Average Error: 29.4 → 23.4
Time: 28.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.88847562666282 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\log x}{x \cdot \left(n \cdot n\right)} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right) + \frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -6.88847562666282 \cdot 10^{-20}:\\
\;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\
\;\;\;\;\left(\frac{\log x}{x \cdot \left(n \cdot n\right)} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right) + \frac{1}{x \cdot n}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\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 r3841260 = x;
        double r3841261 = 1.0;
        double r3841262 = r3841260 + r3841261;
        double r3841263 = n;
        double r3841264 = r3841261 / r3841263;
        double r3841265 = pow(r3841262, r3841264);
        double r3841266 = pow(r3841260, r3841264);
        double r3841267 = r3841265 - r3841266;
        return r3841267;
}
↓
double f(double x, double n) {
        double r3841268 = 1.0;
        double r3841269 = n;
        double r3841270 = r3841268 / r3841269;
        double r3841271 = -6.88847562666282e-20;
        bool r3841272 = r3841270 <= r3841271;
        double r3841273 = x;
        double r3841274 = r3841273 + r3841268;
        double r3841275 = pow(r3841274, r3841270);
        double r3841276 = exp(r3841275);
        double r3841277 = pow(r3841273, r3841270);
        double r3841278 = exp(r3841277);
        double r3841279 = r3841276 / r3841278;
        double r3841280 = log(r3841279);
        double r3841281 = cbrt(r3841280);
        double r3841282 = r3841275 - r3841277;
        double r3841283 = cbrt(r3841282);
        double r3841284 = r3841283 * r3841283;
        double r3841285 = r3841283 * r3841284;
        double r3841286 = cbrt(r3841285);
        double r3841287 = r3841281 * r3841286;
        double r3841288 = exp(r3841282);
        double r3841289 = log(r3841288);
        double r3841290 = cbrt(r3841289);
        double r3841291 = r3841287 * r3841290;
        double r3841292 = 1.8205393655820836e-32;
        bool r3841293 = r3841270 <= r3841292;
        double r3841294 = log(r3841273);
        double r3841295 = r3841269 * r3841269;
        double r3841296 = r3841273 * r3841295;
        double r3841297 = r3841294 / r3841296;
        double r3841298 = 0.5;
        double r3841299 = r3841273 * r3841273;
        double r3841300 = r3841298 / r3841299;
        double r3841301 = r3841300 / r3841269;
        double r3841302 = r3841297 - r3841301;
        double r3841303 = r3841273 * r3841269;
        double r3841304 = r3841268 / r3841303;
        double r3841305 = r3841302 + r3841304;
        double r3841306 = r3841293 ? r3841305 : r3841291;
        double r3841307 = r3841272 ? r3841291 : r3841306;
        return r3841307;
}

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 (/ 1 n) < -6.88847562666282e-20 or 1.8205393655820836e-32 < (/ 1 n)

    1. Initial program 11.7

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

    3. Applied add-cube-cbrt11.7

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

    5. Applied add-log-exp11.8

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

    6. Applied add-log-exp11.8

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

    7. Applied diff-log11.8

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

    9. Applied add-log-exp11.8

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

    11. Applied add-cube-cbrt11.8

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

    if -6.88847562666282e-20 < (/ 1 n) < 1.8205393655820836e-32

    1. Initial program 44.1

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

    2. Taylor expanded around inf 32.9

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

    3. Simplified32.9

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

  4. Final simplification23.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.88847562666282 \cdot 10^{-20}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.8205393655820836 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\log x}{x \cdot \left(n \cdot n\right)} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right) + \frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))