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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 0.9959105341518036:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;x \le 1.461949344627711 \cdot 10^{+137}:\\ \;\;\;\;(\left(-{x}^{\left(\frac{\frac{2}{3}}{n}\right)}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{if}\;x \le 1.4268505780422436 \cdot 10^{+221}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Derivation

Split input into 4 regimes
if x < 0.9959105341518036
1. Initial program 46.7
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 60.0
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^{2}}{{n}^{2}} + 1\right) - \frac{\log \left(\frac{1}{x}\right)}{n}\right)}\]
3. Applied simplify15.1
  \[\leadsto \color{blue}{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - (\left(\frac{\frac{1}{2}}{n \cdot n}\right) \cdot \left(\log x \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*}\]
if 0.9959105341518036 < x < 1.461949344627711e+137
1. Initial program 31.0
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-cube-cbrt31.0
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}\]
4. Taylor expanded around 0 31.0
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{e^{\frac{2}{3} \cdot \frac{\log x}{n}}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\]
5. Applied simplify31.0
  \[\leadsto \color{blue}{(\left(-{x}^{\left(\frac{\frac{2}{3}}{n}\right)}\right) \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right))_*}\]
if 1.461949344627711e+137 < x < 1.4268505780422436e+221
1. Initial program 17.4
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 21.5
  \[\leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^{2}} + \frac{\log \left(\frac{1}{x}\right)}{{n}^{2} \cdot x}\right)}\]
3. Applied simplify21.5
  \[\leadsto \color{blue}{\frac{\frac{\log x}{n \cdot n}}{x} - \left(\frac{\frac{\frac{1}{2}}{n}}{x \cdot x} - \frac{1}{x \cdot n}\right)}\]
if 1.4268505780422436e+221 < x
1. Initial program 6.4
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-log-exp6.6
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
Recombined 4 regimes into one program.

Error

Derivation

`if x < 0.9959105341518036`

`if 0.9959105341518036 < x < 1.461949344627711e+137`

`if 1.461949344627711e+137 < x < 1.4268505780422436e+221`

`if 1.4268505780422436e+221 < x`

Runtime