if eps < -1.2324208637780791e-05

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   29.6
2. Using strategy rm
   29.6
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.5
4. Using strategy rm
   0.5
5. Applied flip3-- to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
   0.7
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   0.6
7. Applied taylor to get
   \[\frac{{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)}^3 - {\left(\sin x\right)}^3}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \frac{{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x\right)}^3}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   0.6
8. Taylor expanded around 0 to get
   \[\frac{\color{red}{{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
   0.6
9. Applied simplify to get
   \[\frac{{\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \cos \varepsilon\right)}^3 - {\left(\sin x\right)}^3}{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \frac{{\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right)}^3 - {\left(\sin x\right)}^3}{\left(\left(\sin x + \cos x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \sin x\right) \cdot \sin x + {\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right)}^2}\]
   0.6

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10. Applied final simplification

if -1.2324208637780791e-05 < eps < 0.15577836889185903

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   44.4
2. Using strategy rm
   44.4
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   43.6
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   43.6
5. Using strategy rm
   43.6
6. Applied add-cbrt-cube to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}^3}}\]
   50.7
7. Applied taylor to get
   \[\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}^3} \leadsto \varepsilon \cdot \cos x - \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)\]
   0.1
8. 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
9. 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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10. Applied final simplification

if 0.15577836889185903 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   30.1
2. Using strategy rm
   30.1
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. Using strategy rm
   0.4
5. Applied flip-- to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x}}\]
   0.6