if eps < -1.9517138f-10

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   16.5
2. Using strategy rm
   16.5
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.9
4. Using strategy rm
   2.9
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
   4.2
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
   2.9
7. Applied simplify to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}}\]
   3.0
8. Using strategy rm
   3.0
9. Applied add-cbrt-cube to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{\sqrt[3]{{\left((\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*\right)}^3}}}\]
   3.0
10. Applied add-cbrt-cube to get
    \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{\sqrt[3]{{\left((\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3\right)}^3}}}{\sqrt[3]{{\left((\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*\right)}^3}}\]
    3.1
11. Applied cbrt-undiv to get
    \[\color{red}{\frac{\sqrt[3]{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3\right)}^3}}{\sqrt[3]{{\left((\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3\right)}^3}{{\left((\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*\right)}^3}}}\]
    3.0
12. Applied simplify to get
    \[\sqrt[3]{\color{red}{\frac{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3\right)}^3}{{\left((\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^3 - {\left(\cos x\right)}^3}{(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin x \cdot \sin \varepsilon\right) + \left(\cos x \cdot \cos x\right))_*}\right)}^3}}\]
    3.0

if -1.9517138f-10 < eps < 0.0017607713f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   21.3
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   4.0
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   4.0
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
   4.0
5. Applied taylor to get
   \[\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_* \leadsto \varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)\]
   4.0
6. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)} \leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right)}\]
   4.0
7. Applied simplify to get
   \[\varepsilon \cdot \left(\frac{1}{6} \cdot {x}^3 - (\frac{1}{2} * \varepsilon + x)_*\right) \leadsto \left({x}^3 \cdot \frac{1}{6} - (\frac{1}{2} * \varepsilon + x)_*\right) \cdot \varepsilon\]
   4.0

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8. Applied final simplification

if 0.0017607713f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.1
2. Using strategy rm
   14.1
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.9
4. Using strategy rm
   0.9
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   1.3
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   0.9
7. Applied taylor to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   0.9
8. Taylor expanded around 0 to get
   \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   0.9
9. Applied simplify to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left((\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos x \cdot \cos \varepsilon\right))_*\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_*} - \cos x\]
   1.1

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10. Applied final simplification