if x < 0.0011344831f0

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   19.0
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto \left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)\]
   19.0
3. Taylor expanded around 0 to get
   \[\color{red}{\left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)} \leadsto \color{blue}{\left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)}\]
   19.0
4. Applied simplify to get
   \[\color{red}{\left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)} \leadsto \color{blue}{\left(1 - \sqrt[3]{x}\right) + \left(\frac{1}{3} - \frac{1}{9} \cdot x\right) \cdot x}\]
   0.1

if 0.0011344831f0 < x

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   24.2
2. Using strategy rm
   24.2
3. Applied add-exp-log to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}}\]
   24.2
4. Using strategy rm
   24.2
5. Applied flip3-- to get
   \[e^{\log \color{red}{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto e^{\log \color{blue}{\left(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\right)}}\]
   23.9
6. Applied log-div to get
   \[e^{\color{red}{\log \left(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\right)}} \leadsto e^{\color{blue}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}}\]
   23.9
7. Applied simplify to get
   \[e^{\color{red}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)} \leadsto e^{\color{blue}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}\]
   24.2
8. Applied taylor to get
   \[e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)} \leadsto e^{\log \left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}\]
   3.2
9. Taylor expanded around inf to get
   \[e^{\log \color{red}{\left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)} \leadsto e^{\log \color{blue}{\left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}\]
   3.2
10. Applied simplify to get
    \[e^{\log \left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)} \leadsto \frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + {\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2}\]
    3.1

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11. Applied final simplification
12. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + {\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2}} \leadsto \color{blue}{\frac{1}{{x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right)}}\]
    3.1