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

⬇

\[\begin{array}{l} \mathbf{if}\;n \le -9443976.011510903:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \mathbf{if}\;n \le 82130166970.21388:\\ \;\;\;\;\log \left(e^{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes.
if n < -9443976.011510903 or 82130166970.21388 < n
1. Initial program 43.8
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Applied taylor 9.4
  \[\leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
3. Taylor expanded around inf 9.4
  
  \[\leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
4. Applied simplify 0.9
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)}\]
if -9443976.011510903 < n < 82130166970.21388
1. Initial program 2.9
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-log-exp 2.9
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
4. Applied add-log-exp 2.9
  \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
5. Applied diff-log 2.9
  \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
6. Applied simplify 2.9
  \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
7. Using strategy rm
8. Applied add-sqr-sqrt 3.0
  \[\leadsto \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)\]
9. Applied add-sqr-sqrt 2.9
  \[\leadsto \log \left(e^{\color{blue}{{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\right)\]
10. Applied difference-of-squares 2.9
  \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Error

Derivation

`if n < -9443976.011510903 or 82130166970.21388 < n`

`if -9443976.011510903 < n < 82130166970.21388`

Runtime