if eps < -0.025611678f0 or 0.18414813f0 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.2
2. Using strategy rm
   14.2
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.4
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.4
5. Using strategy rm
   0.4
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   0.4
7. Applied simplify to get
   \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
   0.4

if -0.025611678f0 < eps < 0.18414813f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   19.5
2. Using strategy rm
   19.5
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   12.8
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   12.8
5. Using strategy rm
   12.8
6. Applied add-cube-cbrt to get
   \[\color{red}{\sin x \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   19.0
7. Applied taylor to get
   \[{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3 + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   12.3
8. Taylor expanded around 0 to get
   \[\color{red}{\left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{\left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   12.3
9. Applied simplify to get
   \[\left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \left(\cos x \cdot \sin \varepsilon - \left(\sin x - \sin x\right)\right) - \left(\sin x \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\]
   0.1

---

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