Error

Derivation

1. Initial program 17.0

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

2. Simplified17.0

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

4. Applied cos-diff3.7

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

6. Applied add-log-exp3.7

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

7. Taylor expanded around -inf 3.7

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

8. Simplified3.7

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

10. Applied *-un-lft-identity3.7

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

11. Applied log-prod3.7

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

12. Applied distribute-lft-in3.7

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

13. Simplified3.6

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

14. Final simplification3.6

    \[\leadsto R \cdot 0 + \cos^{-1} \left((\left(\sin \phi_1\right) \cdot \left(\sin \phi_2\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot (\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right))_*\right)\right))_*\right) \cdot R\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))