Average Error: 24.3 → 24.2
Time: 2.6m
Precision: 64
Internal Precision: 1408
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\log \left(\sqrt{e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right) + \left(\log \left(\sqrt{e^{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)}}\right) - \log \left(\sqrt{e^{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}}\right)\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right)\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 24.3

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

  3. Applied add-log-exp24.3

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

  5. Applied add-sqr-sqrt24.3

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

  6. Applied log-prod24.3

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

  8. Applied div-sub24.3

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

  9. Applied sin-diff24.3

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

  10. Applied exp-diff24.3

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

  11. Applied sqrt-div24.3

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

  12. Applied log-div24.3

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

  14. Applied div-sub24.3

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

  15. Applied sin-diff24.2

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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))))))))))