Average Error: 33.3 → 24.1
Time: 13.4s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.096355036664333621:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 6.8280679940979408 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\left(\sqrt[3]{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}\right)} \cdot \sqrt[3]{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}\right)}\right) \cdot \sqrt[3]{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}\right)}}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -0.096355036664333621:\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}}\\

\mathbf{elif}\;\frac{1}{n} \le 6.8280679940979408 \cdot 10^{-9}:\\
\;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\

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

\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 ((((double) (1.0 / n)) <= -0.09635503666433362)) {
		VAR = ((double) cbrt(((double) pow(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) exp(((double) (((double) log(x)) * ((double) (1.0 / n)))))))), 3.0))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 6.828067994097941e-09)) {
			VAR_1 = ((double) (((double) (((double) (((double) (1.0 / n)) / x)) + ((double) (1.0 * ((double) (0.0 - ((double) (((double) log(((double) (1.0 / x)))) / ((double) (x * ((double) pow(n, 2.0)))))))))))) - ((double) (((double) (0.5 / n)) / ((double) pow(x, 2.0))))));
		} else {
			VAR_1 = ((double) cbrt(((double) exp(((double) (((double) (((double) cbrt(((double) log(((double) pow(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) exp(((double) (((double) log(x)) * ((double) (1.0 / n)))))))), 3.0)))))) * ((double) cbrt(((double) log(((double) pow(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) exp(((double) (((double) log(x)) * ((double) (1.0 / n)))))))), 3.0)))))))) * ((double) cbrt(((double) log(((double) pow(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) exp(((double) (((double) log(x)) * ((double) (1.0 / n)))))))), 3.0))))))))))));
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if (/ 1.0 n) < -0.096355036664333621

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

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

    4. Applied pow-exp0.2

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

    6. Applied add-cbrt-cube0.3

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

    7. Simplified0.3

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

    if -0.096355036664333621 < (/ 1.0 n) < 6.8280679940979408e-9

    1. Initial program 45.5

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

    2. Taylor expanded around inf 33.2

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

    3. Simplified32.7

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}}\]

    if 6.8280679940979408e-9 < (/ 1.0 n)

    1. Initial program 6.7

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

    3. Applied add-exp-log6.7

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

    4. Applied pow-exp6.7

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

    6. Applied add-cbrt-cube6.7

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

    7. Simplified6.7

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

    9. Applied add-exp-log6.7

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

    10. Applied pow-exp6.7

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

    11. Simplified6.7

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

    13. Applied add-cube-cbrt6.7

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

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.096355036664333621:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 6.8280679940979408 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{e^{\left(\sqrt[3]{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}\right)} \cdot \sqrt[3]{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}\right)}\right) \cdot \sqrt[3]{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log x \cdot \frac{1}{n}}\right)}^{3}\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))