Error

Derivation

1. Initial program 37.6

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

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. Taylor expanded around -inf 3.8

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

7. Applied distribute-rgt-in3.8

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

8. Applied cos-sum0.2

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

9. Applied exp-diff0.2

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

10. Applied log-div0.2

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

11. Simplified0.1

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

13. Applied add-log-exp0.1

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

14. Applied exp-to-pow0.1

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

15. Applied log-pow0.1

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

16. Applied prod-diff0.1

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

17. Applied distribute-lft-in0.1

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

18. Simplified0.1

    \[\leadsto \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot (\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_2\right)\right) + \left(-\log \left(e^{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right))_* + \color{blue}{0}\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

19. Final simplification0.1

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

Runtime

Time bar (total: 57.5s)Debug logProfile

herbie shell --seed 2018273 +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))))))