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

↓

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

Error

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

Derivation

Initial program 37.1
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
Applied simplify3.5
\[\leadsto \color{blue}{\sqrt{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R}\]
Using strategy rm
Applied add-log-exp3.6
\[\leadsto \sqrt{\left(\color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]
Using strategy rm
Applied expm1-log1p-u3.6
\[\leadsto \sqrt{\left(\log \color{blue}{\left((e^{\log_* (1 + e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)})} - 1)^*\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

Runtime

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))))))