Derivation

Split input into 2 regimes
if n < -395407852.70067936 or 97.12882204625622 < n
1. Initial program 44.5
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Initial simplification44.5
  \[\leadsto {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
3. Using strategy rm
4. Applied add-exp-log44.5
  \[\leadsto {\color{blue}{\left(e^{\log \left(1 + x\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
5. Applied pow-exp44.5
  \[\leadsto \color{blue}{e^{\log \left(1 + x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
6. Simplified44.5
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
7. Using strategy rm
8. Applied add-log-exp44.5
  \[\leadsto e^{\frac{\log_* (1 + x)}{n}} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
9. Applied add-log-exp44.5
  \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\log_* (1 + x)}{n}}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
10. Applied diff-log44.6
  \[\leadsto \color{blue}{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
11. Simplified44.5
  \[\leadsto \log \color{blue}{\left(e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
12. Using strategy rm
13. Applied rem-log-exp44.5
  \[\leadsto \color{blue}{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\]
14. Taylor expanded around inf 33.0
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]
15. Simplified32.9
  \[\leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\frac{\log x}{n \cdot n}}{x}}\]
if -395407852.70067936 < n < 97.12882204625622
1. Initial program 2.8
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Initial simplification2.8
  \[\leadsto {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
3. Using strategy rm
4. Applied add-exp-log2.9
  \[\leadsto {\color{blue}{\left(e^{\log \left(1 + x\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
5. Applied pow-exp2.9
  \[\leadsto \color{blue}{e^{\log \left(1 + x\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
6. Simplified1.7
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
7. Using strategy rm
8. Applied add-log-exp1.9
  \[\leadsto e^{\frac{\log_* (1 + x)}{n}} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
9. Applied add-log-exp2.0
  \[\leadsto \color{blue}{\log \left(e^{e^{\frac{\log_* (1 + x)}{n}}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
10. Applied diff-log2.0
  \[\leadsto \color{blue}{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
11. Simplified1.9
  \[\leadsto \log \color{blue}{\left(e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
12. Using strategy rm
13. Applied rem-log-exp1.7
  \[\leadsto \color{blue}{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\]
Recombined 2 regimes into one program.
Final simplification23.9
\[\leadsto \begin{array}{l} \mathbf{if}\;n \le -395407852.70067936 \lor \neg \left(n \le 97.12882204625622\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\frac{\log x}{n \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

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Derivation

`if n < -395407852.70067936 or 97.12882204625622 < n`

`if -395407852.70067936 < n < 97.12882204625622`

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