if eps < -6.450162f-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
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.6
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   2.6
5. Using strategy rm
   2.6
6. Applied add-cbrt-cube to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}^3}}\]
   2.6

if -6.450162f-09 < eps < 0.0018547908f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   21.6
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   3.7
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   3.7
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
   3.8

if 0.0018547908f0 < 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
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.9
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   0.9
5. Using strategy rm
   0.9
6. Applied add-cbrt-cube to get
   \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{red}{\sin \varepsilon} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   0.9
7. Applied add-cbrt-cube to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right)\]
   0.9
8. Applied cbrt-unprod to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sqrt[3]{{\left(\sin x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   1.0