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

↓

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

Error

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

Derivation

Initial program 0.2
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
Applied simplify0.2
\[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{(\left(-\sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) + \left(\cos delta\right))_*} + \lambda_1}\]
Using strategy rm
Applied log1p-expm1-u0.2
\[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{(\left(-\sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \color{blue}{\left(\log_* (1 + (e^{(\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*} - 1)^*)\right)}\right)\right) + \left(\cos delta\right))_*} + \lambda_1\]
Using strategy rm
Applied associate-*r*0.2
\[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta}}{(\left(-\sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\log_* (1 + (e^{(\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*} - 1)^*)\right)\right)\right) + \left(\cos delta\right))_*} + \lambda_1\]

Runtime

herbie shell --seed 2018178 +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))))))))))