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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `lambda1`

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Derivation

Initial program 0.1
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
Taylor expanded around inf 0.1
\[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta - \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin delta \cdot \left(\sin \phi_1 \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)}}\]
Final simplification0.1
\[\leadsto \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \left(\left(\sin \phi_1 \cdot \left(\cos theta \cdot \cos \phi_1\right)\right) \cdot \sin delta + \cos delta \cdot {\left(\sin \phi_1\right)}^{2}\right)} + \lambda_1\]

Runtime

herbie shell --seed 2018216 
(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))))))))))