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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

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

9. Applied associate-*l*0.3

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

11. Applied flip3-+0.3

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

12. Applied cbrt-div0.2

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

13. Applied flip-+0.2

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

14. Applied cbrt-div0.2

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

15. Applied frac-times0.2

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

16. Simplified0.2

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

17. Final simplification0.2

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

Runtime

Time bar (total: 55.8s)Debug logProfile

herbie shell --seed 2018256 
(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))))))