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Derivation

1. Initial program 36.8

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

2. Initial simplification3.7

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

4. Applied add-log-exp3.8

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

6. Applied log1p-expm1-u3.8

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

8. Applied add-cbrt-cube3.8

   \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\log \left(\log_* (1 + (e^{e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}} - 1)^*)\right) \cdot \log \left(\log_* (1 + (e^{e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}} - 1)^*)\right)\right) \cdot \log \left(\log_* (1 + (e^{e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}} - 1)^*)\right)}}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

9. Final simplification3.8

   \[\leadsto \sqrt{\left(\sqrt[3]{\left(\log \left(\log_* (1 + (e^{e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}} - 1)^*)\right) \cdot \log \left(\log_* (1 + (e^{e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}} - 1)^*)\right)\right) \cdot \log \left(\log_* (1 + (e^{e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}} - 1)^*)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

Reproduce

herbie shell --seed 2018362 +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))))))

Details

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