if eps < -1.075305105072193e-70

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   31.0
2. Using strategy rm
   31.0
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   6.2
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   6.2
5. Using strategy rm
   6.2
6. Applied add-log-exp to get
   \[\color{red}{\sin x \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{\log \left(e^{\sin x \cdot \cos \varepsilon}\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   6.5

if -1.075305105072193e-70 < eps < 7.404354464384737e-68

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   46.2
2. Applied taylor to get
   \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
   10.8
3. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
   10.8
4. Applied simplify to get
   \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right)}\]
   0.1

if 7.404354464384737e-68 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   30.6
2. Using strategy rm
   30.6
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   5.5
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   5.6
5. Using strategy rm
   5.6
6. Applied flip3-- to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
   5.8
7. Applied simplify to get
   \[\sin x \cdot \cos \varepsilon + \frac{\color{red}{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   5.7