if eps < -0.0041207117f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.2
2. Using strategy rm
   14.2
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.9
4. Using strategy rm
   0.9
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x\]
   0.9
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{blue}{(\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \cos x\right))_*}} - \cos x\]
   1.0

if -0.0041207117f0 < eps < 0.037785772f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.3
2. Using strategy rm
   23.3
3. Applied add-cube-cbrt to get
   \[\color{red}{\cos \left(x + \varepsilon\right) - \cos x} \leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^3}\]
   23.3
4. Applied taylor to get
   \[{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^3 \leadsto \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right)\]
   0.1
5. 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
6. Applied simplify to get
   \[\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) - \left(\frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \cos x\right) + \varepsilon \cdot \sin x\right) \leadsto \frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - (\left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x\right) + \left(\sin x \cdot \varepsilon\right))_*\]
   0.2

---

7. Applied final simplification

if 0.037785772f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   15.0
2. Using strategy rm
   15.0
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 log1p-expm1-u to get
   \[\left(\color{red}{\cos x \cdot \cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) - \cos x \leadsto \left(\color{blue}{\log_* (1 + (e^{\cos x \cdot \cos \varepsilon} - 1)^*)} - \sin x \cdot \sin \varepsilon\right) - \cos x\]
   0.9