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Derivation

1. Initial program 13.3

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

2. Initial simplification13.3

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

4. Applied sin-diff6.7

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\]
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 add-cube-cbrt0.3

   \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \phi_1} \cdot \sqrt[3]{\sin \phi_1}\right) \cdot \sqrt[3]{\sin \phi_1}\right)}\right)}\]

9. Applied associate-*r*0.3

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

11. Applied add-cube-cbrt0.4

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\sqrt[3]{\sin \phi_1} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sin \phi_1}} \cdot \sqrt[3]{\sqrt[3]{\sin \phi_1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin \phi_1}}\right)}\right)\right) \cdot \sqrt[3]{\sin \phi_1}\right)}\]

12. Final simplification0.4

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

Runtime

Time bar (total: 41.0s)Debug logProfile

herbie shell --seed 2018256 +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))))))