if eps < -6.349368961717618e-07 or 0.0441511379087146 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   30.1
2. Using strategy rm
   30.1
3. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
   30.1
4. Using strategy rm
   30.1
5. Applied cos-sum to get
   \[\frac{\sin \left(x + \varepsilon\right)}{\color{red}{\cos \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]
   28.2
6. Using strategy rm
   28.2
7. Applied sin-sum to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x\]
   0.6

if -6.349368961717618e-07 < eps < 0.0441511379087146

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.7
2. Using strategy rm
   44.7
3. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
   44.8
4. Using strategy rm
   44.8
5. Applied cos-sum to get
   \[\frac{\sin \left(x + \varepsilon\right)}{\color{red}{\cos \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]
   44.7
6. Using strategy rm
   44.7
7. Applied tan-quot to get
   \[\frac{\sin \left(x + \varepsilon\right)}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \color{red}{\tan x} \leadsto \frac{\sin \left(x + \varepsilon\right)}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}}\]
   44.5
8. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cos x - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \sin x}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x}}\]
   44.5
9. Applied taylor to get
   \[\frac{\sin \left(x + \varepsilon\right) \cdot \cos x - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \sin x}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \leadsto \frac{\varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x}\]
   0.3
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\varepsilon}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\varepsilon}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x}\]
    0.3
11. Applied simplify to get
    \[\frac{\varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \leadsto \frac{\varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) \cdot \cos x}\]
    0.3

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12. Applied final simplification