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Derivation

1. Initial program 13.5

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]

2. Initial simplification13.5

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\]
3. Using strategy rm

4. Applied sin-diff6.9

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\]
5. Using strategy rm

6. Applied cos-diff0.2

   \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\]
7. Using strategy rm

8. Applied expm1-log1p-u0.2

   \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{(e^{\log_* (1 + \sin \lambda_1 \cdot \cos \lambda_2)} - 1)^*} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\]
9. Using strategy rm

10. Applied flip-+0.2

    \[\leadsto \tan^{-1}_* \frac{\left((e^{\log_* (1 + \sin \lambda_1 \cdot \cos \lambda_2)} - 1)^* - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \color{blue}{\frac{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}{\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \lambda_2}} \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\]

11. Applied associate-*l/0.2

    \[\leadsto \tan^{-1}_* \frac{\left((e^{\log_* (1 + \sin \lambda_1 \cdot \cos \lambda_2)} - 1)^* - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \color{blue}{\frac{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}{\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \lambda_2}}}\]

12. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left((e^{\log_* (1 + \cos \lambda_2 \cdot \sin \lambda_1)} - 1)^* - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 \cdot \cos \phi_1 - \frac{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) - \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)}{\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_2 \cdot \sin \lambda_1}}\]

Runtime

Time bar (total: 43.7s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))