Average Error: 28.9 → 21.7
Time: 20.1s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -946530.043801923166:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)} \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right) \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\\ \mathbf{elif}\;n \le 20201670204046.1445:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -946530.043801923166:\\
\;\;\;\;\left(\sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)} \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right) \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\\

\mathbf{elif}\;n \le 20201670204046.1445:\\
\;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\

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

\end{array}
double f(double x, double n) {
        double r271 = x;
        double r272 = 1.0;
        double r273 = r271 + r272;
        double r274 = n;
        double r275 = r272 / r274;
        double r276 = pow(r273, r275);
        double r277 = pow(r271, r275);
        double r278 = r276 - r277;
        return r278;
}
↓
double f(double x, double n) {
        double r279 = n;
        double r280 = -946530.0438019232;
        bool r281 = r279 <= r280;
        double r282 = 1.0;
        double r283 = r282 / r279;
        double r284 = x;
        double r285 = r283 / r284;
        double r286 = 0.5;
        double r287 = r286 / r279;
        double r288 = 2.0;
        double r289 = pow(r284, r288);
        double r290 = r287 / r289;
        double r291 = log(r284);
        double r292 = r291 * r282;
        double r293 = pow(r279, r288);
        double r294 = r284 * r293;
        double r295 = r292 / r294;
        double r296 = r290 - r295;
        double r297 = r285 - r296;
        double r298 = cbrt(r297);
        double r299 = r298 * r298;
        double r300 = r299 * r298;
        double r301 = 20201670204046.145;
        bool r302 = r279 <= r301;
        double r303 = r284 + r282;
        double r304 = pow(r303, r283);
        double r305 = pow(r284, r283);
        double r306 = sqrt(r305);
        double r307 = r306 * r306;
        double r308 = r304 - r307;
        double r309 = r289 * r279;
        double r310 = r286 / r309;
        double r311 = exp(r310);
        double r312 = log(r311);
        double r313 = r312 - r295;
        double r314 = r285 - r313;
        double r315 = r302 ? r308 : r314;
        double r316 = r281 ? r300 : r315;
        return r316;
}

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 < -946530.0438019232

    1. Initial program 44.4

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

    2. Taylor expanded around inf 31.7

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

    3. Simplified31.1

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

    5. Applied add-cube-cbrt31.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)} \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right) \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}}\]

    if -946530.0438019232 < n < 20201670204046.145

    1. Initial program 8.3

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

    3. Applied add-sqr-sqrt8.3

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

    if 20201670204046.145 < n

    1. Initial program 45.0

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

    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

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

    5. Applied add-log-exp32.4

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

    6. Simplified32.4

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

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -946530.043801923166:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)} \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\right) \cdot \sqrt[3]{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\\ \mathbf{elif}\;n \le 20201670204046.1445:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))