if eps < -7.175399107131671e-12 or 45926.76670985681 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   30.3
2. Using strategy rm
   30.3
3. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
   30.4
4. Using strategy rm
   30.4
5. Applied cotan-quot to get
   \[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x\]
   30.4
6. Applied associate-/r/ to get
   \[\color{red}{\frac{1}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x \leadsto \color{blue}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)} - \tan x\]
   30.4
7. Using strategy rm
   30.4
8. Applied sin-sum to get
   \[\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \tan x \leadsto \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x\]
   28.6
9. Applied distribute-lft-in to get
   \[\color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x \leadsto \color{blue}{\left(\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \tan x\]
   28.6
10. Applied associate--l+ to get
    \[\color{red}{\left(\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \tan x} \leadsto \color{blue}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right) - \tan x\right)}\]
    28.6
11. Applied simplify to get
    \[\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \color{red}{\left(\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right) - \tan x\right)} \leadsto \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \color{blue}{\left(\frac{\cos x \cdot \sin \varepsilon}{\cos \left(\varepsilon + x\right)} - \tan x\right)}\]
    28.6

if -7.175399107131671e-12 < eps < 45926.76670985681

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.9
2. Using strategy rm
   44.9
3. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
   45.1
4. Using strategy rm
   45.1
5. Applied add-cube-cbrt to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3}\]
   45.4
6. Applied taylor to get
   \[{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
   0.2
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)}\]
   0.2
8. Applied simplify to get
   \[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left((\left(\frac{\varepsilon}{\cos x}\right) * \left(\frac{\sin x}{\frac{\cos x}{\sin x}}\right) + \left(\sin x \cdot \frac{{\varepsilon}^2}{\cos x}\right))_* + (\left(\frac{{\varepsilon}^2}{{\left(\cos x\right)}^3}\right) * \left({\left(\sin x\right)}^3\right) + \left(\frac{1}{3} \cdot {\varepsilon}^3\right))_*\right) + (\left(\frac{{\varepsilon}^3}{{\left(\cos x\right)}^{4}}\right) * \left({\left(\sin x\right)}^{4}\right) + \left((\left(\frac{\frac{4}{3}}{\cos x}\right) * \left(\frac{\sin x \cdot \sin x}{\frac{\cos x}{{\varepsilon}^3}}\right) + \varepsilon)_*\right))_*\]
   0.2

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9. Applied final simplification