Error

Derivation

1. Initial program 0.1

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

2. Initial simplification0.1

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

4. Applied log1p-expm1-u0.2

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

6. Applied add-cbrt-cube0.2

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

7. Final simplification0.2

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

Runtime

Time bar (total: 28.1s)Debug logProfile

herbie shell --seed 2018290 +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))))))))))