if eps < -1.5175483707585876e-61 or 1.5975906962049632e-26 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   30.1
2. Using strategy rm
   30.1
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   30.0
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   30.1
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   30.1
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   30.1
7. Using strategy rm
   30.1
8. Applied sin-sum to get
   \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   28.3
9. Applied distribute-lft-in to get
   \[\frac{\color{red}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   28.3
10. Applied associate--l+ to get
    \[\frac{\color{red}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    28.3

if -1.5175483707585876e-61 < eps < 1.5975906962049632e-26

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   46.3
2. Applied taylor to get
   \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
   18.4
3. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
   18.4
4. Applied simplify to get
   \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
   18.4