if eps < -4.037641697290413e-05

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   31.1
2. Using strategy rm
   31.1
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 flip3-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
   1.1
7. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
   1.1

if -4.037641697290413e-05 < eps < 0.0016565824112741959

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   49.7
2. Using strategy rm
   49.7
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   49.2
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   49.2
5. Using strategy rm
   49.2
6. Applied add-cube-cbrt to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{{\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}^3}\]
   49.2
7. Applied taylor to get
   \[{\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
   0.1
8. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)}\]
   0.1
9. Applied simplify to get
   \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right) \leadsto \left(\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \frac{1}{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\]
   0.1

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\left(\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \frac{1}{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x\right)\right) - \sin x \cdot \varepsilon} \leadsto \color{blue}{\left(\sin x \cdot \left(\varepsilon \cdot \frac{1}{6}\right) - \frac{1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot \sin x}\]
    0.1

if 0.0016565824112741959 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   29.0
2. Using strategy rm
   29.0
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.8
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
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right)\]
   1.0
8. Applied cbrt-unprod to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sqrt[3]{{\left(\sin x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   0.9