Average Error: 29.3 → 22.1
Time: 2.4m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -24784678.5745056308805942535400390625:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{0.5}{x \cdot x}}{n}\right) + \frac{\log x \cdot 1}{x \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \le 0.6224037324982906183024056190333794802427:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{x \cdot x}\right) - \left(-\frac{\frac{\log x}{x}}{n} \cdot \frac{1}{n}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -24784678.5745056308805942535400390625:\\
\;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{0.5}{x \cdot x}}{n}\right) + \frac{\log x \cdot 1}{x \cdot \left(n \cdot n\right)}\\

\mathbf{elif}\;n \le 0.6224037324982906183024056190333794802427:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\

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

\end{array}
double f(double x, double n) {
        double r3550194 = x;
        double r3550195 = 1.0;
        double r3550196 = r3550194 + r3550195;
        double r3550197 = n;
        double r3550198 = r3550195 / r3550197;
        double r3550199 = pow(r3550196, r3550198);
        double r3550200 = pow(r3550194, r3550198);
        double r3550201 = r3550199 - r3550200;
        return r3550201;
}
↓
double f(double x, double n) {
        double r3550202 = n;
        double r3550203 = -24784678.57450563;
        bool r3550204 = r3550202 <= r3550203;
        double r3550205 = 1.0;
        double r3550206 = x;
        double r3550207 = r3550205 / r3550206;
        double r3550208 = r3550207 / r3550202;
        double r3550209 = 0.5;
        double r3550210 = r3550206 * r3550206;
        double r3550211 = r3550209 / r3550210;
        double r3550212 = r3550211 / r3550202;
        double r3550213 = r3550208 - r3550212;
        double r3550214 = log(r3550206);
        double r3550215 = r3550214 * r3550205;
        double r3550216 = r3550202 * r3550202;
        double r3550217 = r3550206 * r3550216;
        double r3550218 = r3550215 / r3550217;
        double r3550219 = r3550213 + r3550218;
        double r3550220 = 0.6224037324982906;
        bool r3550221 = r3550202 <= r3550220;
        double r3550222 = r3550206 + r3550205;
        double r3550223 = r3550205 / r3550202;
        double r3550224 = cbrt(r3550223);
        double r3550225 = r3550224 * r3550224;
        double r3550226 = pow(r3550222, r3550225);
        double r3550227 = pow(r3550226, r3550224);
        double r3550228 = pow(r3550206, r3550223);
        double r3550229 = r3550227 - r3550228;
        double r3550230 = exp(r3550229);
        double r3550231 = log(r3550230);
        double r3550232 = cbrt(r3550231);
        double r3550233 = r3550232 * r3550232;
        double r3550234 = r3550233 * r3550232;
        double r3550235 = r3550223 / r3550206;
        double r3550236 = r3550209 / r3550202;
        double r3550237 = r3550236 / r3550210;
        double r3550238 = r3550235 - r3550237;
        double r3550239 = r3550214 / r3550206;
        double r3550240 = r3550239 / r3550202;
        double r3550241 = r3550240 * r3550223;
        double r3550242 = -r3550241;
        double r3550243 = r3550238 - r3550242;
        double r3550244 = r3550221 ? r3550234 : r3550243;
        double r3550245 = r3550204 ? r3550219 : r3550244;
        return r3550245;
}

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 n < -24784678.57450563

    1. Initial program 45.0

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

    3. Applied add-cube-cbrt45.0

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

    4. Applied pow-unpow45.0

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

    5. Taylor expanded around inf 33.0

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

    6. Simplified32.2

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

    if -24784678.57450563 < n < 0.6224037324982906

    1. Initial program 8.1

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

    3. Applied add-cube-cbrt8.1

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

    4. Applied pow-unpow8.1

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

    6. Applied add-log-exp8.2

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

    8. Applied add-cube-cbrt8.2

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

    if 0.6224037324982906 < n

    1. Initial program 44.8

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

    3. Applied add-cube-cbrt44.8

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

    4. Applied pow-unpow44.8

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

    6. Applied add-log-exp44.9

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

    7. Taylor expanded around inf 33.1

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

    8. Simplified32.5

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{x \cdot x}\right) - \frac{1}{n} \cdot \frac{\frac{-\log x}{x}}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -24784678.5745056308805942535400390625:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{0.5}{x \cdot x}}{n}\right) + \frac{\log x \cdot 1}{x \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \le 0.6224037324982906183024056190333794802427:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{{\left({\left(x + 1\right)}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{n}}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{0.5}{n}}{x \cdot x}\right) - \left(-\frac{\frac{\log x}{x}}{n} \cdot \frac{1}{n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))