if eps < -0.008718856f0 or 0.18414813f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.5
2. Using strategy rm
   14.5
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.8
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   0.8
5. Using strategy rm
   0.8
6. Applied add-cbrt-cube to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^3}}\]
   0.8

if -0.008718856f0 < eps < 0.18414813f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.0
2. Using strategy rm
   23.0
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.4
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   18.4
5. Using strategy rm
   18.4
6. Applied add-cube-cbrt to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{{\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}^3}\]
   18.4
7. Applied taylor to get
   \[{\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
   0.1
8. 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
9. 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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10. Applied final simplification
11. 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