if eps < -2.2300634f-26 or 6.368611f-18 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   15.3
2. Using strategy rm
   15.3
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   2.8
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   2.8
5. Using strategy rm
   2.8
6. Applied flip3-- to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
   6.4
7. Applied simplify to get
   \[\sin x \cdot \cos \varepsilon + \frac{\color{red}{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   2.8
8. Using strategy rm
   2.8
9. Applied flip-- to get
   \[\sin x \cdot \cos \varepsilon + \frac{\color{red}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{\color{blue}{\frac{{\left({\left(\sin \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\sin x\right)}^3\right)}^2}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 + {\left(\sin x\right)}^3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   2.8
10. Applied associate-/l/ to get
    \[\sin x \cdot \cos \varepsilon + \color{red}{\frac{\frac{{\left({\left(\sin \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\sin x\right)}^3\right)}^2}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 + {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left({\left(\sin \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\sin x\right)}^3\right)}^2}{\left({\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)\right) \cdot \left({\left(\sin \varepsilon \cdot \cos x\right)}^3 + {\left(\sin x\right)}^3\right)}}\]
    2.8
11. Applied simplify to get
    \[\sin x \cdot \cos \varepsilon + \frac{{\left({\left(\sin \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\sin x\right)}^3\right)}^2}{\color{red}{\left({\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)\right) \cdot \left({\left(\sin \varepsilon \cdot \cos x\right)}^3 + {\left(\sin x\right)}^3\right)}} \leadsto \sin x \cdot \cos \varepsilon + \frac{{\left({\left(\sin \varepsilon \cdot \cos x\right)}^3\right)}^2 - {\left({\left(\sin x\right)}^3\right)}^2}{\color{blue}{\left({\left(\sin \varepsilon \cdot \cos x\right)}^2 + \left(\sin x + \sin \varepsilon \cdot \cos x\right) \cdot \sin x\right) \cdot \left({\left(\sin x\right)}^3 + {\left(\sin \varepsilon \cdot \cos x\right)}^3\right)}}\]
    2.8

if -2.2300634f-26 < eps < 6.368611f-18

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   23.3
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