if eps < -4.9196116f-11

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   17.0
2. Using strategy rm
   17.0
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   3.1
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   3.1
5. Using strategy rm
   3.1
6. Applied add-cbrt-cube to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left(\cos \varepsilon\right)}^3}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   3.1
7. Applied add-cbrt-cube to get
   \[\color{red}{\cos x} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3}} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   3.2
8. Applied cbrt-unprod to get
   \[\color{red}{\sqrt[3]{{\left(\cos x\right)}^3} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   3.2
9. Using strategy rm
   3.2
10. Applied add-cube-cbrt to get
    \[\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3} - \left(\color{red}{\sin x \cdot \sin \varepsilon} + \cos x\right) \leadsto \sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3} - \left(\color{blue}{{\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3} + \cos x\right)\]
    3.3

if -4.9196116f-11 < eps < 4.7023132f-11

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   20.5
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   4.1
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   4.1
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
   4.1

if 4.7023132f-11 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   16.7
2. Using strategy rm
   16.7
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.9
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   2.9
5. Using strategy rm
   2.9
6. Applied flip-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
   2.9