if eps < -1.3544471110926727e-95 or 9.071891940860134e-92 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   31.5
2. Using strategy rm
   31.5
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   7.7
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   7.7
5. Using strategy rm
   7.7
6. Applied sub-neg to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)}\]
   7.7
7. Applied associate-+r+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \left(-\sin x\right)}\]
   7.7
8. Applied simplify to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} + \left(-\sin x\right) \leadsto \color{blue}{(\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x\right))_*} + \left(-\sin x\right)\]
   7.7

if -1.3544471110926727e-95 < eps < 9.071891940860134e-92

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   48.4
2. Applied taylor to get
   \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
   10.6
3. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
   10.6
4. Applied simplify to get
   \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \frac{1}{2}}\]
   0.1