if eps < -0.0032873317f0 or 0.004007106f0 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.3
2. Using strategy rm
   14.3
3. Applied tan-quot to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
   14.2

if -0.0032873317f0 < eps < 0.004007106f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   19.9
2. Using strategy rm
   19.9
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   20.3
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   19.9
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   19.9
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   19.9
7. Applied taylor to get
   \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)\]
   15.5
8. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)}\]
   15.5
9. Applied simplify to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon}\]
   15.4
10. Applied taylor to get
    \[\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon \leadsto 0 + \varepsilon\]
    0
11. Taylor expanded around 0 to get
    \[\color{red}{0} + \varepsilon \leadsto \color{blue}{0} + \varepsilon\]
    0
12. Applied simplify to get
    \[0 + \varepsilon \leadsto \varepsilon\]
    0

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13. Applied final simplification