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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `R`

Derivation

Initial program 16.7
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
Simplified16.7
\[\leadsto \color{blue}{R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)}\]
Using strategy rm
Applied cos-diff3.8
\[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
Using strategy rm
Applied add-log-exp3.8
\[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\log \left(e^{\sin \lambda_1 \cdot \sin \lambda_2}\right)}\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
Final simplification3.8
\[\leadsto \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\log \left(e^{\sin \lambda_1 \cdot \sin \lambda_2}\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_1 \cdot \sin \phi_2\right))_*\right) \cdot R\]

Reproduce

herbie shell --seed 2019093 +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))