Average Error: 32.7 → 23.5
Time: 22.0s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.40843517291627578 \cdot 10^{-17}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 6.9300284984344972 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(\frac{1}{{n}^{2}} \cdot \frac{0}{x} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {x}^{\left(2 \cdot \frac{1}{n}\right)}}{\left(\sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}}\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -6.40843517291627578 \cdot 10^{-17}:\\
\;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 6.9300284984344972 \cdot 10^{-10}:\\
\;\;\;\;\left(\left(\frac{1}{{n}^{2}} \cdot \frac{0}{x} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {x}^{\left(2 \cdot \frac{1}{n}\right)}}{\left(\sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\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)) <= -6.408435172916276e-17)) {
		VAR = ((double) (((double) cbrt(((double) pow(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))), 3.0)))) - ((double) pow(x, ((double) (1.0 / n))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 6.930028498434497e-10)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (1.0 / ((double) pow(n, 2.0)))) * ((double) (0.0 / x)))) - ((double) (1.0 * ((double) (((double) log(((double) (1.0 / x)))) / ((double) (x * ((double) pow(n, 2.0)))))))))) - ((double) (0.5 * ((double) (1.0 / ((double) (((double) pow(x, 2.0)) * n)))))))) + ((double) (((double) (1.0 / n)) / x))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))) * ((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))))) - ((double) (((double) pow(x, ((double) (2.0 * ((double) (1.0 / n)))))) * ((double) pow(x, ((double) (2.0 * ((double) (1.0 / n)))))))))) / ((double) (((double) (((double) cbrt(((double) (((double) (((double) pow(x, ((double) (1.0 / n)))) + ((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))))) * ((double) (((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))) + ((double) pow(x, ((double) (2.0 * ((double) (1.0 / n)))))))))))) * ((double) cbrt(((double) (((double) (((double) pow(x, ((double) (1.0 / n)))) + ((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))))) * ((double) (((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))) + ((double) pow(x, ((double) (2.0 * ((double) (1.0 / n)))))))))))))) * ((double) cbrt(((double) (((double) (((double) pow(x, ((double) (1.0 / n)))) + ((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))))) * ((double) (((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))) + ((double) pow(x, ((double) (2.0 * ((double) (1.0 / n))))))))))))))));
		}
		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) < -6.40843517291627578e-17

    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-cbrt-cube3.4

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

    4. Simplified3.4

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

    if -6.40843517291627578e-17 < (/ 1.0 n) < 6.9300284984344972e-10

    1. Initial program 45.0

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

    3. Applied add-cbrt-cube45.0

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

    4. Simplified45.0

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

    5. Taylor expanded around inf 32.5

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

    6. Simplified31.7

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

    if 6.9300284984344972e-10 < (/ 1.0 n)

    1. Initial program 6.2

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

    3. Applied add-cbrt-cube6.2

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

    4. Simplified6.2

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

    6. Applied flip--6.2

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

    7. Simplified6.1

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

    8. Simplified6.1

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

    10. Applied flip--6.1

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

    11. Applied associate-/l/6.1

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

    13. Applied add-cube-cbrt6.1

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.40843517291627578 \cdot 10^{-17}:\\ \;\;\;\;\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 6.9300284984344972 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(\frac{1}{{n}^{2}} \cdot \frac{0}{x} - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right) + \frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)} \cdot {x}^{\left(2 \cdot \frac{1}{n}\right)}}{\left(\sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)} \cdot \sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\right) \cdot \sqrt[3]{\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))