if eps < -3.0129406f-05

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   14.9
2. Using strategy rm
   14.9
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.5
4. Using strategy rm
   1.5
5. Applied log1p-expm1-u to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)}\right) - \cos x\]
   1.5
6. Using strategy rm
   1.5
7. Applied add-cube-cbrt to get
   \[\left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\sin x \cdot \color{red}{\sin \varepsilon}} - 1)^*)\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\sin x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3}} - 1)^*)\right) - \cos x\]
   1.6
8. Applied add-cube-cbrt to get
   \[\left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\color{red}{\sin x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3} - 1)^*)\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3} - 1)^*)\right) - \cos x\]
   1.6
9. Applied cube-unprod to get
   \[\left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3}} - 1)^*)\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3}} - 1)^*)\right) - \cos x\]
   1.6

if -3.0129406f-05 < eps < 0.024397144f0

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   22.7
2. Using strategy rm
   22.7
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   18.5
4. Using strategy rm
   18.5
5. Applied log1p-expm1-u to get
   \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)}\right) - \cos x\]
   18.5
6. Applied taylor to get
   \[\left(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right) - \cos x \leadsto -\left(\frac{1}{2} \cdot {\varepsilon}^2 + \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right)\]
   0.2
7. Taylor expanded around 0 to get
   \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right)} \leadsto \color{blue}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right)}\]
   0.2
8. Applied simplify to get
   \[\color{red}{-\left(\frac{1}{2} \cdot {\varepsilon}^2 + \log_* (1 + (e^{\sin x \cdot \sin \varepsilon} - 1)^*)\right)} \leadsto \color{blue}{(\left(-\sin x\right) * \left(\sin \varepsilon\right) + \left(\left(-\varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)\right))_*}\]
   0.1
9. Applied simplify to get
   \[(\left(-\sin x\right) * \left(\sin \varepsilon\right) + \color{red}{\left(\left(-\varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)\right)})_* \leadsto (\left(-\sin x\right) * \left(\sin \varepsilon\right) + \color{blue}{\left({\varepsilon}^2 \cdot \left(-\frac{1}{2}\right)\right)})_*\]
   0.1

if 0.024397144f0 < 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
   \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon}} - \cos x \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon\right)}^2}{\color{blue}{(\left(\sin x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \cos x\right))_*}} - \cos x\]
   0.9