Average Error: 0.8 → 0.3
Time: 31.1s
Precision: 64
Internal Precision: 1344
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \lambda_1 \cdot \sin \left(-\lambda_2\right) + \left(\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}} \cdot \sqrt[3]{\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}}\right)\right) \cdot \sqrt[3]{\sin \lambda_1 \cdot \cos \lambda_2}\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \sin \left(-\lambda_2\right) \cdot \sin \lambda_1\right) + \left(\cos \phi_1\right))_*}\]

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.8

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

  2. Initial simplification0.8

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

  4. Applied sub-neg0.8

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

  5. Applied cos-sum0.8

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

  6. Simplified0.8

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

  8. Applied sub-neg0.8

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

  9. Applied sin-sum0.2

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

  10. Simplified0.2

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

  12. Applied add-cube-cbrt0.3

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

  14. Applied add-cube-cbrt0.3

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

  15. Final simplification0.3

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

Runtime

Time bar (total: 31.1s)Debug logProfile

herbie shell --seed 2018220 +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)))))))