if eps < -0.0011342058f0 or 0.25760502f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.5
2. Using strategy rm
   14.5
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 pow1 to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.9
7. Applied pow1 to get
   \[\color{red}{\cos x} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.9
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.0

if -0.0011342058f0 < eps < 0.25760502f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.3
2. Using strategy rm
   23.3
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.7
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   18.7
5. Using strategy rm
   18.7
6. Applied add-log-exp to get
   \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{red}{\cos x}\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{blue}{\log \left(e^{\cos x}\right)}\right)\]
   20.6
7. Applied add-log-exp to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x \cdot \sin \varepsilon} + \log \left(e^{\cos x}\right)\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)} + \log \left(e^{\cos x}\right)\right)\]
   22.7
8. Applied sum-log to get
   \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\log \left(e^{\sin x \cdot \sin \varepsilon}\right) + \log \left(e^{\cos x}\right)\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)}\]
   22.7
9. Applied add-log-exp to get
   \[\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right) \leadsto \color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)\]
   22.5
10. Applied diff-log to get
    \[\color{red}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)} \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}}\right)}\]
    22.5
11. Applied simplify to get
    \[\log \color{red}{\left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}}\right)} \leadsto \log \color{blue}{\left(e^{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon}\right)}\]
    22.4
12. Applied taylor to get
    \[\log \left(e^{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon}\right) \leadsto \log \left(e^{\frac{-1}{2} \cdot {\varepsilon}^2 - \sin x \cdot \sin \varepsilon}\right)\]
    22.2
13. Taylor expanded around 0 to get
    \[\log \left(e^{\color{red}{\frac{-1}{2} \cdot {\varepsilon}^2} - \sin x \cdot \sin \varepsilon}\right) \leadsto \log \left(e^{\color{blue}{\frac{-1}{2} \cdot {\varepsilon}^2} - \sin x \cdot \sin \varepsilon}\right)\]
    22.2
14. Applied simplify to get
    \[\log \left(e^{\frac{-1}{2} \cdot {\varepsilon}^2 - \sin x \cdot \sin \varepsilon}\right) \leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{2} - \sin \varepsilon \cdot \sin x\]
    0.2

---

15. Applied final simplification