Error

Derivation

1. Initial program 0.9

   \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]

2. Initial simplification0.9

   \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*} + \lambda_1\]
3. Using strategy rm

4. Applied sub-neg0.9

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

5. Applied sin-sum0.8

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

6. Applied distribute-lft-in0.8

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

8. Applied sub-neg0.8

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

9. Applied cos-sum0.2

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

10. Simplified0.2

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

12. Applied add-log-exp0.2

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

14. Applied add-cube-cbrt0.2

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

15. Applied log-prod0.2

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

16. Final simplification0.2

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

Runtime

Time bar (total: 37.8s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))