Initial program 16.8
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
- Using strategy
rm Applied cos-diff3.7
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R\]
- Using strategy
rm Applied add-log-exp3.7
\[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right)} \cdot R\]
Taylor expanded around inf 3.7
\[\leadsto \log \color{blue}{\left(e^{\cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)} \cdot R\]
Simplified3.7
\[\leadsto \log \color{blue}{\left(e^{\cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \left(\sin \phi_1 \cdot \sin \phi_2\right))_*\right)}\right)} \cdot R\]
- Using strategy
rm Applied log1p-expm1-u3.7
\[\leadsto \log \left(e^{\cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) + \color{blue}{\left(\log_* (1 + (e^{\sin \phi_1 \cdot \sin \phi_2} - 1)^*)\right)})_*\right)}\right) \cdot R\]
Final simplification3.7
\[\leadsto \log \left(e^{\cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\log_* (1 + (e^{\sin \phi_2 \cdot \sin \phi_1} - 1)^*)\right))_*\right)}\right) \cdot R\]
