if eps < -0.0010371751f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.5
2. Using strategy rm
   14.5
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.0
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   1.0
5. Using strategy rm
   1.0
6. Applied flip-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
   1.0
7. Using strategy rm
   1.0
8. Applied flip-- to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right)}^2 - {\left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}^2}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   1.0

if -0.0010371751f0 < eps < 0.028850537f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.3
2. Using strategy rm
   23.3
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.5
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   18.5
5. Using strategy rm
   18.5
6. Applied flip-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
   18.5
7. Using strategy rm
   18.5
8. Applied add-cube-cbrt to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}\right)}^3}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   18.5
9. Applied taylor to get
   \[\frac{{\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}\right)}^3}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) - \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   0.2
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) - \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) - \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    0.2
11. Applied simplify to get
    \[\frac{\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) - \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{\left(\left(\frac{1}{3} \cdot {\varepsilon}^3\right) \cdot \left(\cos x \cdot \sin x\right) - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin x\right) - \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x + \left(\varepsilon \cdot 2\right) \cdot \sin x\right)}{\left(\cos x + \cos \varepsilon \cdot \cos x\right) + \sin \varepsilon \cdot \sin x}\]
    0.2

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{\left(\left(\frac{1}{3} \cdot {\varepsilon}^3\right) \cdot \left(\cos x \cdot \sin x\right) - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin x\right) - \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x + \left(\varepsilon \cdot 2\right) \cdot \sin x\right)}{\left(\cos x + \cos \varepsilon \cdot \cos x\right) + \sin \varepsilon \cdot \sin x}} \leadsto \color{blue}{\frac{\left(\left({\varepsilon}^3 \cdot \frac{1}{3}\right) \cdot \left(\sin x \cdot \cos x\right) - \left(\sin x \cdot \varepsilon\right) \cdot \left(\sin x \cdot \varepsilon\right)\right) - \left(\cos x \cdot \varepsilon\right) \cdot \left(\sin x \cdot 2 + \cos x \cdot \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon + \cos x\right) + \sin x \cdot \sin \varepsilon}}\]
    0.1

if 0.028850537f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.9
2. Using strategy rm
   14.9
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.7
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   0.7
5. Using strategy rm
   0.7
6. Applied pow1 to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.7
7. Applied pow1 to get
   \[\color{red}{\cos x} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.7
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   0.8