Initial program 37.1
\[\sin \left(x + \varepsilon\right) - \sin x\]
- Using strategy
rm Applied diff-sin37.4
\[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
Simplified15.6
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
Taylor expanded around inf 15.6
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
Simplified15.5
\[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left((\frac{1}{2} \cdot \varepsilon + x)_*\right)\right)}\]
- Using strategy
rm Applied fma-udef15.5
\[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\]
Applied cos-sum0.3
\[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}\right)\]
- Using strategy
rm Applied fma-neg0.3
\[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x\right) + \left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right))_*}\right)\]
Final simplification0.3
\[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot (\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x\right) + \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-\sin x\right)\right))_*\right)\]
