if eps < -4.7317332f-09

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.0
2. Using strategy rm
   14.0
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   13.8
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   14.0
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   14.0
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   14.0
7. Using strategy rm
   14.0
8. Applied div-sub to get
   \[\color{red}{\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \color{blue}{\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \frac{\cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   14.0
9. Applied simplify to get
   \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{red}{\frac{\cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{blue}{\frac{1}{\cot x}}\]
   13.9
10. Using strategy rm
    13.9
11. Applied cotan-tan to get
    \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \color{red}{\cot x}} - \frac{1}{\cot x} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \color{blue}{\frac{1}{\tan x}}} - \frac{1}{\cot x}\]
    13.8
12. Applied cotan-tan to get
    \[\frac{\cot x}{\color{red}{\cot \left(x + \varepsilon\right)} \cdot \frac{1}{\tan x}} - \frac{1}{\cot x} \leadsto \frac{\cot x}{\color{blue}{\frac{1}{\tan \left(x + \varepsilon\right)}} \cdot \frac{1}{\tan x}} - \frac{1}{\cot x}\]
    13.8
13. Applied frac-times to get
    \[\frac{\cot x}{\color{red}{\frac{1}{\tan \left(x + \varepsilon\right)} \cdot \frac{1}{\tan x}}} - \frac{1}{\cot x} \leadsto \frac{\cot x}{\color{blue}{\frac{1 \cdot 1}{\tan \left(x + \varepsilon\right) \cdot \tan x}}} - \frac{1}{\cot x}\]
    13.8
14. Applied associate-/r/ to get
    \[\color{red}{\frac{\cot x}{\frac{1 \cdot 1}{\tan \left(x + \varepsilon\right) \cdot \tan x}}} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\cot x}{1 \cdot 1} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right)} - \frac{1}{\cot x}\]
    13.8
15. Applied simplify to get
    \[\color{red}{\frac{\cot x}{1 \cdot 1}} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right) - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\cot x}{1}} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right) - \frac{1}{\cot x}\]
    13.8

if -4.7317332f-09 < eps < 1.8738475f-09

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   20.8
2. Using strategy rm
   20.8
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   20.8
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   20.8
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   20.8
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   20.8
7. Applied taylor to get
   \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)\]
   4.9
8. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)}\]
   4.9
9. Applied simplify to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{(\left(x \cdot \varepsilon\right) * \left(\varepsilon + x\right) + \varepsilon)_*}\]
   0.1

if 1.8738475f-09 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.8
2. Using strategy rm
   14.8
3. Applied tan-quot to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
   14.6