Initial program 36.7
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied diff-sin37.0
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified14.6
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
- Using strategy
rm Applied expm1-log1p-u14.6
\[\leadsto 2 \cdot \left(\color{blue}{(e^{\log_* (1 + \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
Taylor expanded around -inf 14.6
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
Simplified14.6
\[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)\right)}\]
- Using strategy
rm Applied fma-udef14.6
\[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\]
Applied cos-sum0.3
\[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\right)\]
Final simplification0.3
\[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)\]
