if eps < -0.00030713418f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.6
2. Using strategy rm
   14.6
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.2
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   1.2
5. Using strategy rm
   1.2
6. Applied pow1 to get
   \[\cos x \cdot \color{red}{\cos \varepsilon} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \cos x \cdot \color{blue}{{\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.2
7. Applied pow1 to get
   \[\color{red}{\cos x} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x\right)}^{1}} \cdot {\left(\cos \varepsilon\right)}^{1} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.2
8. Applied pow-prod-down to get
   \[\color{red}{{\left(\cos x\right)}^{1} \cdot {\left(\cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \leadsto \color{blue}{{\left(\cos x \cdot \cos \varepsilon\right)}^{1}} - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\]
   1.2

if -0.00030713418f0 < eps < 0.0017607713f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   22.8
2. Using strategy rm
   22.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   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 flip-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
   18.7
7. Using strategy rm
   18.7
8. Applied add-exp-log to get
   \[\color{red}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}} \leadsto \color{blue}{e^{\log \left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}}\]
   21.1
9. Applied simplify to get
   \[e^{\color{red}{\log \left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}} \leadsto e^{\color{blue}{\log \left(\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon\right)}}\]
   17.1
10. Applied taylor to get
    \[e^{\log \left(\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon\right)} \leadsto e^{\log \left(\frac{-1}{2} \cdot {\varepsilon}^2 - \sin x \cdot \sin \varepsilon\right)}\]
    17.3
11. Taylor expanded around 0 to get
    \[e^{\log \left(\color{red}{\frac{-1}{2} \cdot {\varepsilon}^2} - \sin x \cdot \sin \varepsilon\right)} \leadsto e^{\log \left(\color{blue}{\frac{-1}{2} \cdot {\varepsilon}^2} - \sin x \cdot \sin \varepsilon\right)}\]
    17.3
12. Applied simplify to get
    \[e^{\log \left(\frac{-1}{2} \cdot {\varepsilon}^2 - \sin x \cdot \sin \varepsilon\right)} \leadsto \left(\varepsilon \cdot \frac{-1}{2}\right) \cdot \varepsilon - \sin \varepsilon \cdot \sin x\]
    0.1

---

13. Applied final simplification

if 0.0017607713f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.1
2. Using strategy rm
   14.1
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   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 add-cube-cbrt to get
   \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{red}{\sin \varepsilon} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3} + \cos x\right)\]
   1.0
7. Applied add-cube-cbrt to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 + \cos x\right)\]
   1.1
8. Applied cube-unprod to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3} + \cos x\right)\]
   1.1