Initial program 37.6
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.6
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
- Using strategy
rm Applied sin-sum22.3
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
Applied associate--l+0.4
\[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \color{blue}{1 \cdot \sin x}\right)\]
Applied prod-diff0.4
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left((\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_* + (\left(-\sin x\right) \cdot 1 + \left(\sin x \cdot 1\right))_*\right)}\]
Applied associate-+r+0.4
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + (\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*\right) + (\left(-\sin x\right) \cdot 1 + \left(\sin x \cdot 1\right))_*}\]
Simplified0.4
\[\leadsto \left(\sin \varepsilon \cdot \cos x + (\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*\right) + \color{blue}{(-1 \cdot \left(\sin x\right) + \left(\sin x\right))_*}\]
- Using strategy
rm Applied fma-def0.4
\[\leadsto \color{blue}{(\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left((\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*\right))_*} + (-1 \cdot \left(\sin x\right) + \left(\sin x\right))_*\]
- Using strategy
rm Applied log1p-expm1-u0.4
\[\leadsto (\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \color{blue}{\left(\log_* (1 + (e^{(\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x \cdot 1\right))_*} - 1)^*)\right)})_* + (-1 \cdot \left(\sin x\right) + \left(\sin x\right))_*\]
Final simplification0.4
\[\leadsto (-1 \cdot \left(\sin x\right) + \left(\sin x\right))_* + (\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left(\log_* (1 + (e^{(\left(\cos \varepsilon\right) \cdot \left(\sin x\right) + \left(-\sin x\right))_*} - 1)^*)\right))_*\]
