Average Error: 29.2 → 21.7
Time: 16.6s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -14349362.023440707:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(\left(\frac{1}{n} + \frac{\log x}{{n}^{2}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{elif}\;n \le 59709498.1110491082:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -14349362.023440707:\\
\;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(\left(\frac{1}{n} + \frac{\log x}{{n}^{2}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\

\mathbf{elif}\;n \le 59709498.1110491082:\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\

\end{array}
double f(double x, double n) {
        double r56282 = x;
        double r56283 = 1.0;
        double r56284 = r56282 + r56283;
        double r56285 = n;
        double r56286 = r56283 / r56285;
        double r56287 = pow(r56284, r56286);
        double r56288 = pow(r56282, r56286);
        double r56289 = r56287 - r56288;
        return r56289;
}
↓
double f(double x, double n) {
        double r56290 = n;
        double r56291 = -14349362.023440707;
        bool r56292 = r56290 <= r56291;
        double r56293 = 1.0;
        double r56294 = x;
        double r56295 = r56293 / r56294;
        double r56296 = sqrt(r56295);
        double r56297 = 1.0;
        double r56298 = r56297 / r56290;
        double r56299 = log(r56294);
        double r56300 = 2.0;
        double r56301 = pow(r56290, r56300);
        double r56302 = r56299 / r56301;
        double r56303 = r56298 + r56302;
        double r56304 = r56303 * r56296;
        double r56305 = r56296 * r56304;
        double r56306 = 0.5;
        double r56307 = pow(r56294, r56300);
        double r56308 = r56307 * r56290;
        double r56309 = r56306 / r56308;
        double r56310 = r56305 - r56309;
        double r56311 = 59709498.11104911;
        bool r56312 = r56290 <= r56311;
        double r56313 = r56294 + r56293;
        double r56314 = r56293 / r56290;
        double r56315 = pow(r56313, r56314);
        double r56316 = 3.0;
        double r56317 = pow(r56315, r56316);
        double r56318 = cbrt(r56317);
        double r56319 = pow(r56294, r56314);
        double r56320 = r56318 - r56319;
        double r56321 = -r56299;
        double r56322 = r56321 / r56301;
        double r56323 = r56298 - r56322;
        double r56324 = r56295 * r56323;
        double r56325 = exp(r56308);
        double r56326 = log(r56325);
        double r56327 = r56306 / r56326;
        double r56328 = r56324 - r56327;
        double r56329 = r56312 ? r56320 : r56328;
        double r56330 = r56292 ? r56310 : r56329;
        return r56330;
}

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 < -14349362.023440707

    1. Initial program 44.5

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

    2. Taylor expanded around inf 32.0

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

    3. Simplified31.4

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

    5. Applied add-sqr-sqrt31.5

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

    6. Applied associate-*l*31.5

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

    7. Simplified31.5

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

    if -14349362.023440707 < n < 59709498.11104911

    1. Initial program 8.5

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

    3. Applied add-cbrt-cube8.5

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

    4. Simplified8.6

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

    if 59709498.11104911 < n

    1. Initial program 44.3

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

    2. Taylor expanded around inf 31.8

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

    3. Simplified31.3

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

    5. Applied add-log-exp31.2

      \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{\color{blue}{\log \left(e^{{x}^{2} \cdot n}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -14349362.023440707:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(\left(\frac{1}{n} + \frac{\log x}{{n}^{2}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{elif}\;n \le 59709498.1110491082:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020044 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))