Average Error: 29.4 → 22.0
Time: 31.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -48075474.504670314:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \mathbf{elif}\;n \le 55908.36668550803:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -48075474.504670314:\\
\;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\

\mathbf{elif}\;n \le 55908.36668550803:\\
\;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\

\end{array}
double f(double x, double n) {
        double r2444226 = x;
        double r2444227 = 1.0;
        double r2444228 = r2444226 + r2444227;
        double r2444229 = n;
        double r2444230 = r2444227 / r2444229;
        double r2444231 = pow(r2444228, r2444230);
        double r2444232 = pow(r2444226, r2444230);
        double r2444233 = r2444231 - r2444232;
        return r2444233;
}
↓
double f(double x, double n) {
        double r2444234 = n;
        double r2444235 = -48075474.504670314;
        bool r2444236 = r2444234 <= r2444235;
        double r2444237 = 1.0;
        double r2444238 = r2444237 / r2444234;
        double r2444239 = x;
        double r2444240 = r2444238 / r2444239;
        double r2444241 = log(r2444239);
        double r2444242 = r2444241 / r2444239;
        double r2444243 = r2444234 * r2444234;
        double r2444244 = r2444242 / r2444243;
        double r2444245 = 0.5;
        double r2444246 = r2444239 * r2444239;
        double r2444247 = r2444245 / r2444246;
        double r2444248 = r2444247 / r2444234;
        double r2444249 = r2444244 - r2444248;
        double r2444250 = r2444240 + r2444249;
        double r2444251 = 55908.36668550803;
        bool r2444252 = r2444234 <= r2444251;
        double r2444253 = r2444237 + r2444239;
        double r2444254 = pow(r2444253, r2444238);
        double r2444255 = pow(r2444239, r2444238);
        double r2444256 = r2444254 - r2444255;
        double r2444257 = cbrt(r2444256);
        double r2444258 = sqrt(r2444254);
        double r2444259 = sqrt(r2444255);
        double r2444260 = r2444258 - r2444259;
        double r2444261 = r2444259 + r2444258;
        double r2444262 = r2444260 * r2444261;
        double r2444263 = cbrt(r2444262);
        double r2444264 = r2444263 * r2444257;
        double r2444265 = exp(r2444264);
        double r2444266 = log(r2444265);
        double r2444267 = r2444257 * r2444266;
        double r2444268 = r2444252 ? r2444267 : r2444250;
        double r2444269 = r2444236 ? r2444250 : r2444268;
        return r2444269;
}

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 n < -48075474.504670314 or 55908.36668550803 < n

    1. Initial program 45.6

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

    2. Taylor expanded around inf 33.2

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

    3. Simplified32.4

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

    if -48075474.504670314 < n < 55908.36668550803

    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 add-log-exp8.3

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

    4. Applied add-log-exp8.3

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

    5. Applied diff-log8.3

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

    6. Simplified8.3

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

    8. Applied add-cube-cbrt8.3

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

    9. Applied exp-prod8.3

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

    10. Applied log-pow8.3

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

    12. Applied add-sqr-sqrt8.3

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

    13. Applied add-sqr-sqrt8.3

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

    14. Applied difference-of-squares8.3

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -48075474.504670314:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \mathbf{elif}\;n \le 55908.36668550803:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{x}}{n \cdot n} - \frac{\frac{\frac{1}{2}}{x \cdot x}}{n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))