if eps < -1.2324208637780791e-05 or 0.15577836889185903 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.3
2. Using strategy rm
   30.3
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
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   1.1
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   1.0

if -1.2324208637780791e-05 < eps < 0.15577836889185903

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   49.2
2. Using strategy rm
   49.2
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   48.4
4. Using strategy rm
   48.4
5. Applied add-cube-cbrt to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{{\left(\sqrt[3]{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}^3}\]
   48.4
6. Applied taylor to get
   \[{\left(\sqrt[3]{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
   0.1
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)}\]
   0.1
8. Applied simplify to get
   \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right) \leadsto \left(\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \frac{1}{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\]
   0.1

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9. Applied final simplification
10. Applied simplify to get
    \[\color{red}{\left(\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \frac{1}{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x\right)\right) - \sin x \cdot \varepsilon} \leadsto \color{blue}{\left(\sin x \cdot \left(\varepsilon \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x}\]
    0.1