if eps < -6.3877197f-21

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.9
2. Using strategy rm
   14.9
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   2.3
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   2.3
5. Using strategy rm
   2.3
6. Applied pow1 to get
   \[\sin x \cdot \color{red}{\cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \sin x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   2.3
7. Applied pow1 to get
   \[\color{red}{\sin x} \cdot {\left(\cos \varepsilon\right)}^{1} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sin x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   2.3
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\sin x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sin x \cdot \cos \varepsilon\right)}^{1}} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   2.4

if -6.3877197f-21 < eps < 1.4901758f-22

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   23.9
2. Applied taylor to get
   \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
   5.9
3. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
   5.9
4. Applied simplify to get
   \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right)}\]
   0.2

if 1.4901758f-22 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   15.4
2. Using strategy rm
   15.4
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   3.0
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   3.0
5. Using strategy rm
   3.0
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   3.0