if eps < -0.00020185983f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.2
2. Using strategy rm
   14.2
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.5
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 flip3-- to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
   0.7
7. Applied simplify to get
   \[\sin x \cdot \cos \varepsilon + \frac{\color{red}{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   0.5
8. Applied simplify to get
   \[\sin x \cdot \cos \varepsilon + \frac{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}{\color{red}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}} \leadsto \sin x \cdot \cos \varepsilon + \frac{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}{\color{blue}{(\left(\sin \varepsilon \cdot \cos x\right) * \left((\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\sin x\right))_*\right) + \left({\left(\sin x\right)}^2\right))_*}}\]
   0.7

if -0.00020185983f0 < eps < 8.7403685f-05

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   20.3
2. Using strategy rm
   20.3
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   13.6
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   13.6
5. Using strategy rm
   13.6
6. Applied expm1-log1p-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{(e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*} - \sin x\right)\]
   13.6
7. Applied taylor to get
   \[\sin x \cdot \cos \varepsilon + \left((e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^* - \sin x\right) \leadsto (e^{\log_* (1 + \sin \varepsilon \cdot \cos x)} - 1)^*\]
   0.1
8. Taylor expanded around 0 to get
   \[\color{red}{(e^{\log_* (1 + \sin \varepsilon \cdot \cos x)} - 1)^*} \leadsto \color{blue}{(e^{\log_* (1 + \sin \varepsilon \cdot \cos x)} - 1)^*}\]
   0.1
9. Applied simplify to get
   \[\color{red}{(e^{\log_* (1 + \sin \varepsilon \cdot \cos x)} - 1)^*} \leadsto \color{blue}{\cos x \cdot \sin \varepsilon}\]
   0.1

if 8.7403685f-05 < 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 sub-neg to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)}\]
   0.5
7. Applied associate-+r+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \left(-\sin x\right)}\]
   0.5
8. Applied simplify to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} + \left(-\sin x\right) \leadsto \color{blue}{(\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x\right))_*} + \left(-\sin x\right)\]
   0.6