Error

Derivation

1. Initial program 0.2

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

2. Initial simplification0.2

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

4. Applied log1p-expm1-u0.2

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

5. Taylor expanded around inf 0.2

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

6. Simplified0.2

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

7. Taylor expanded around inf 0.2

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

8. Simplified0.1

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

10. Applied log1p-expm1-u0.1

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

11. Applied expm1-log1p0.1

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

12. Simplified0.1

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

13. Final simplification0.1

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\log_* (1 + (e^{(\left(-\sin \phi_1\right) \cdot \left(\cos theta \cdot \sin delta\right) + \left(\cos delta \cdot \cos \phi_1\right))_* \cdot \cos \phi_1} - 1)^*)}\]

Runtime

Time bar (total: 41.2s)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))