Initial program 0.8
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
- Using strategy
rm Applied sin-diff0.8
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
- Using strategy
rm Applied cos-diff0.2
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\]
- Using strategy
rm Applied add-log-exp0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 + \color{blue}{\log \left(e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)}}\]
Applied add-log-exp0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\log \left(e^{\cos \phi_1}\right)} + \log \left(e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)}\]
Applied sum-log0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\log \left(e^{\cos \phi_1} \cdot e^{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)}}\]
Simplified0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \color{blue}{\left(e^{(\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_*\right) \cdot \left(\cos \phi_2\right) + \left(\cos \phi_1\right))_*}\right)}}\]
- Using strategy
rm Applied fma-udef0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\color{blue}{(\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_* \cdot \cos \phi_2 + \cos \phi_1}}\right)}\]
Applied exp-sum0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \color{blue}{\left(e^{(\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_* \cdot \cos \phi_2} \cdot e^{\cos \phi_1}\right)}}\]
Applied log-prod0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\log \left(e^{(\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right))_* \cdot \cos \phi_2}\right) + \log \left(e^{\cos \phi_1}\right)}}\]
Simplified0.3
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\cos \phi_2 \cdot (\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right))_*} + \log \left(e^{\cos \phi_1}\right)}\]
Final simplification0.3
\[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\log \left(e^{\cos \phi_1}\right) + \cos \phi_2 \cdot (\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right))_*} + \lambda_1\]
