Error

Derivation

1. Initial program 13.8

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

3. Applied sin-diff7.0

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

5. Applied cos-diff0.2

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

7. Applied flip3-+0.2

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

8. Applied associate-*r/0.2

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

9. Simplified0.2

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

11. Applied flip3-+0.2

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

12. Applied associate-*r/0.2

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

13. Simplified0.2

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

14. Final simplification0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018257 +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))))))