if eps < -0.00030713418f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.8
2. Using strategy rm
   13.8
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 sub-neg to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)}\]
   0.4
7. Applied associate-+r+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \left(-\sin x\right)}\]
   0.4
8. Applied simplify to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} + \left(-\sin x\right) \leadsto \color{blue}{(\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x\right))_*} + \left(-\sin x\right)\]
   0.5

if -0.00030713418f0 < eps < 2.180514f-09

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   20.1
2. Using strategy rm
   20.1
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   13.9
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   13.9
5. Using strategy rm
   13.9
6. Applied add-cube-cbrt to get
   \[\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{red}{\sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3} - \sin x\right)\]
   14.2
7. Applied add-cube-cbrt to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\sqrt[3]{\cos x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 - \sin x\right)\]
   14.2
8. Applied cube-unprod to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{{\left(\sqrt[3]{\cos x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3} - \sin x\right)\]
   14.2
9. Applied taylor to get
   \[\sin x \cdot \cos \varepsilon + \left({\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3 - \sin x\right) \leadsto {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\cos x}\right)}^3\]
   0.8
10. Taylor expanded around 0 to get
    \[\color{red}{{\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\cos x}\right)}^3} \leadsto \color{blue}{{\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\cos x}\right)}^3}\]
    0.8
11. Applied simplify to get
    \[{\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\cos x}\right)}^3 \leadsto \sin \varepsilon \cdot \cos x\]
    0.1

---

12. Applied final simplification

if 2.180514f-09 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.8
2. Using strategy rm
   13.8
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.7
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.7
5. Using strategy rm
   0.7
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   0.7
7. Applied simplify to get
   \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
   0.7