Average Error: 29.4 → 22.3
Time: 27.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.5351028006273464976416676108783576637506 \lor \neg \left(\frac{1}{n} \le 4.133628459687642410799750514629715336321 \cdot 10^{-12}\right):\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + 1 \cdot \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.5351028006273464976416676108783576637506 \lor \neg \left(\frac{1}{n} \le 4.133628459687642410799750514629715336321 \cdot 10^{-12}\right):\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\

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

\end{array}
double f(double x, double n) {
        double r58280 = x;
        double r58281 = 1.0;
        double r58282 = r58280 + r58281;
        double r58283 = n;
        double r58284 = r58281 / r58283;
        double r58285 = pow(r58282, r58284);
        double r58286 = pow(r58280, r58284);
        double r58287 = r58285 - r58286;
        return r58287;
}
↓
double f(double x, double n) {
        double r58288 = 1.0;
        double r58289 = n;
        double r58290 = r58288 / r58289;
        double r58291 = -0.5351028006273465;
        bool r58292 = r58290 <= r58291;
        double r58293 = 4.1336284596876424e-12;
        bool r58294 = r58290 <= r58293;
        double r58295 = !r58294;
        bool r58296 = r58292 || r58295;
        double r58297 = x;
        double r58298 = r58297 + r58288;
        double r58299 = pow(r58298, r58290);
        double r58300 = pow(r58297, r58290);
        double r58301 = 3.0;
        double r58302 = pow(r58300, r58301);
        double r58303 = cbrt(r58302);
        double r58304 = r58299 - r58303;
        double r58305 = r58288 / r58297;
        double r58306 = r58305 / r58289;
        double r58307 = log(r58297);
        double r58308 = 2.0;
        double r58309 = pow(r58289, r58308);
        double r58310 = r58297 * r58309;
        double r58311 = r58307 / r58310;
        double r58312 = r58288 * r58311;
        double r58313 = r58306 + r58312;
        double r58314 = 0.5;
        double r58315 = pow(r58297, r58308);
        double r58316 = r58315 * r58289;
        double r58317 = r58314 / r58316;
        double r58318 = r58313 - r58317;
        double r58319 = r58296 ? r58304 : r58318;
        return r58319;
}

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 (/ 1.0 n) < -0.5351028006273465 or 4.1336284596876424e-12 < (/ 1.0 n)

    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-cube8.6

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

    4. Simplified8.6

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

    if -0.5351028006273465 < (/ 1.0 n) < 4.1336284596876424e-12

    1. Initial program 44.6

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

    2. Taylor expanded around inf 32.9

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

    3. Simplified32.3

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

    5. Applied sub-neg32.3

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

    6. Applied distribute-lft-in32.3

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

    7. Simplified32.3

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

    8. Simplified32.3

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

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.5351028006273464976416676108783576637506 \lor \neg \left(\frac{1}{n} \le 4.133628459687642410799750514629715336321 \cdot 10^{-12}\right):\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + 1 \cdot \frac{\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))