Average Error: 30.1 → 22.8
Time: 14.8s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -8.677989282032493185982098635872228342691 \cdot 10^{-8}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \log \left(\sqrt{e^{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.779819658150885208826442400589736548601 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -8.677989282032493185982098635872228342691 \cdot 10^{-8}:\\
\;\;\;\;\left(\log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \log \left(\sqrt{e^{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)\\

\mathbf{elif}\;\frac{1}{n} \le 3.779819658150885208826442400589736548601 \cdot 10^{-21}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\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)}}\\

\end{array}
double f(double x, double n) {
        double r88231 = x;
        double r88232 = 1.0;
        double r88233 = r88231 + r88232;
        double r88234 = n;
        double r88235 = r88232 / r88234;
        double r88236 = pow(r88233, r88235);
        double r88237 = pow(r88231, r88235);
        double r88238 = r88236 - r88237;
        return r88238;
}
↓
double f(double x, double n) {
        double r88239 = 1.0;
        double r88240 = n;
        double r88241 = r88239 / r88240;
        double r88242 = -8.677989282032493e-08;
        bool r88243 = r88241 <= r88242;
        double r88244 = x;
        double r88245 = r88244 + r88239;
        double r88246 = pow(r88245, r88241);
        double r88247 = pow(r88244, r88241);
        double r88248 = r88246 - r88247;
        double r88249 = exp(r88248);
        double r88250 = sqrt(r88249);
        double r88251 = sqrt(r88250);
        double r88252 = log(r88251);
        double r88253 = r88252 + r88252;
        double r88254 = 2.0;
        double r88255 = r88241 / r88254;
        double r88256 = pow(r88245, r88255);
        double r88257 = sqrt(r88247);
        double r88258 = r88256 + r88257;
        double r88259 = r88256 - r88257;
        double r88260 = r88258 * r88259;
        double r88261 = exp(r88260);
        double r88262 = sqrt(r88261);
        double r88263 = log(r88262);
        double r88264 = r88253 + r88263;
        double r88265 = 3.779819658150885e-21;
        bool r88266 = r88241 <= r88265;
        double r88267 = r88241 / r88244;
        double r88268 = 0.5;
        double r88269 = r88268 / r88240;
        double r88270 = pow(r88244, r88254);
        double r88271 = r88269 / r88270;
        double r88272 = log(r88244);
        double r88273 = r88272 * r88239;
        double r88274 = pow(r88240, r88254);
        double r88275 = r88244 * r88274;
        double r88276 = r88273 / r88275;
        double r88277 = r88271 - r88276;
        double r88278 = r88267 - r88277;
        double r88279 = cbrt(r88248);
        double r88280 = r88279 * r88279;
        double r88281 = r88280 * r88279;
        double r88282 = r88266 ? r88278 : r88281;
        double r88283 = r88243 ? r88264 : r88282;
        return r88283;
}

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 (/ 1.0 n) < -8.677989282032493e-08

    1. Initial program 0.8

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

    3. Applied add-log-exp1.0

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

    4. Applied add-log-exp0.9

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

    5. Applied diff-log0.9

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

    6. Simplified0.9

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

    8. Applied add-sqr-sqrt0.9

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

    9. Applied log-prod0.9

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

    11. Applied add-sqr-sqrt0.9

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

    12. Applied sqr-pow0.9

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

    13. Applied difference-of-squares0.9

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

    15. Applied add-sqr-sqrt0.9

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

    16. Applied sqrt-prod0.9

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

    17. Applied log-prod0.9

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

    if -8.677989282032493e-08 < (/ 1.0 n) < 3.779819658150885e-21

    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 33.3

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

    3. Simplified32.6

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

    if 3.779819658150885e-21 < (/ 1.0 n)

    1. Initial program 27.1

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

    3. Applied add-cube-cbrt27.1

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

  4. Final simplification22.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -8.677989282032493185982098635872228342691 \cdot 10^{-8}:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt{\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right) + \log \left(\sqrt{e^{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.779819658150885208826442400589736548601 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))