if eps < -0.003667146f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.8
2. Using strategy rm
   13.8
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.4
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.4
5. Using strategy rm
   0.4
6. Applied add-cube-cbrt to get
   \[\sin x \cdot \color{red}{\cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \sin x \cdot \color{blue}{{\left(\sqrt[3]{\cos \varepsilon}\right)}^3} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   0.6
7. Applied add-cube-cbrt to get
   \[\color{red}{\sin x} \cdot {\left(\sqrt[3]{\cos \varepsilon}\right)}^3 + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\cos \varepsilon}\right)}^3 + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   0.7
8. Applied cube-unprod to get
   \[\color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\cos \varepsilon}\right)}^3} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\cos \varepsilon}\right)}^3} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   0.7

if -0.003667146f0 < eps < 0.0023221506f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   20.2
2. Using strategy rm
   20.2
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   13.2
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   13.2
5. Using strategy rm
   13.2
6. Applied add-cube-cbrt to get
   \[\color{red}{\sin x \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   20.1
7. Applied taylor to get
   \[{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon}\right)}^3 + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   12.9
8. Taylor expanded around 0 to get
   \[\color{red}{\left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{\left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   12.9
9. Applied simplify to get
   \[\left(\sin x - \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot \sin x\right)\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \left(\cos x \cdot \sin \varepsilon - \left(\sin x - \sin x\right)\right) - \left(\sin x \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\]
   0.1

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\left(\cos x \cdot \sin \varepsilon - \left(\sin x - \sin x\right)\right) - \left(\sin x \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \leadsto \color{blue}{\sin \varepsilon \cdot \cos x - \varepsilon \cdot \left(\left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\]
    0.1

if 0.0023221506f0 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.2
2. Using strategy rm
   14.2
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.4
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.4
5. Using strategy rm
   0.4
6. Applied expm1-log1p-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{(e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*} - \sin x\right)\]
   0.5
7. Using strategy rm
   0.5
8. Applied sub-neg to get
   \[\sin x \cdot \cos \varepsilon + \color{red}{\left((e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^* - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\left((e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^* + \left(-\sin x\right)\right)}\]
   0.5
9. Applied associate-+r+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left((e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^* + \left(-\sin x\right)\right)} \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + (e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*\right) + \left(-\sin x\right)}\]
   0.5
10. Applied simplify to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + (e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*\right)} + \left(-\sin x\right) \leadsto \color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin \varepsilon \cdot \cos x\right))_*} + \left(-\sin x\right)\]
    0.5