if eps < -0.003667146f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   13.9
2. Using strategy rm
   13.9
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   13.8
4. Using strategy rm
   13.8
5. Applied add-cube-cbrt to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3} - \frac{1}{\cot x}\]
   14.1

if -0.003667146f0 < eps < 0.33526045f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   20.2
2. Using strategy rm
   20.2
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   20.5
4. Using strategy rm
   20.5
5. Applied add-cube-cbrt to get
   \[\color{red}{\tan \left(x + \varepsilon\right) - \frac{1}{\cot x}} \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \frac{1}{\cot x}}\right)}^3}\]
   20.7
6. Applied taylor to get
   \[{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \frac{1}{\cot x}}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
   0.2
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 0.33526045f0 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.4
2. Using strategy rm
   14.4
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   14.3
4. Using strategy rm
   14.3
5. Applied add-cube-cbrt to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3} - \frac{1}{\cot x}\]
   14.5