if eps < -1.2734516787323002e-34 or 639.2159073015437 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   29.8
2. Using strategy rm
   29.8
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   29.8
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   29.8
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   29.9
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   29.9
7. Using strategy rm
   29.9
8. Applied sin-sum to get
   \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   28.0

if -1.2734516787323002e-34 < eps < -4.478153436892399e-243 or 3.932045143122947e-267 < eps < 639.2159073015437

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.0
2. Using strategy rm
   44.0
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   44.0
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   44.1
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   44.2
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   44.2
7. Using strategy rm
   44.2
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}^3}}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   51.2
9. Applied taylor to get
   \[\frac{\sqrt[3]{{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}^3}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   0.9
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    0.9
11. Applied simplify to get
    \[\frac{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cot x} \cdot \left(\frac{\cos x}{\frac{\sin x}{\cos x}} + \sin x\right)\]
    13.1

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12. Applied final simplification

if -4.478153436892399e-243 < eps < 3.932045143122947e-267

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   55.7
2. Using strategy rm
   55.7
3. Applied add-cbrt-cube to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\sqrt[3]{{\left(\tan \left(x + \varepsilon\right)\right)}^3}} - \tan x\]
   59.9
4. Applied taylor to get
   \[\sqrt[3]{{\left(\tan \left(x + \varepsilon\right)\right)}^3} - \tan x \leadsto \left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right) - \tan x\]
   56.3
5. Taylor expanded around 0 to get
   \[\color{red}{\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right)} - \tan x \leadsto \color{blue}{\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right)} - \tan x\]
   56.3
6. Applied simplify to get
   \[\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right) - \tan x \leadsto \left(\frac{{\varepsilon}^2}{\cos x} \cdot \sin x + \varepsilon\right) + \left(\left(\left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^3 \cdot {\varepsilon}^2}{{\left(\cos x\right)}^3}\right) + \frac{\frac{\varepsilon}{\frac{\cos x}{\sin x}}}{\frac{\cos x}{\sin x}}\right) - \tan x\right)\]
   26.9

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7. Applied final simplification