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Derivation

1. Initial program 16.4

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

3. Applied cos-diff3.6

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

4. Applied distribute-lft-in3.6

   \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R\]
5. Using strategy rm

6. Applied add-cbrt-cube3.6

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

7. Applied add-cbrt-cube3.6

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

8. Applied cbrt-unprod3.6

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

9. Applied add-cbrt-cube3.6

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

10. Applied add-cbrt-cube3.6

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

11. Applied cbrt-unprod3.6

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

12. Applied cbrt-unprod3.6

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

13. Applied simplify3.6

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

15. Applied acos-asin3.7

    \[\leadsto \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \sqrt[3]{\left({\left(\sin \lambda_2\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right) \cdot \left({\left(\sin \lambda_1\right)}^{3} \cdot {\left(\cos \phi_2\right)}^{3}\right)}\right)\right)\right)} \cdot R\]
16. Using strategy rm

17. Applied add-log-exp3.7

    \[\leadsto \left(\frac{\pi}{2} - \color{blue}{\log \left(e^{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \sqrt[3]{\left({\left(\sin \lambda_2\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right) \cdot \left({\left(\sin \lambda_1\right)}^{3} \cdot {\left(\cos \phi_2\right)}^{3}\right)}\right)\right)}\right)}\right) \cdot R\]

18. Applied add-log-exp3.7

    \[\leadsto \left(\color{blue}{\log \left(e^{\frac{\pi}{2}}\right)} - \log \left(e^{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \sqrt[3]{\left({\left(\sin \lambda_2\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right) \cdot \left({\left(\sin \lambda_1\right)}^{3} \cdot {\left(\cos \phi_2\right)}^{3}\right)}\right)\right)}\right)\right) \cdot R\]

19. Applied diff-log3.7

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \sqrt[3]{\left({\left(\sin \lambda_2\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right) \cdot \left({\left(\sin \lambda_1\right)}^{3} \cdot {\left(\cos \phi_2\right)}^{3}\right)}\right)\right)}}\right)} \cdot R\]

20. Applied simplify3.7

    \[\leadsto \log \color{blue}{\left(e^{\frac{\pi}{2} - \sin^{-1} \left(\sqrt[3]{\left({\left(\sin \lambda_2\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right) \cdot \left({\left(\cos \phi_2\right)}^{3} \cdot {\left(\sin \lambda_1\right)}^{3}\right)} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)} \cdot R\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))