if eps < -3.2784791869173944e-06

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   28.8
2. Using strategy rm
   28.8
3. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
   28.9
4. Using strategy rm
   28.9
5. Applied cos-sum to get
   \[\frac{\sin \left(x + \varepsilon\right)}{\color{red}{\cos \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]
   27.1

if -3.2784791869173944e-06 < eps < 5.4458409913370574e-05

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.5
2. Using strategy rm
   44.5
3. Applied add-cbrt-cube to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right)\right)}^3}} - \tan x\]
   52.1
4. Applied taylor to get
   \[\sqrt[3]{{\left(\tan \left(x + \varepsilon\right)\right)}^3} - \tan x \leadsto \left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right) - \tan x\]
   44.1
5. Taylor expanded around 0 to get
   \[\color{red}{\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right)} - \tan x \leadsto \color{blue}{\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right)} - \tan x\]
   44.1
6. Applied simplify to get
   \[\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right) - \tan x \leadsto \left(\frac{{\varepsilon}^2}{\cos x} \cdot \sin x + \varepsilon\right) + \left(\left(\left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^3 \cdot {\varepsilon}^2}{{\left(\cos x\right)}^3}\right) + \frac{\frac{\varepsilon}{\frac{\cos x}{\sin x}}}{\frac{\cos x}{\sin x}}\right) - \tan x\right)\]
   27.3

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7. Applied final simplification

if 5.4458409913370574e-05 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   29.1
2. Using strategy rm
   29.1
3. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
   29.1
4. Using strategy rm
   29.1
5. Applied sin-sum to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
   27.2