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. Applied simplify17.0

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

4. Applied sub-neg17.0

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

5. Applied cos-sum3.9

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

6. Applied simplify3.9

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

8. Applied acos-asin4.0

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

9. Applied simplify4.0

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

11. Applied flip--4.1

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

12. Applied associate-*l/4.1

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018198 +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))