if eps < -0.0017290534f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   14.0
2. Using strategy rm
   14.0
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 log1p-expm1-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)} - \sin x\right)\]
   0.6

if -0.0017290534f0 < eps < 0.0541411f0

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   19.9
2. Using strategy rm
   19.9
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   13.3
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   13.3
5. Using strategy rm
   13.3
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   13.3
7. Applied simplify to get
   \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
   13.3
8. Using strategy rm
   13.3
9. Applied add-cube-cbrt to get
   \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}\right)}^3}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
   16.8
10. Applied taylor to get
    \[\frac{{\left(\sqrt[3]{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}\right)}^3}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
    2.1
11. Taylor expanded around 0 to get
    \[\frac{\color{red}{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\color{blue}{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}\]
    2.1
12. Applied simplify to get
    \[\frac{2 \cdot \left(\varepsilon \cdot \left(\sin x \cdot \cos x\right)\right) - \left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \cos x\right)\right) + \left({\varepsilon}^2 \cdot {\left(\sin x\right)}^2 + {\varepsilon}^2 \cdot {\left(\cos x\right)}^2\right)\right)}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x} \leadsto \frac{\left(\left(\varepsilon \cdot 2\right) \cdot \cos x\right) \cdot \sin x - (\left(\left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x \cdot \sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 1\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon}\]
    2.1

---

13. Applied final simplification
14. Applied simplify to get
    \[\color{red}{\frac{\left(\left(\varepsilon \cdot 2\right) \cdot \cos x\right) \cdot \sin x - (\left(\left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) * \left(\cos x \cdot \sin x\right) + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 1\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon}} \leadsto \color{blue}{\frac{\sin x \cdot \left(\cos x \cdot \left(2 \cdot \varepsilon\right)\right) - (\left(\left(\varepsilon \cdot \frac{1}{3}\right) \cdot {\varepsilon}^2\right) * \left(\cos x \cdot \sin x\right) + \left({\varepsilon}^2\right))_*}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \cos x \cdot \sin \varepsilon}}\]
    2.1

if 0.0541411f0 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.4
2. Using strategy rm
   13.4
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.3
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.3
5. Using strategy rm
   0.3
6. Applied add-log-exp to get
   \[\color{red}{\sin x \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{\log \left(e^{\sin x \cdot \cos \varepsilon}\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
   0.5