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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 0.9992593994234809:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \left(\frac{\frac{1}{2}}{n} \cdot \log x + 1\right) \cdot \frac{\log x}{n}\\ \mathbf{if}\;x \le 1.5081096997880613 \cdot 10^{+35}:\\ \;\;\;\;\log \left(e^{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \mathbf{if}\;x \le 3.2151683887379773 \cdot 10^{+78}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{if}\;x \le 4.7580896216649 \cdot 10^{+98}:\\ \;\;\;\;\log \left(e^{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \mathbf{if}\;x \le 7.847367044553675 \cdot 10^{+174}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \end{array}\]

Derivation

Split input into 4 regimes
if x < 0.9992593994234809
1. Initial program 47.0
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 59.9
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^{2}}{{n}^{2}} + 1\right) - \frac{\log \left(\frac{1}{x}\right)}{n}\right)}\]
3. Applied simplify15.1
  \[\leadsto \color{blue}{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \left(\frac{\frac{1}{2}}{n} \cdot \log x + 1\right) \cdot \frac{\log x}{n}}\]
if 0.9992593994234809 < x < 1.5081096997880613e+35 or 3.2151683887379773e+78 < x < 4.7580896216649e+98
1. Initial program 34.1
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-log-exp34.1
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
4. Applied add-log-exp34.1
  \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
5. Applied diff-log34.1
  \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
6. Applied simplify34.1
  \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
7. Using strategy rm
8. Applied add-sqr-sqrt34.1
  \[\leadsto \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\]
9. Applied add-sqr-sqrt34.1
  \[\leadsto \log \left(e^{{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\]
10. Applied unpow-prod-down34.1
  \[\leadsto \log \left(e^{\color{blue}{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\]
11. Applied difference-of-squares34.1
  \[\leadsto \log \left(e^{\color{blue}{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)\]
if 1.5081096997880613e+35 < x < 3.2151683887379773e+78 or 4.7580896216649e+98 < x < 7.847367044553675e+174
1. Initial program 27.3
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around -inf 63.0
  \[\leadsto \color{blue}{\left(\frac{\log -1}{{n}^{2} \cdot x} + \frac{1}{n \cdot x}\right) - \left(\frac{\log \left(\frac{-1}{x}\right)}{{n}^{2} \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^{2}}\right)}\]
3. Applied simplify21.9
  \[\leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}}\]
if 7.847367044553675e+174 < x
1. Initial program 9.3
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-log-exp9.4
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
4. Applied add-log-exp9.3
  \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
5. Applied diff-log9.3
  \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
6. Applied simplify9.3
  \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
7. Using strategy rm
8. Applied add-sqr-sqrt9.4
  \[\leadsto \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\]
9. Applied add-sqr-sqrt9.4
  \[\leadsto \log \left(e^{{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\]
10. Applied unpow-prod-down9.4
  \[\leadsto \log \left(e^{\color{blue}{{\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\]
11. Applied difference-of-squares9.4
  \[\leadsto \log \left(e^{\color{blue}{\left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{1 + x}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)\]
Recombined 4 regimes into one program.

Error

Derivation

`if x < 0.9992593994234809`

`if 0.9992593994234809 < x < 1.5081096997880613e+35 or 3.2151683887379773e+78 < x < 4.7580896216649e+98`

`if 1.5081096997880613e+35 < x < 3.2151683887379773e+78 or 4.7580896216649e+98 < x < 7.847367044553675e+174`

`if 7.847367044553675e+174 < x`

Runtime