if eps < -0.00024292355f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.8
2. Using strategy rm
   13.8
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.5
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 expm1-log1p-u to get
   \[\color{red}{\sin x \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{(e^{\log_* (1 + \sin x \cdot \cos \varepsilon)} - 1)^*} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   0.6

if -0.00024292355f0 < eps < 1.6179318f-10

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   20.9
2. Using strategy rm
   20.9
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   14.6
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   14.6
5. Using strategy rm
   14.6
6. Applied expm1-log1p-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{(e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*} - \sin x\right)\]
   14.6
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 1.6179318f-10 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.4
2. Using strategy rm
   14.4
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.6
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.6
5. Using strategy rm
   0.6
6. Applied expm1-log1p-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{(e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*} - \sin x\right)\]
   0.7