Average Error: 29.2 → 22.2
Time: 27.4s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -17527377755.61773681640625 \lor \neg \left(n \le 2504539914490964992\right):\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{0.5}{x \cdot n} - \frac{0.25}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) - \frac{0.25 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -17527377755.61773681640625 \lor \neg \left(n \le 2504539914490964992\right):\\
\;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{0.5}{x \cdot n} - \frac{0.25}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) - \frac{0.25 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right)\\

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

\end{array}
double f(double x, double n) {
        double r67193 = x;
        double r67194 = 1.0;
        double r67195 = r67193 + r67194;
        double r67196 = n;
        double r67197 = r67194 / r67196;
        double r67198 = pow(r67195, r67197);
        double r67199 = pow(r67193, r67197);
        double r67200 = r67198 - r67199;
        return r67200;
}
↓
double f(double x, double n) {
        double r67201 = n;
        double r67202 = -17527377755.617737;
        bool r67203 = r67201 <= r67202;
        double r67204 = 2.504539914490965e+18;
        bool r67205 = r67201 <= r67204;
        double r67206 = !r67205;
        bool r67207 = r67203 || r67206;
        double r67208 = x;
        double r67209 = 1.0;
        double r67210 = r67208 + r67209;
        double r67211 = r67209 / r67201;
        double r67212 = 2.0;
        double r67213 = r67211 / r67212;
        double r67214 = pow(r67210, r67213);
        double r67215 = pow(r67208, r67213);
        double r67216 = r67214 + r67215;
        double r67217 = 0.5;
        double r67218 = r67208 * r67201;
        double r67219 = r67217 / r67218;
        double r67220 = 0.25;
        double r67221 = pow(r67208, r67212);
        double r67222 = r67221 * r67201;
        double r67223 = exp(r67222);
        double r67224 = log(r67223);
        double r67225 = r67220 / r67224;
        double r67226 = r67219 - r67225;
        double r67227 = log(r67208);
        double r67228 = -r67227;
        double r67229 = r67220 * r67228;
        double r67230 = pow(r67201, r67212);
        double r67231 = r67208 * r67230;
        double r67232 = r67229 / r67231;
        double r67233 = r67226 - r67232;
        double r67234 = r67216 * r67233;
        double r67235 = r67214 - r67215;
        double r67236 = r67216 * r67235;
        double r67237 = r67207 ? r67234 : r67236;
        return r67237;
}

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 < -17527377755.617737 or 2.504539914490965e+18 < n

    1. Initial program 45.4

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

    3. Applied sqr-pow45.5

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

    4. Applied sqr-pow45.4

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

    5. Applied difference-of-squares45.4

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

    6. Taylor expanded around inf 32.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

    7. Simplified32.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\frac{0.5}{x \cdot n} - \frac{0.25}{{x}^{2} \cdot n}\right) - \frac{0.25 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right)}\]
    8. Using strategy rm

    9. Applied add-log-exp32.9

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{0.5}{x \cdot n} - \frac{0.25}{\color{blue}{\log \left(e^{{x}^{2} \cdot n}\right)}}\right) - \frac{0.25 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right)\]

    if -17527377755.617737 < n < 2.504539914490965e+18

    1. Initial program 9.0

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

    3. Applied sqr-pow9.1

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

    4. Applied sqr-pow9.0

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

    5. Applied difference-of-squares9.0

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

  4. Final simplification22.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -17527377755.61773681640625 \lor \neg \left(n \le 2504539914490964992\right):\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\frac{0.5}{x \cdot n} - \frac{0.25}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) - \frac{0.25 \cdot \left(-\log x\right)}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))