Average Error: 32.5 → 23.7
Time: 13.7s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -119587.819115493898 \lor \neg \left(n \le 4360947995.2746449\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -119587.819115493898 \lor \neg \left(n \le 4360947995.2746449\right):\\
\;\;\;\;\frac{1}{n \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\

\end{array}
double code(double x, double n) {
	return ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n))))));
}
↓
double code(double x, double n) {
	double VAR;
	if (((n <= -119587.8191154939) || !(n <= 4360947995.274645))) {
		VAR = ((double) (1.0 / ((double) (n * x))));
	} else {
		VAR = ((double) (((double) (((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))) + ((double) sqrt(((double) log(((double) exp(((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))))))))))) * ((double) cbrt(((double) pow(((double) (((double) sqrt(((double) pow(((double) (1.0 + x)), ((double) (1.0 / n)))))) - ((double) pow(((double) sqrt(x)), ((double) (1.0 / n)))))), 3.0))))));
	}
	return VAR;
}

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 < -119587.819115493898 or 4360947995.2746449 < 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 64.0

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

    3. Simplified32.3

      \[\leadsto \color{blue}{\left(\frac{1}{x \cdot n} + 0\right) - 0}\]

    if -119587.819115493898 < n < 4360947995.2746449

    1. Initial program 3.4

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

    3. Applied add-sqr-sqrt3.4

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

    4. Applied unpow-prod-down3.4

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

    5. Applied add-sqr-sqrt3.4

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

    6. Applied difference-of-squares3.4

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

    7. Simplified3.4

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

    9. Applied add-log-exp3.5

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

    11. Applied add-cbrt-cube3.5

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

    12. Simplified3.5

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

  4. Final simplification23.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -119587.819115493898 \lor \neg \left(n \le 4360947995.2746449\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \sqrt[3]{{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))