if eps < -0.0035937247f0 or 0.3662591f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.7
2. Using strategy rm
   14.7
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
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}}\]
   0.8
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
7. Using strategy rm
   0.9
8. Applied flip3-- to get
   \[\frac{{\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}^2 - {\left(\cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{{\color{blue}{\left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}\right)}}^2 - {\left(\cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
   1.8
9. Applied square-div to get
   \[\frac{\color{red}{{\left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}\right)}^2} - {\left(\cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}\right)}^2}{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)\right)}^2}} - {\left(\cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
   1.8
10. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}\right)}^2}}{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)\right)}^2} - {\left(\cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\frac{\color{blue}{{\left({\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3\right)}^2}}{{\left({\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)\right)}^2} - {\left(\cos x\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
    0.9

if -0.0035937247f0 < eps < 0.3662591f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   22.9
2. Using strategy rm
   22.9
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.5
4. Using strategy rm
   18.5
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}}\]
   18.5
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
   18.5
7. Using strategy rm
   18.5
8. Applied add-cube-cbrt to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}\right)}^3}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
   18.5
9. Applied taylor to get
   \[\frac{{\left(\sqrt[3]{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}\right)}^3}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
   0.2
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\color{blue}{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
    0.2
11. Applied simplify to get
    \[\frac{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{(\left(\sin x \cdot \sin x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\left(\frac{4}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin x\right)\right))_* - \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x + \left(2 \cdot \varepsilon\right) \cdot \sin x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
    0.2

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{(\left(\sin x \cdot \sin x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\left(\frac{4}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin x\right)\right))_* - \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x + \left(2 \cdot \varepsilon\right) \cdot \sin x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}} \leadsto \color{blue}{\frac{(\left(\sin x \cdot \sin x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\sin x \cdot \left(\varepsilon \cdot \frac{4}{3}\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right))_* - (\left(\cos x\right) * \varepsilon + \left(\sin x \cdot 2\right))_* \cdot \left(\varepsilon \cdot \cos x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
    0.2