if x < 1.5606351f-15

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   18.3
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\]
   18.3
3. Taylor expanded around 0 to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{e^{\frac{\log x}{n}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{e^{\frac{\log x}{n}}}\]
   18.3

if 1.5606351f-15 < x < 0.0005054283f0

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   28.6
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
   13.9
3. Taylor expanded around inf to get
   \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   13.9
4. Applied taylor to get
   \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
   13.9
5. Taylor expanded around 0 to get
   \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
   13.9
6. Applied simplify to get
   \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
   13.3

---

7. Applied final simplification

if 0.0005054283f0 < x < 218.70316f0

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   19.9
2. Using strategy rm
   19.9
3. Applied add-cube-cbrt to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
   19.9
4. Using strategy rm
   19.9
5. Applied add-log-exp to get
   \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}^3\]
   20.3

if 218.70316f0 < x < 5.814338f+18

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   25.0
2. Using strategy rm
   25.0
3. Applied add-cube-cbrt to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
   25.0
4. Using strategy rm
   25.0
5. Applied add-log-exp to get
   \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}^3\]
   25.0
6. Applied taylor to get
   \[{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\sqrt[3]{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)\right)}^3\]
   24.0
7. Taylor expanded around inf to get
   \[{\left(\log \left(e^{\sqrt[3]{\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\sqrt[3]{\color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}}\right)\right)}^3\]
   24.0
8. Applied simplify to get
   \[\color{red}{{\left(\log \left(e^{\sqrt[3]{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)\right)}^3} \leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\frac{\log x}{n \cdot x}}{n}}\]
   14.5
9. Applied taylor to get
   \[\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\frac{\log x}{n \cdot x}}{n} \leadsto \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-1 \cdot \frac{\log x}{n \cdot x}}{n}\]
   0.3
10. Taylor expanded around inf to get
    \[\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\color{red}{-1 \cdot \frac{\log x}{n \cdot x}}}{n} \leadsto \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\color{blue}{-1 \cdot \frac{\log x}{n \cdot x}}}{n}\]
    0.3
11. Applied simplify to get
    \[\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-1 \cdot \frac{\log x}{n \cdot x}}{n} \leadsto \left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\log x}{x \cdot n} \cdot \frac{-1}{n}\]
    0.3

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\log x}{x \cdot n} \cdot \frac{-1}{n}} \leadsto \color{blue}{\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{-\log x}{n \cdot \left(n \cdot x\right)}}\]
    0.3

if 5.814338f+18 < x

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   5.3
2. Using strategy rm
   5.3
3. Applied add-cube-cbrt to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
   5.3
4. Using strategy rm
   5.3
5. Applied add-log-exp to get
   \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}^3\]
   5.3