Average Error: 24.1 → 24.0
Time: 58.5s
Precision: 64
Internal Precision: 1344
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{(\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right) + \left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right))_*}}{\sqrt{(\left(-\left(\cos \left(\frac{\lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1}{2}\right) - \sin \left(\frac{\lambda_2}{2}\right) \cdot \cos \left(\frac{\lambda_1}{2}\right)\right)\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\frac{\lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1}{2}\right) - \sin \left(\frac{\lambda_2}{2}\right) \cdot \cos \left(\frac{\lambda_1}{2}\right)\right)\right) + \left(\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right)\right))_*}}\]

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 24.1

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

  2. Initial simplification24.1

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

  4. Applied div-sub24.1

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

  5. Applied sin-diff24.3

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

  7. Applied div-sub24.3

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

  8. Applied sin-diff23.8

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

  10. Applied add-cbrt-cube24.0

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

  11. Final simplification24.0

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

Runtime

Time bar (total: 58.5s)Debug logProfile

herbie shell --seed 2018242 +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))))))))))