if eps < -6.349368961717618e-07

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.8
2. Using strategy rm
   30.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.0
4. Using strategy rm
   1.0
5. Applied add-log-exp to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)}\right) - \cos x\]
   1.1

if -6.349368961717618e-07 < eps < 0.0441511379087146

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   48.8
2. Using strategy rm
   48.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   48.3
4. Using strategy rm
   48.3
5. Applied add-cube-cbrt to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{{\left(\sqrt[3]{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}^3}\]
   48.3
6. Applied taylor to get
   \[{\left(\sqrt[3]{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
   0.1
7. 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
8. 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

---

9. Applied final simplification
10. 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 0.0441511379087146 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.4
2. Using strategy rm
   30.4
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. Using strategy rm
   0.8
5. Applied add-log-exp to get
   \[\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{red}{\cos x} \leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{\log \left(e^{\cos x}\right)}\]
   0.9
6. Applied add-log-exp to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \log \left(e^{\cos x}\right) \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)}\right) - \log \left(e^{\cos x}\right)\]
   1.0
7. Applied add-log-exp to get
   \[\left(\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right) \leadsto \left(\color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right)\]
   1.1
8. Applied diff-log to get
   \[\color{red}{\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right)} - \log \left(e^{\cos x}\right) \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right)} - \log \left(e^{\cos x}\right)\]
   1.2
9. Applied diff-log to get
   \[\color{red}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right) - \log \left(e^{\cos x}\right)} \leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]
   1.3
10. Applied simplify to get
    \[\log \color{red}{\left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)} \leadsto \log \color{blue}{\left(e^{\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon}\right)}\]
    1.0