if eps < -9.748260542427816e-17

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.9
2. Using strategy rm
   30.9
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-log-exp to get
   \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{red}{\cos x}\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{blue}{\log \left(e^{\cos x}\right)}\right)\]
   2.7
7. Applied add-log-exp to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x \cdot \sin \varepsilon} + \log \left(e^{\cos x}\right)\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)} + \log \left(e^{\cos x}\right)\right)\]
   2.8
8. Applied sum-log to get
   \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\log \left(e^{\sin x \cdot \sin \varepsilon}\right) + \log \left(e^{\cos x}\right)\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)}\]
   2.8
9. Applied add-log-exp to get
   \[\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right) \leadsto \color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)\]
   3.0
10. Applied diff-log to get
    \[\color{red}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)} \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}}\right)}\]
    3.1
11. Applied simplify to get
    \[\log \color{red}{\left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}}\right)} \leadsto \log \color{blue}{\left(e^{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon}\right)}\]
    2.8

if -9.748260542427816e-17 < eps < 0.00044197627601391903

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   46.0
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   6.9
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   6.9
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
   6.9

if 0.00044197627601391903 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.9
2. Using strategy rm
   30.9
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.9
4. Using strategy rm
   0.9
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
   1.1
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
   1.0