if eps < -3.0221453f-10 or 6.204036f-16 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.6
2. Using strategy rm
   14.6
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.9
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.9
5. Using strategy rm
   0.9
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   0.9
7. Applied simplify to get
   \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
   1.0

if -3.0221453f-10 < eps < 6.204036f-16

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   21.2
2. Using strategy rm
   21.2
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   17.7
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   17.7
5. Using strategy rm
   17.7
6. Applied add-cube-cbrt to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\sqrt[3]{\cos x \cdot \sin \varepsilon}\right)}^3} - \sin x\right)\]
   17.9
7. Applied taylor to get
   \[\sin x \cdot \cos \varepsilon + \left({\left(\sqrt[3]{\cos x \cdot \sin \varepsilon}\right)}^3 - \sin x\right) \leadsto \varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
   3.1
8. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
   3.1
9. Applied simplify to get
   \[\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right) \leadsto \cos x \cdot \varepsilon - (\left(\frac{1}{6} \cdot {\varepsilon}^3\right) * \left(\cos x\right) + \left(\left(x \cdot \frac{1}{2}\right) \cdot {\varepsilon}^2\right))_*\]
   3.1

---

10. Applied final simplification