Error

Derivation

1. Initial program 16.6

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

2. Simplified16.6

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

4. Applied cos-diff4.1

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

5. Taylor expanded around -inf 4.1

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

6. Simplified4.1

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

8. Applied log1p-expm1-u4.1

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

10. Applied add-log-exp4.1

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

11. Final simplification4.1

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

Reproduce

herbie shell --seed 2019004 +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))

Details

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