if eps < -0.0017290534f0 or 0.0541411f0 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.7
2. Using strategy rm
   13.7
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.4
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.4
5. Using strategy rm
   0.4
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   0.4
7. Using strategy rm
   0.4
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \frac{\color{blue}{\sqrt[3]{{\left({\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}^3}}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.5

if -0.0017290534f0 < eps < 0.0541411f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   19.9
2. Using strategy rm
   19.9
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   13.3
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   13.3
5. Using strategy rm
   13.3
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   13.3
7. Using strategy rm
   13.3
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \frac{\color{blue}{\sqrt[3]{{\left({\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}^3}}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   22.3
9. Applied taylor to get
   \[\frac{\sqrt[3]{{\left({\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}^3}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \frac{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   2.1
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \frac{\color{blue}{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    2.1
11. Applied simplify to get
    \[\frac{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \frac{\left(\cos x \cdot \sin x\right) \cdot \left(\varepsilon \cdot 2 - \frac{1}{3} \cdot {\varepsilon}^3\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot 1}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    2.1

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12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{\left(\cos x \cdot \sin x\right) \cdot \left(\varepsilon \cdot 2 - \frac{1}{3} \cdot {\varepsilon}^3\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot 1}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \color{blue}{\frac{\left(\sin x \cdot \left(\cos x \cdot \varepsilon\right)\right) \cdot \left(2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) - \varepsilon \cdot \varepsilon}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
    2.1