if eps < -0.0011422609f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.6
2. Using strategy rm
   14.6
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.0
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   1.0
5. Using strategy rm
   1.0
6. Applied pow1 to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.0
7. Applied pow1 to get
   \[\color{red}{\cos x} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.0
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.1

if -0.0011422609f0 < eps < 2.8713222f-07

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   21.5
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   3.5
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   3.5
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
   3.5

if 2.8713222f-07 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   15.7
2. Using strategy rm
   15.7
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.3
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   2.3
5. Using strategy rm
   2.3
6. Applied flip-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
   2.3
7. Using strategy rm
   2.3
8. Applied flip-- to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^2}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   2.3