if eps < -0.00030713418f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.0
2. Using strategy rm
   14.0
3. Applied add-cube-cbrt to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3} - \tan x\]
   14.1

if -0.00030713418f0 < eps < 3.6169324f-06

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   19.6
2. Applied taylor to get
   \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
   9.0
3. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
   9.0
4. Applied simplify to get
   \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
   9.0

if 3.6169324f-06 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   13.7
2. Using strategy rm
   13.7
3. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
   13.5
4. Using strategy rm
   13.5
5. Applied flip-- to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}}\]
   13.5
6. Using strategy rm
   13.5
7. Applied tan-quot to get
   \[\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\color{red}{\left(\tan x\right)}}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\color{blue}{\left(\frac{\sin x}{\cos x}\right)}}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
   13.5
8. Applied square-div to get
   \[\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - \color{red}{{\left(\frac{\sin x}{\cos x}\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - \color{blue}{\frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
   13.5
9. Applied square-div to get
   \[\frac{\color{red}{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2} - \frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{\color{blue}{\frac{{1}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2}} - \frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
   13.5
10. Applied frac-sub to get
    \[\frac{\color{red}{\frac{{1}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2} - \frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{\color{blue}{\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2}}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
    13.5
11. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}} \leadsto \color{blue}{\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{\left(\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x\right) \cdot \left({\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2\right)}}\]
    13.5