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Derivation

1. Initial program 37.1

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

2. Initial simplification3.8

   \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]
3. Using strategy rm

4. Applied expm1-log1p-u3.8

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

5. Taylor expanded around inf 3.9

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

6. Simplified3.8

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

7. Final simplification3.8

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

Runtime

Time bar (total: 18.6s)Debug logProfile

herbie shell --seed 2018274 +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))))))