if eps < -0.008720061f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.7
2. Using strategy rm
   14.7
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 add-log-exp to get
   \[\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{red}{\cos x} \leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{\log \left(e^{\cos x}\right)}\]
   1.1
6. Applied add-log-exp to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \log \left(e^{\cos x}\right) \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)}\right) - \log \left(e^{\cos x}\right)\]
   1.2
7. Applied add-log-exp to get
   \[\left(\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right) \leadsto \left(\color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right)\]
   1.4
8. Applied diff-log to get
   \[\color{red}{\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right)} - \log \left(e^{\cos x}\right) \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right)} - \log \left(e^{\cos x}\right)\]
   1.4
9. Applied diff-log to get
   \[\color{red}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right) - \log \left(e^{\cos x}\right)} \leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]
   1.4
10. Applied simplify to get
    \[\log \color{red}{\left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)} \leadsto \log \color{blue}{\left(e^{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)}\]
    1.2

if -0.008720061f0 < eps < 0.019318156f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   23.2
2. Using strategy rm
   23.2
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.5
4. Using strategy rm
   18.5
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}}\]
   18.5
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
   18.5
7. Using strategy rm
   18.5
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\color{blue}{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2\right)}^3}}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
   18.8
9. Applied taylor to get
   \[\frac{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2\right)}^3}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
   0.2
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{\color{blue}{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
    0.2
11. Applied simplify to get
    \[\frac{\left(\frac{4}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\sin x\right)}^2\right) - \left(2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x} \leadsto \frac{(\left(\sin x \cdot \sin x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\left(\frac{4}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin x\right)\right))_* - \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x + \left(2 \cdot \varepsilon\right) \cdot \sin x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}\]
    0.2

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{(\left(\sin x \cdot \sin x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\left(\frac{4}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\cos x \cdot \sin x\right)\right))_* - \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \cos x + \left(2 \cdot \varepsilon\right) \cdot \sin x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}} \leadsto \color{blue}{\frac{(\left(\sin x \cdot \sin x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\sin x \cdot \left(\varepsilon \cdot \frac{4}{3}\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right))_* - (\left(\cos x\right) * \varepsilon + \left(\sin x \cdot 2\right))_* \cdot \left(\varepsilon \cdot \cos x\right)}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
    0.2

if 0.019318156f0 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.5
2. Using strategy rm
   14.5
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   0.8
4. Using strategy rm
   0.8
5. Applied flip-- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x\]
   0.8
6. Applied simplify to get
   \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{blue}{(\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \cos x\right))_*}} - \cos x\]
   1.0