Average Error: 13.7 → 0.2
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot (\left(\cos \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right))_* + 0}{(\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \left((\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right))_*\right) + \left(\cos \phi_1 \cdot \sin \phi_2\right))_*}\]

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 13.7

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

  3. Applied sub-neg13.7

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

  4. Applied sin-sum7.0

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

  5. Applied simplify7.0

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

  7. Applied sub-neg7.0

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

  8. Applied cos-sum0.2

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

  9. Applied simplify0.2

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

  10. Taylor expanded around inf 0.2

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

  11. Applied simplify0.2

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

  13. Applied prod-diff0.2

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

  14. Applied distribute-lft-in0.2

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

  15. Applied simplify0.2

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))