if eps < -3.1000053f-10

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.4
2. Using strategy rm
   14.4
3. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
   14.2

if -3.1000053f-10 < eps < 1.5375762f-06

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   20.6
2. Using strategy rm
   20.6
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   20.6
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   20.6
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   20.6
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   20.6
7. Applied taylor to get
   \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)\]
   4.6
8. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)}\]
   4.6
9. Applied simplify to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon}\]
   0.1

if 1.5375762f-06 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.2
2. Using strategy rm
   14.2
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   14.0
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   14.2
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   14.2
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   14.2
7. Using strategy rm
   14.2
8. Applied add-cube-cbrt to get
   \[\color{red}{\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\right)}^3}\]
   14.5