Error

Derivation

1. Initial program 13.2

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

3. Applied sub-neg13.2

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

4. Applied sin-sum6.7

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

5. Simplified6.7

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

7. Applied sub-neg6.7

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

8. Applied cos-sum0.2

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

9. Simplified0.2

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

11. Applied add-cbrt-cube0.2

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

12. Applied add-cbrt-cube0.2

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

13. Applied add-cbrt-cube0.4

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

14. Applied cbrt-unprod0.4

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

15. Applied cbrt-unprod0.4

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

16. Simplified0.4

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

18. Applied add-cube-cbrt0.4

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

19. Applied unpow-prod-down0.4

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

20. Simplified0.4

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

21. Final simplification0.4

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

Runtime

Time bar (total: 48.9s)Debug logProfile

herbie shell --seed 2018221 +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))))))