Error

Derivation

1. Initial program 16.5

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

2. Applied simplify16.5

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

4. Applied sub-neg16.5

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

5. Applied cos-sum3.8

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

6. Applied simplify3.8

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

8. Applied add-log-exp3.9

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

9. Applied simplify3.9

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

11. Applied add-sqr-sqrt4.0

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

12. Applied log-prod4.0

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

14. Applied pow1/24.0

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

15. Applied log-pow3.9

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

16. Applied simplify3.9

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

Runtime

Time bar (total: 51.0s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))