if eps < -9.280756734832815e-07 or 1.265053560136945e-13 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   29.2
2. Using strategy rm
   29.2
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   29.1
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   29.2
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   29.2
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   29.2
7. Using strategy rm
   29.2
8. Applied sin-sum to get
   \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   27.5
9. Applied distribute-lft-in to get
   \[\frac{\color{red}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   27.5
10. Applied associate--l+ to get
    \[\frac{\color{red}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    27.5

if -9.280756734832815e-07 < eps < 1.265053560136945e-13

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.9
2. Using strategy rm
   44.9
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   45.0
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   45.1
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   45.2
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   45.2
7. Using strategy rm
   45.2
8. Applied cos-sum to get
   \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{red}{\cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   45.1
9. Applied associate--r- to get
   \[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   42.5
10. Using strategy rm
    42.5
11. Applied add-cube-cbrt to get
    \[\color{red}{\frac{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\right)}^3}\]
    42.8
12. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
    0.2
13. 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
14. Applied simplify to get
    \[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right)\]
    0.2

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15. Applied final simplification