if eps < -2.320036f-13

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   17.1
2. Using strategy rm
   17.1
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   3.6
4. Using strategy rm
   3.6
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   5.6
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   3.6
7. Using strategy rm
   3.6
8. Applied add-cbrt-cube to get
   \[\color{red}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\right)}^3}}\]
   3.6

if -2.320036f-13 < eps < 0.18811941f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.2
2. Using strategy rm
   23.2
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   23.0
4. Using strategy rm
   23.0
5. Applied add-log-exp to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)}\right) - \cos x\]
   23.2
6. Applied add-log-exp to get
   \[\left(\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \cos x \leadsto \left(\color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \cos x\]
   23.2
7. Applied diff-log to get
   \[\color{red}{\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right)} - \cos x \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right)} - \cos x\]
   23.2
8. Applied simplify to get
   \[\log \color{red}{\left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right)} - \cos x \leadsto \log \color{blue}{\left(e^{\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon}\right)} - \cos x\]
   23.2
9. Applied taylor to get
   \[\log \left(e^{\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   0.0
10. Taylor expanded around 0 to get
    \[\color{red}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
    0.0
11. Applied simplify to get
    \[\frac{1}{24} \cdot {\varepsilon}^{4} - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right) \leadsto \frac{1}{24} \cdot {\varepsilon}^{4} - (\left(\varepsilon \cdot \varepsilon\right) * \frac{1}{2} + \left(x \cdot \varepsilon\right))_*\]
    0.0

---

12. Applied final simplification

if 0.18811941f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.8
2. Using strategy rm
   14.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.7
4. Using strategy rm
   0.7
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   1.0
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   0.7
7. Using strategy rm
   0.7
8. Applied add-cbrt-cube to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2} + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{\color{blue}{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^3}} + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   0.8