. Applied add-log-exp3.2

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

    5. Applied add-sqr-sqrt3.2

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

    6. Applied log-prod3.2

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

    8. Applied add-cube-cbrt3.3

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

    9. Applied sqrt-prod3.3

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

    10. Applied log-prod3.3

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

    12. Applied add-cube-cbrt3.3

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

    13. Applied sqrt-prod3.3

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

    14. Applied log-prod3.3

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

    if -7.830689015669747e-22 < (/ 1.0 n) < 8.232517658085415e-08

    1. Initial program 45.3

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

    2. Taylor expanded around inf 32.6

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

    3. Simplified32.5

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

    if 8.232517658085415e-08 < (/ 1.0 n)

    1. Initial program 24.4

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

    3. Applied add-log-exp24.4

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

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.830689015669747194744665829427838275581 \cdot 10^{-22}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \left(\log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 8.232517658085414805800372013352972366818 \cdot 10^{-8}:\\ \;\;\;\;\frac{-0.5}{x \cdot \left(x \cdot n\right)} + \left(\frac{1}{x \cdot n} + \frac{\log x \cdot 1}{\left(x \cdot n\right) \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))