if eps < -6.974478371048821e-14 or 2.3610750006627307e-19 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   29.7
2. Using strategy rm
   29.7
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   29.6
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   29.7
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   29.7
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   29.7
7. Using strategy rm
   29.7
8. Applied sin-sum to get
   \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   27.7

if -6.974478371048821e-14 < eps < 2.3610750006627307e-19

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.4
2. Applied taylor to get
   \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
   20.3
3. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
   20.3
4. Applied taylor to get
   \[\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right) \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + 0\right)\]
   14.4
5. Taylor expanded around inf to get
   \[\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + \color{red}{0}\right) \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + \color{blue}{0}\right)\]
   14.4
6. Applied simplify to get
   \[\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + 0\right) \leadsto (\left(x \cdot x\right) * \left({\varepsilon}^3\right) + \varepsilon)_*\]
   14.4

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7. Applied final simplification
8. Applied simplify to get
   \[\color{red}{(\left(x \cdot x\right) * \left({\varepsilon}^3\right) + \varepsilon)_*} \leadsto \color{blue}{(\left({x}^2\right) * \left({\varepsilon}^3\right) + \varepsilon)_*}\]
   14.4