if eps < -0.003667146f0 or 3.458902f-08 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.0
2. Using strategy rm
   14.0
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.5
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.5
5. Using strategy rm
   0.5
6. Applied pow1 to get
   \[\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{red}{\sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{blue}{{\left(\sin \varepsilon\right)}^{1}} - \sin x\right)\]
   0.5
7. Applied pow1 to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x} \cdot {\left(\sin \varepsilon\right)}^{1} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\sin \varepsilon\right)}^{1} - \sin x\right)\]
   0.5
8. Applied pow-prod-down to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\sin \varepsilon\right)}^{1}} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\cos x \cdot \sin \varepsilon\right)}^{1}} - \sin x\right)\]
   0.6

if -0.003667146f0 < eps < 3.458902f-08

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   20.7
2. Using strategy rm
   20.7
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   14.0
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   14.0
5. Using strategy rm
   14.0
6. Applied add-cube-cbrt to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\right)}^3}\]
   14.3
7. Applied taylor to get
   \[{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\right)}^3 \leadsto \varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)\]
   0.1
8. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} \leadsto \color{blue}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)}\]
   0.1
9. Applied simplify to get
   \[\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) \leadsto \left(\varepsilon - \frac{1}{6} \cdot {\varepsilon}^3\right) \cdot \cos x - \left(\sin x \cdot \frac{1}{2}\right) \cdot {\varepsilon}^2\]
   0.1

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10. Applied final simplification