if eps < -2.7212152f-06 or 1.03416316f-08 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.4
2. Using strategy rm
   14.4
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   14.2
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   14.3
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   14.3
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   14.3
7. Using strategy rm
   14.3
8. Applied add-cube-cbrt to get
   \[\frac{\cot x - \color{red}{\cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \color{blue}{{\left(\sqrt[3]{\cot \left(\varepsilon + x\right)}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   14.2

if -2.7212152f-06 < eps < 1.03416316f-08

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   21.2
2. Using strategy rm
   21.2
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   21.0
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   21.1
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   21.1
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   21.1
7. Applied taylor to get
   \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   0.3
8. Taylor expanded around 0 to get
   \[\frac{\color{red}{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   0.3
9. Applied simplify to get
   \[\frac{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{(\frac{1}{6} * \left(x \cdot \varepsilon\right) + \left(\frac{\varepsilon}{x}\right))_* - \frac{{\varepsilon}^3}{\frac{x}{\frac{1}{6}}}}{\cos \left(\varepsilon + x\right) \cdot \cot x}\]
   0.3

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10. Applied final simplification