Average Error: 29.5 → 22.7
Time: 31.5s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -490576324.36488575:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \mathbf{elif}\;n \le 1.180746203929133 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -490576324.36488575:\\
\;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\

\mathbf{elif}\;n \le 1.180746203929133 \cdot 10^{+18}:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\

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

\end{array}
double f(double x, double n) {
        double r3135170 = x;
        double r3135171 = 1.0;
        double r3135172 = r3135170 + r3135171;
        double r3135173 = n;
        double r3135174 = r3135171 / r3135173;
        double r3135175 = pow(r3135172, r3135174);
        double r3135176 = pow(r3135170, r3135174);
        double r3135177 = r3135175 - r3135176;
        return r3135177;
}
↓
double f(double x, double n) {
        double r3135178 = n;
        double r3135179 = -490576324.36488575;
        bool r3135180 = r3135178 <= r3135179;
        double r3135181 = 1.0;
        double r3135182 = x;
        double r3135183 = r3135182 * r3135178;
        double r3135184 = r3135181 / r3135183;
        double r3135185 = 0.5;
        double r3135186 = r3135185 / r3135178;
        double r3135187 = r3135182 * r3135182;
        double r3135188 = r3135186 / r3135187;
        double r3135189 = log(r3135182);
        double r3135190 = r3135178 * r3135178;
        double r3135191 = r3135189 / r3135190;
        double r3135192 = r3135191 / r3135182;
        double r3135193 = r3135188 - r3135192;
        double r3135194 = r3135184 - r3135193;
        double r3135195 = 1.180746203929133e+18;
        bool r3135196 = r3135178 <= r3135195;
        double r3135197 = r3135181 + r3135182;
        double r3135198 = r3135181 / r3135178;
        double r3135199 = pow(r3135197, r3135198);
        double r3135200 = pow(r3135182, r3135198);
        double r3135201 = r3135199 - r3135200;
        double r3135202 = exp(r3135201);
        double r3135203 = log(r3135202);
        double r3135204 = cbrt(r3135203);
        double r3135205 = 2.0;
        double r3135206 = r3135198 / r3135205;
        double r3135207 = pow(r3135182, r3135206);
        double r3135208 = sqrt(r3135199);
        double r3135209 = r3135207 + r3135208;
        double r3135210 = sqrt(r3135208);
        double r3135211 = r3135206 / r3135205;
        double r3135212 = pow(r3135182, r3135211);
        double r3135213 = r3135210 + r3135212;
        double r3135214 = r3135210 - r3135212;
        double r3135215 = r3135213 * r3135214;
        double r3135216 = r3135209 * r3135215;
        double r3135217 = cbrt(r3135216);
        double r3135218 = r3135204 * r3135217;
        double r3135219 = cbrt(r3135201);
        double r3135220 = r3135218 * r3135219;
        double r3135221 = r3135196 ? r3135220 : r3135194;
        double r3135222 = r3135180 ? r3135194 : r3135221;
        return r3135222;
}

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 < -490576324.36488575 or 1.180746203929133e+18 < n

    1. Initial program 45.7

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

    2. Taylor expanded around inf 33.3

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

    3. Simplified33.3

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

    if -490576324.36488575 < n < 1.180746203929133e+18

    1. Initial program 9.4

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

    3. Applied add-cube-cbrt9.4

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

    5. Applied sqr-pow9.4

      \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

    6. Applied add-sqr-sqrt9.4

      \[\leadsto \left(\sqrt[3]{\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

    7. Applied difference-of-squares9.4

      \[\leadsto \left(\sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]
    8. Using strategy rm

    9. Applied sqr-pow9.5

      \[\leadsto \left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \color{blue}{{x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

    10. Applied add-sqr-sqrt9.5

      \[\leadsto \left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

    11. Applied sqrt-prod9.5

      \[\leadsto \left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\color{blue}{\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]

    12. Applied difference-of-squares9.5

      \[\leadsto \left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\sqrt{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\]
    13. Using strategy rm

    14. Applied add-log-exp9.6

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

    15. Applied add-log-exp9.6

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

    16. Applied diff-log9.6

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

    17. Simplified9.6

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -490576324.36488575:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \mathbf{elif}\;n \le 1.180746203929133 \cdot 10^{+18}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} + {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right) \cdot \left(\sqrt{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{\frac{\log x}{n \cdot n}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))