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Derivation

1. Initial program 37.2

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

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. Taylor expanded around inf 3.8

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

5. Applied distribute-lft-in3.8

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

6. Applied cos-sum0.1

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

8. Applied add-cube-cbrt0.2

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

9. Applied associate-*r*0.2

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

10. Final simplification0.2

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

Runtime

Time bar (total: 59.1s)Debug logProfile

herbie shell --seed 2018353 +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))))))