if eps < -9.268573f-09

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   15.9
2. Using strategy rm
   15.9
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.5
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   2.5
5. Using strategy rm
   2.5
6. Applied pow1 to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   2.5
7. Applied pow1 to get
   \[\color{red}{\cos x} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   2.5
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   2.5

if -9.268573f-09 < eps < 0.25760502f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   21.7
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   3.9
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   3.9
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
   3.9

if 0.25760502f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.4
2. Using strategy rm
   14.4
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.7
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   0.7
5. Using strategy rm
   0.7
6. Applied pow1 to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.7
7. Applied pow1 to get
   \[\color{red}{\cos x} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.7
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.8