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

↓

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

Error

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

Derivation

Initial program 16.8
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
Using strategy rm
Applied cos-diff3.9
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R\]
Taylor expanded around -inf 3.9
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)}\right) \cdot R\]
Simplified3.9
\[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot (\left(\sin \lambda_1\right) \cdot \left(\sin \lambda_2\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right))_*}\right) \cdot R\]
Final simplification3.9
\[\leadsto \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + (\left(\sin \lambda_1\right) \cdot \left(\sin \lambda_2\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_* \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\]

Runtime

herbie shell --seed 2018230 +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))