if eps < -0.0041207117f0 or 0.004007106f0 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.2
2. Using strategy rm
   14.2
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 flip3-- to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
   0.6
7. Applied simplify to get
   \[\sin x \cdot \cos \varepsilon + \frac{\color{red}{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   0.4
8. Using strategy rm
   0.4
9. Applied add-cbrt-cube to get
   \[\sin x \cdot \cos \varepsilon + \frac{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - \color{red}{{\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - \color{blue}{\sqrt[3]{{\left({\left(\sin x\right)}^3\right)}^3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   0.4

if -0.0041207117f0 < eps < 0.004007106f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   19.8
2. Using strategy rm
   19.8
3. Applied add-cube-cbrt to get
   \[\color{red}{\sin \left(x + \varepsilon\right) - \sin x} \leadsto \color{blue}{{\left(\sqrt[3]{\sin \left(x + \varepsilon\right) - \sin x}\right)}^3}\]
   20.0
4. Applied taylor to get
   \[{\left(\sqrt[3]{\sin \left(x + \varepsilon\right) - \sin x}\right)}^3 \leadsto \varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)\]
   0.1
5. Taylor expanded around 0 to get
   \[\color{red}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} \leadsto \color{blue}{\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)}\]
   0.1
6. Applied simplify to get
   \[\varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) \leadsto \left(\varepsilon - \frac{1}{6} \cdot {\varepsilon}^3\right) \cdot \cos x - \left(\sin x \cdot \frac{1}{2}\right) \cdot {\varepsilon}^2\]
   0.1

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7. Applied final simplification