if eps < -0.081347406f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   15.4
2. Using strategy rm
   15.4
3. Applied tan-quot to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
   15.3
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
   15.3
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
   15.3
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   15.3
7. Using strategy rm
   15.3
8. Applied cotan-quot to get
   \[\frac{\cos x - \color{red}{\cot \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \color{blue}{\frac{\cos \left(\varepsilon + x\right)}{\sin \left(\varepsilon + x\right)}} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   15.4
9. Applied taylor to get
   \[\frac{\cos x - \frac{\cos \left(\varepsilon + x\right)}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   14.3
10. Taylor expanded around inf to get
    \[\frac{\cos x - \frac{\color{red}{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \frac{\color{blue}{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    14.3
11. Applied simplify to get
    \[\frac{\cos x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \frac{\cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\frac{\sin \left(\varepsilon + x\right)}{\sin x}}}{\cos x \cdot \cot \left(\varepsilon + x\right)}\]
    14.3

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12. Applied final simplification

if -0.081347406f0 < eps < 0.07298134f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   19.7
2. Using strategy rm
   19.7
3. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
   20.1
4. Using strategy rm
   20.1
5. Applied add-cube-cbrt to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3}\]
   20.2
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

if 0.07298134f0 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   13.4
2. Using strategy rm
   13.4
3. Applied tan-quot to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
   13.3
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
   13.3
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
   13.3
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   13.3
7. Using strategy rm
   13.3
8. Applied add-cube-cbrt to get
   \[\frac{\cos x - \color{red}{\cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \color{blue}{{\left(\sqrt[3]{\cot \left(\varepsilon + x\right) \cdot \sin x}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   13.3