if eps < -6.974478371048821e-14

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.8
2. Using strategy rm
   30.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.0
4. Using strategy rm
   2.0
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   2.2
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   2.1
7. Using strategy rm
   2.1
8. Applied pow3 to get
   \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^{3}} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   2.2

if -6.974478371048821e-14 < eps < 0.00018005451351709028

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   49.8
2. Using strategy rm
   49.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   49.7
4. Using strategy rm
   49.7
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   49.7
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   49.7
7. Using strategy rm
   49.7
8. Applied pow3 to get
   \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x\right)}^3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^{3}} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   49.7
9. Applied taylor to get
   \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \varepsilon \cdot \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 \cdot x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 + \left({\varepsilon}^2 \cdot {\left(\sin x \cdot \sin \varepsilon\right)}^3 + \varepsilon \cdot x\right)\right)\right)\]
   0.0
10. Taylor expanded around 0 to get
    \[\color{red}{\varepsilon \cdot \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 \cdot x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 + \left({\varepsilon}^2 \cdot {\left(\sin x \cdot \sin \varepsilon\right)}^3 + \varepsilon \cdot x\right)\right)\right)} \leadsto \color{blue}{\varepsilon \cdot \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 \cdot x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 + \left({\varepsilon}^2 \cdot {\left(\sin x \cdot \sin \varepsilon\right)}^3 + \varepsilon \cdot x\right)\right)\right)}\]
    0.0
11. Applied simplify to get
    \[\varepsilon \cdot \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 \cdot x\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^3 + \left({\varepsilon}^2 \cdot {\left(\sin x \cdot \sin \varepsilon\right)}^3 + \varepsilon \cdot x\right)\right)\right) \leadsto \left(x \cdot \varepsilon\right) \cdot {\left(\sin x \cdot \sin \varepsilon\right)}^3 - \left((\left({\varepsilon}^2\right) * \frac{1}{2} + \left({\left(\sin x \cdot \sin \varepsilon\right)}^3\right))_* + (\left({\left(\sin x \cdot \sin \varepsilon\right)}^3\right) * \left({\varepsilon}^2\right) + \left(x \cdot \varepsilon\right))_*\right)\]
    0.0

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12. Applied final simplification

if 0.00018005451351709028 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.5
2. Using strategy rm
   30.5
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.9
4. Using strategy rm
   0.9
5. Applied flip3-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}} - \cos x\]
   1.1
6. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\]
   1.0
7. Using strategy rm
   1.0
8. Applied add-cbrt-cube to get
   \[\color{red}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\right)}^3}}\]
   1.3
9. Applied taylor to get
   \[\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} - \cos x\right)}^3} \leadsto \sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3}\]
   1.4
10. Taylor expanded around 0 to get
    \[\sqrt[3]{\color{red}{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3}}\]
    1.4
11. Applied simplify to get
    \[\sqrt[3]{{\left(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)} - \left(\cos x + \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{{\left(\cos x\right)}^2 \cdot {\left(\cos \varepsilon\right)}^2 + \left(\sin x \cdot \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) + {\left(\sin x\right)}^2 \cdot {\left(\sin \varepsilon\right)}^2\right)}\right)\right)}^3} \leadsto \left(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left(\cos x \cdot \cos \varepsilon\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_* + {\left(\sin x \cdot \sin \varepsilon\right)}^2} - \cos x\right) - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^3}{(\left(\sin x \cdot \sin \varepsilon\right) * \left(\cos x \cdot \cos \varepsilon\right) + \left({\left(\cos x \cdot \cos \varepsilon\right)}^2\right))_* + {\left(\sin x \cdot \sin \varepsilon\right)}^2}\]
    1.0

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12. Applied final simplification