if eps < -0.00261488f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.3
2. Using strategy rm
   14.3
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.0
4. Using strategy rm
   1.0
5. Applied add-cbrt-cube to get
   \[\left(\cos x \cdot \color{red}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) - \cos x \leadsto \left(\cos x \cdot \color{blue}{\sqrt[3]{{\left(\cos \varepsilon\right)}^3}} - \sin x \cdot \sin \varepsilon\right) - \cos x\]
   1.0
6. Applied add-cbrt-cube to get
   \[\left(\color{red}{\cos x} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3} - \sin x \cdot \sin \varepsilon\right) - \cos x \leadsto \left(\color{blue}{\sqrt[3]{{\left(\cos x\right)}^3}} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3} - \sin x \cdot \sin \varepsilon\right) - \cos x\]
   1.0
7. Applied cbrt-unprod to get
   \[\left(\color{red}{\sqrt[3]{{\left(\cos x\right)}^3} \cdot \sqrt[3]{{\left(\cos \varepsilon\right)}^3}} - \sin x \cdot \sin \varepsilon\right) - \cos x \leadsto \left(\color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\cos \varepsilon\right)}^3}} - \sin x \cdot \sin \varepsilon\right) - \cos x\]
   1.0

if -0.00261488f0 < eps < 0.25760502f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.3
2. Using strategy rm
   23.3
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.7
4. Using strategy rm
   18.7
5. Applied add-cube-cbrt to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x \cdot \sin \varepsilon}\right)}^3}\right) - \cos x\]
   18.7
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
    \[\color{red}{\frac{1}{6} \cdot \left({\varepsilon}^3 \cdot \sin x\right) - \varepsilon \cdot \left(\sin x + \frac{1}{2} \cdot \varepsilon\right)} \leadsto \color{blue}{{\varepsilon}^3 \cdot \left(\frac{1}{6} \cdot \sin x\right) - \varepsilon \cdot (\frac{1}{2} * \varepsilon + \left(\sin x\right))_*}\]
    0.2

if 0.25760502f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.6
2. Using strategy rm
   14.6
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.8
4. Using strategy rm
   0.8
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}}\]
   0.7
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
   0.9