if eps < -1.9234536f-05

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   15.1
2. Using strategy rm
   15.1
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.6
4. Using strategy rm
   1.6
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
   1.8
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
   1.6
7. Applied simplify to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}}\]
   1.7
8. Using strategy rm
   1.7
9. Applied log1p-expm1-u to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - \color{red}{{\left(\cos x\right)}^3}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - \color{blue}{\log_* (1 + (e^{{\left(\cos x\right)}^3} - 1)^*)}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}\]
   1.8

if -1.9234536f-05 < eps < 5.083976f-14

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   20.7
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   4.0
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   4.0
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
   4.0

if 5.083976f-14 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   16.8
2. Using strategy rm
   16.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   3.6
4. Using strategy rm
   3.6
5. Applied log1p-expm1-u to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)}\right) - \cos x\]
   3.7