Initial program 37.0
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied diff-sin37.3
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified14.8
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
- Using strategy
rm Applied log1p-expm1-u14.9
\[\leadsto 2 \cdot \left(\color{blue}{\log_* (1 + (e^{\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} - 1)^*)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Taylor expanded around inf 14.8
\[\leadsto 2 \cdot \left(\color{blue}{\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Simplified14.8
\[\leadsto 2 \cdot \left(\color{blue}{\cos \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
- Using strategy
rm Applied fma-udef14.8
\[\leadsto 2 \cdot \left(\cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Applied cos-sum0.3
\[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Final simplification0.3
\[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
