if eps < -0.0012318226924060028 or 27.89344249548887 < 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
   \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   27.9

if -0.0012318226924060028 < eps < 27.89344249548887

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.5
2. Using strategy rm
   44.5
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   44.6
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   44.7
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   44.8
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   44.8
7. Using strategy rm
   44.8
8. Applied cos-sum to get
   \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{red}{\cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   44.5
9. Applied associate--r- to get
   \[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   41.9
10. Using strategy rm
    41.9
11. Applied add-log-exp to get
    \[\frac{\color{red}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right)} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\log \left(e^{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right)} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    56.4
12. Applied taylor to get
    \[\frac{\log \left(e^{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right) + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    1.1
13. Taylor expanded around 0 to get
    \[\frac{\color{red}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    1.1
14. Applied simplify to get
    \[\frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right) + \sin \varepsilon \cdot \sin x}{\cos \left(\varepsilon + x\right) \cdot \cot x}\]
    1.1

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15. Applied final simplification