if eps < -9.280756734832815e-07

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   28.7
2. Using strategy rm
   28.7
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   28.6
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   28.7
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   28.7
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   28.7
7. Using strategy rm
   28.7
8. Applied sin-sum to get
   \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   27.0

if -9.280756734832815e-07 < eps < 4.601804552032629e-281 or 2.8946473862233273e-230 < eps < 1.265053560136945e-13

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.1
2. Using strategy rm
   44.1
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   44.2
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   44.3
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   44.4
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   44.4
7. Using strategy rm
   44.4
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}^3}}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   51.1
9. Applied taylor to get
   \[\frac{\sqrt[3]{{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}^3}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   0.5
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    0.5
11. Applied simplify to get
    \[\frac{\varepsilon \cdot \left(\frac{{\left(\cos x\right)}^2}{\sin x} + \sin x\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cot x} \cdot \left(\frac{\cos x}{\frac{\sin x}{\cos x}} + \sin x\right)\]
    14.5

---

12. Applied final simplification

if 4.601804552032629e-281 < eps < 2.8946473862233273e-230

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   53.7
2. Using strategy rm
   53.7
3. Applied add-cube-cbrt to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3} - \tan x\]
   55.5
4. Applied taylor to get
   \[{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3 - \tan x \leadsto \left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right) - \tan x\]
   54.1
5. Taylor expanded around 0 to get
   \[\color{red}{\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right)} - \tan x \leadsto \color{blue}{\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right)} - \tan x\]
   54.1
6. Applied simplify to get
   \[\left(\varepsilon + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{\sin x}{\cos x}\right)\right)\right)\right) - \tan x \leadsto \left(\frac{{\varepsilon}^2}{\cos x} \cdot \sin x + \varepsilon\right) + \left(\left(\left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^3 \cdot {\varepsilon}^2}{{\left(\cos x\right)}^3}\right) + \frac{\frac{\varepsilon}{\frac{\cos x}{\sin x}}}{\frac{\cos x}{\sin x}}\right) - \tan x\right)\]
   27.4

---

7. Applied final simplification

if 1.265053560136945e-13 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   29.7
2. Using strategy rm
   29.7
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   29.6
4. Applied tan-quot to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   29.7
5. Applied frac-sub to get
   \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
   29.7
6. Applied simplify to get
   \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   29.7
7. Using strategy rm
   29.7
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
9. Applied distribute-lft-in to get
   \[\frac{\color{red}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
   28.0
10. Applied associate--l+ to get
    \[\frac{\color{red}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    28.0