Average Error: 33.5 → 24.7
Time: 20.5s
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 -3.0289444609869303 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -\sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.80942107347966552 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}, \left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\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 -3.0289444609869303 \cdot 10^{-15}:\\
\;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -\sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\

\mathbf{elif}\;\frac{1}{n} \le 1.80942107347966552 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}, \left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\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)) <= -3.0289444609869303e-15)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))) * ((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))))) * ((double) cbrt(((double) (((double) (((double) fma(((double) pow(((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) cbrt(((double) (x + 1.0)))))), ((double) (1.0 / n)))), ((double) pow(((double) cbrt(((double) (x + 1.0)))), ((double) (1.0 / n)))), ((double) -(((double) (((double) sqrt(((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))) * ((double) (((double) pow(((double) cbrt(x)), ((double) (1.0 / n)))) * ((double) sqrt(((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))))))))))) + ((double) fma(((double) -(((double) sqrt(((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))))), ((double) (((double) pow(((double) cbrt(x)), ((double) (1.0 / n)))) * ((double) sqrt(((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))))), ((double) (((double) sqrt(((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))) * ((double) (((double) pow(((double) cbrt(x)), ((double) (1.0 / n)))) * ((double) sqrt(((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))))))))))) + ((double) (((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))) * ((double) (((double) -(((double) pow(((double) cbrt(x)), ((double) (1.0 / n)))))) + ((double) pow(((double) cbrt(x)), ((double) (1.0 / n))))))))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / n)) <= 1.8094210734796655e-08)) {
			VAR_1 = ((double) fma(1.0, ((double) (1.0 / ((double) (x * n)))), ((double) -(((double) fma(0.5, ((double) (1.0 / ((double) (((double) pow(x, 2.0)) * n)))), ((double) (1.0 * ((double) (((double) log(((double) (1.0 / x)))) / ((double) (x * ((double) pow(n, 2.0))))))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) (((double) (((double) pow(((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))), 3.0)) - ((double) pow(((double) pow(x, ((double) (1.0 / n)))), 3.0)))) / ((double) fma(((double) pow(x, ((double) (1.0 / n)))), ((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) + ((double) pow(x, ((double) (1.0 / n)))))), ((double) pow(((double) (x + 1.0)), ((double) (2.0 * ((double) (1.0 / n)))))))))))) * ((double) cbrt(((double) (((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))) - ((double) pow(x, ((double) (1.0 / n)))))))))) * ((double) cbrt(((double) (((double) (((double) fma(1.0, ((double) pow(((double) (x + 1.0)), ((double) (1.0 / n)))), ((double) -(((double) (((double) (((double) pow(((double) cbrt(((double) cbrt(x)))), ((double) (1.0 / n)))) * ((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))) * ((double) pow(((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))), ((double) (1.0 / n)))))))))) + ((double) fma(((double) -(((double) (((double) pow(((double) cbrt(((double) cbrt(x)))), ((double) (1.0 / n)))) * ((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))))), ((double) pow(((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))), ((double) (1.0 / n)))), ((double) (((double) (((double) pow(((double) cbrt(((double) cbrt(x)))), ((double) (1.0 / n)))) * ((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))))) * ((double) pow(((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))), ((double) (1.0 / n)))))))))) + ((double) (((double) pow(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) (1.0 / n)))) * ((double) (((double) -(((double) pow(((double) cbrt(x)), ((double) (1.0 / n)))))) + ((double) pow(((double) cbrt(x)), ((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) < -3.0289444609869303e-15

    1. Initial program 3.6

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

    3. Applied add-cube-cbrt3.6

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

    5. Applied add-cube-cbrt3.6

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

    6. Applied unpow-prod-down3.6

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

    7. Applied *-un-lft-identity3.6

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

    8. Applied unpow-prod-down3.6

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

    9. Applied prod-diff3.6

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

    10. Simplified3.6

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

    11. Simplified3.6

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

    13. Applied add-sqr-sqrt3.6

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

    14. Applied associate-*r*3.6

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

    15. Applied add-cube-cbrt3.6

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

    16. Applied unpow-prod-down3.6

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

    17. Applied prod-diff3.6

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

    if -3.0289444609869303e-15 < (/ 1.0 n) < 1.8094210734796655e-08

    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 33.0

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

    3. Simplified33.0

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

    if 1.8094210734796655e-08 < (/ 1.0 n)

    1. Initial program 6.6

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

    3. Applied add-cube-cbrt6.6

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

    5. Applied add-cube-cbrt6.6

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

    6. Applied unpow-prod-down6.6

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

    7. Applied *-un-lft-identity6.6

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

    8. Applied unpow-prod-down6.6

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

    9. Applied prod-diff6.6

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

    10. Simplified6.6

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

    11. Simplified6.6

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

    13. Applied add-cube-cbrt6.6

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

    14. Applied cbrt-prod6.6

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

    15. Applied unpow-prod-down6.6

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

    16. Applied associate-*l*6.6

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

    17. Applied *-un-lft-identity6.6

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

    18. Applied unpow-prod-down6.6

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

    19. Applied prod-diff6.6

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

    20. Simplified6.6

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

    22. Applied flip3--6.7

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

    23. Simplified6.7

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

  4. Final simplification24.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.0289444609869303 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -\sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \left({\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \sqrt{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\right)\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.80942107347966552 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}, \left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))