if eps < -0.0011422609f0

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. Using strategy rm
   1.0
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x\]
   1.0
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{blue}{(\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \cos x\right))_*}} - \cos x\]
   1.2

if -0.0011422609f0 < eps < 0.0032539368f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   22.8
2. Using strategy rm
   22.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.4
4. Using strategy rm
   18.4
5. Applied add-cube-cbrt to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3}\right) - \cos x\]
   18.4
6. Applied taylor to get
   \[\left(\cos x \cdot \cos \varepsilon - {\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3\right) - \cos x \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right)\]
   0.1
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right)}\]
   0.1
8. Applied simplify to get
   \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot \sin x\right) \leadsto \frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \varepsilon \cdot \left(\sin x + \frac{1}{2} \cdot \varepsilon\right)\]
   0.1

---

9. Applied final simplification
10. Applied simplify to get
    \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \varepsilon \cdot \left(\sin x + \frac{1}{2} \cdot \varepsilon\right)} \leadsto \color{blue}{{\varepsilon}^3 \cdot \left(\frac{1}{6} \cdot \sin x\right) - \varepsilon \cdot (\frac{1}{2} * \varepsilon + \left(\sin x\right))_*}\]
    0.1

if 0.0032539368f0 < 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
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.0
4. Using strategy rm
   1.0
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   1.3
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   1.0
7. Using strategy rm
   1.0
8. Applied expm1-log1p-u to get
   \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{(e^{\log_* (1 + {\left(\cos \varepsilon \cdot \cos x\right)}^3)} - 1)^*} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   1.2