\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

↓

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot (e^{\log_* (1 + \cos \left(\frac{\phi_2 + \phi_1}{2}\right))} - 1)^*\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]

Error

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

Derivation

Initial program 36.7
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
Simplified3.7
\[\leadsto \color{blue}{\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R}\]
Using strategy rm
Applied expm1-log1p-u3.7
\[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{(e^{\log_* (1 + \cos \left(\frac{\phi_2 + \phi_1}{2}\right))} - 1)^*}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]
Final simplification3.7
\[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot (e^{\log_* (1 + \cos \left(\frac{\phi_2 + \phi_1}{2}\right))} - 1)^*\right)^2 + \left(\phi_1 - \phi_2\right)^2}^*\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))