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

⬇

\[\begin{array}{l} \mathbf{if}\;n \le -4951856969506.348:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{1}{4} \cdot \log x}{n \cdot \left(n \cdot x\right)}\right) - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{if}\;n \le 212327.6258399754:\\ \;\;\;\;{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\right)}^3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{1}{4} \cdot \log x}{n \cdot \left(n \cdot x\right)}\right) - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if n < -4951856969506.348 or 212327.6258399754 < n
1. Initial program 43.7
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-sqr-sqrt 43.7
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\]
4. Applied add-sqr-sqrt 43.7
  \[\leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2\]
5. Applied difference-of-squares 43.7
  \[\leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
6. Applied taylor 9.2
  \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)\]
7. Taylor expanded around inf 9.2
  
  \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}\]
8. Applied simplify 8.4
  \[\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{1}{4} \cdot \log x}{n \cdot \left(n \cdot x\right)}\right) - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right) \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)}\]
if -4951856969506.348 < n < 212327.6258399754
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-sqr-sqrt 3.7
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\]
4. Applied add-sqr-sqrt 3.6
  \[\leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2\]
5. Applied difference-of-squares 3.6
  \[\leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
6. Using strategy rm
7. Applied add-cube-cbrt 3.6
  \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}^3}\]
8. Using strategy rm
9. Applied add-cube-cbrt 3.6
  \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\right)}}^3\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Error

Derivation

`if n < -4951856969506.348 or 212327.6258399754 < n`

`if -4951856969506.348 < n < 212327.6258399754`

Runtime