if eps < -2011.657242358063 or 1.888112624037396e+21 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   29.9
2. Using strategy rm
   29.9
3. Applied add-log-exp to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\log \left(e^{\tan x}\right)}\]
   29.8
4. Applied add-log-exp to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \log \left(e^{\tan x}\right) \leadsto \color{blue}{\log \left(e^{\tan \left(x + \varepsilon\right)}\right)} - \log \left(e^{\tan x}\right)\]
   30.1
5. Applied diff-log to get
   \[\color{red}{\log \left(e^{\tan \left(x + \varepsilon\right)}\right) - \log \left(e^{\tan x}\right)} \leadsto \color{blue}{\log \left(\frac{e^{\tan \left(x + \varepsilon\right)}}{e^{\tan x}}\right)}\]
   30.1
6. Applied simplify to get
   \[\log \color{red}{\left(\frac{e^{\tan \left(x + \varepsilon\right)}}{e^{\tan x}}\right)} \leadsto \log \color{blue}{\left(e^{\tan \left(x + \varepsilon\right) - \tan x}\right)}\]
   30.1

if -2011.657242358063 < eps < 1.888112624037396e+21

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   44.7
2. Using strategy rm
   44.7
3. Applied tan-quot to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
   44.8
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
   44.9
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
   44.9
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   44.9
7. Using strategy rm
   44.9
8. Applied add-cbrt-cube to get
   \[\color{red}{\frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\right)}^3}}\]
   52.2
9. Applied simplify to get
   \[\sqrt[3]{\color{red}{{\left(\frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\right)}^3}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}\right)}^3}}\]
   52.2
10. Applied taylor to get
    \[\sqrt[3]{{\left(\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}\right)}^3} \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
    0.2
11. 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
12. 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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13. Applied final simplification