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Derivation

1. Initial program 0.9

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

2. Applied simplify0.9

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

4. Applied sub-neg0.9

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

5. Applied sin-sum0.8

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

6. Applied simplify0.8

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

8. Applied sub-neg0.8

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

9. Applied cos-sum0.2

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

10. Applied simplify0.2

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

12. Applied add-cbrt-cube0.3

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

13. Applied simplify0.3

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

15. Applied add-cube-cbrt0.6

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

16. Applied unpow-prod-down0.6

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

17. Applied simplify0.3

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

18. Applied simplify0.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))