Error

Derivation

1. Initial program 38.0

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

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-rgt-in3.8

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

6. Applied cos-sum0.1

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

8. Applied flip3--0.2

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

9. Applied associate-*r/0.2

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

10. Simplified0.2

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

11. Final simplification0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018354 +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))))))