if eps < -3.5438160870943954e-16

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   32.2
2. Using strategy rm
   32.2
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   2.4
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   2.4
5. Using strategy rm
   2.4
6. Applied flip3-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
   2.6
7. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
   2.6

if -3.5438160870943954e-16 < eps < -3.484092632009674e-67 or 3.9198528401673747e-134 < eps < 0.0016565824112741959

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   61.1
2. Using strategy rm
   61.1
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   60.3
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   60.3
5. Using strategy rm
   60.3
6. Applied add-cbrt-cube to get
   \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{red}{\sin \varepsilon} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   60.3
7. Applied add-cbrt-cube to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right)\]
   60.3
8. Applied cbrt-unprod to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sqrt[3]{{\left(\sin x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   60.3
9. Applied taylor to get
   \[\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3} + \cos x\right) \leadsto -\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)\]
   9.4
10. Taylor expanded around 0 to get
    \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)} \leadsto \color{blue}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right)}\]
    9.4

if -3.484092632009674e-67 < eps < 3.9198528401673747e-134

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   40.6
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   8.7
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   8.7
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]
   8.7

if 0.0016565824112741959 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   29.3
2. Using strategy rm
   29.3
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. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   0.9
5. Using strategy rm
   0.9
6. Applied add-cbrt-cube to get
   \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{red}{\sin \varepsilon} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   0.9
7. Applied add-cbrt-cube to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right)\]
   1.0
8. Applied cbrt-unprod to get
   \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sqrt[3]{{\left(\sin x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
   0.9