if eps < -0.0015323369f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.8
2. Using strategy rm
   14.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.0
4. Using strategy rm
   1.0
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x\]
   1.0
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{blue}{(\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \cos x\right))_*}} - \cos x\]
   1.2

if -0.0015323369f0 < eps < 0.0038476672f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.0
2. Using strategy rm
   23.0
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.9
4. Using strategy rm
   18.9
5. Applied add-cube-cbrt to get
   \[\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{red}{\sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x\]
   18.9
6. Applied add-cube-cbrt to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x\]
   18.9
7. Applied cube-unprod to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x\]
   18.9
8. Applied taylor to get
   \[\left(\cos x \cdot \cos \varepsilon - {\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x \leadsto -\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right)\]
   0.6
9. Taylor expanded around 0 to get
   \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right)} \leadsto \color{blue}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right)}\]
   0.6
10. Applied simplify to get
    \[-\left(\frac{1}{2} \cdot {\varepsilon}^2 + {\left(\sqrt[3]{\sin \varepsilon} \cdot \sqrt[3]{\sin x}\right)}^3\right) \leadsto -(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right))_*\]
    0.2

---

11. Applied final simplification

if 0.0038476672f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.8
2. Using strategy rm
   14.8
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.9
4. Using strategy rm
   0.9
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}}\]
   0.9
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
   1.0