Error

Derivation

1. Initial program 24.0

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

2. Initial simplification23.9

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

4. Applied expm1-log1p-u23.9

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

6. Applied add-log-exp23.9

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

8. Applied log1p-expm1-u23.9

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

10. Applied add-log-exp24.0

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

11. Final simplification24.0

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

Runtime

Time bar (total: 43.2s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* R (* 2 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))) (sqrt (- 1 (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))))))))