Average Error: 3.5 → 3.5
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{(e^{\log_* (1 + (\left(\sin \left(\frac{\pi}{2} - \cos^{-1} \left((\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos delta\right))_*)} - 1)^*}\]

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

  1. Initial program 3.5

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

  2. Initial simplification3.5

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

  4. Applied expm1-log1p-u3.5

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

  6. Applied asin-acos3.5

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

  7. Final simplification3.5

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018304 +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))))))))))