if eps < -0.0032873317f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   13.9
2. Using strategy rm
   13.9
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   13.7
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   13.8
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   13.8
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   13.8
7. Using strategy rm
   13.8
8. Applied add-cube-cbrt to get
   \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \color{red}{\cot x}} \leadsto \frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \color{blue}{{\left(\sqrt[3]{\cot x}\right)}^3}}\]
   14.2
9. Applied add-cube-cbrt to get
   \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\color{red}{\cot \left(x + \varepsilon\right)} \cdot {\left(\sqrt[3]{\cot x}\right)}^3} \leadsto \frac{\cot x - \cot \left(\varepsilon + x\right)}{\color{blue}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)}\right)}^3} \cdot {\left(\sqrt[3]{\cot x}\right)}^3}\]
   14.3
10. Applied cube-unprod to get
    \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\color{red}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)}\right)}^3 \cdot {\left(\sqrt[3]{\cot x}\right)}^3}} \leadsto \frac{\cot x - \cot \left(\varepsilon + x\right)}{\color{blue}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\cot x}\right)}^3}}\]
    14.3
11. Applied add-cube-cbrt to get
    \[\frac{\color{red}{\cot x - \cot \left(\varepsilon + x\right)}}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\cot x}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\cot x - \cot \left(\varepsilon + x\right)}\right)}^3}}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\cot x}\right)}^3}\]
    14.2
12. Applied cube-undiv to get
    \[\color{red}{\frac{{\left(\sqrt[3]{\cot x - \cot \left(\varepsilon + x\right)}\right)}^3}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\cot x}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\cot x - \cot \left(\varepsilon + x\right)}}{\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\cot x}}\right)}^3}\]
    14.2

if -0.0032873317f0 < eps < 0.004007106f0

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   19.9
2. Using strategy rm
   19.9
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   20.3
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   19.9
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   19.9
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   19.9
7. Applied taylor to get
   \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)\]
   15.5
8. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)}\]
   15.5
9. Applied simplify to get
   \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon}\]
   15.4
10. Applied taylor to get
    \[\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon \leadsto 0 + \varepsilon\]
    0
11. Taylor expanded around 0 to get
    \[\color{red}{0} + \varepsilon \leadsto \color{blue}{0} + \varepsilon\]
    0
12. Applied simplify to get
    \[0 + \varepsilon \leadsto \varepsilon\]
    0

---

13. Applied final simplification

if 0.004007106f0 < eps

1. Started with
   \[\tan \left(x + \varepsilon\right) - \tan x\]
   14.7
2. Using strategy rm
   14.7
3. Applied tan-cotan to get
   \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
   14.6
4. Applied tan-cotan to get
   \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
   14.7
5. Applied frac-sub to get
   \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
   14.7
6. Applied simplify to get
   \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   14.7
7. Using strategy rm
   14.7
8. Applied add-cube-cbrt to get
   \[\frac{\cot x - \color{red}{\cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \color{blue}{{\left(\sqrt[3]{\cot \left(\varepsilon + x\right)}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
   14.6