if eps < -0.0001021761f0

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 log1p-expm1-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)} - \sin x\right)\]
   0.6
7. Using strategy rm
   0.6
8. Applied sub-neg to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*) - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*) + \left(-\sin x\right)\right)}\]
   0.6
9. Applied associate-+r+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*) + \left(-\sin x\right)\right)} \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)\right) + \left(-\sin x\right)}\]
   0.6
10. Applied simplify to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)\right)} + \left(-\sin x\right) \leadsto \color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin \varepsilon \cdot \cos x\right))_*} + \left(-\sin x\right)\]
    0.6

if -0.0001021761f0 < eps < 0.004007106f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   19.9
2. Using strategy rm
   19.9
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   13.2
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   13.2
5. Using strategy rm
   13.2
6. Applied log1p-expm1-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)} - \sin x\right)\]
   13.2
7. Applied taylor to get
   \[\sin x \cdot \cos \varepsilon + \left(\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*) - \sin x\right) \leadsto \log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)\]
   0.1
8. Taylor expanded around 0 to get
   \[\color{red}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)} \leadsto \color{blue}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)}\]
   0.1
9. Applied simplify to get
   \[\color{red}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)} \leadsto \color{blue}{\cos x \cdot \sin \varepsilon}\]
   0.1

if 0.004007106f0 < 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
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.4
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