Average Error: 0.1 → 0.2
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\cos delta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{4} + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}\right)\right) + 2 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)\right)\right)\right)}{\cos delta + \left(\sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right) + \cos delta \cdot {\left(\sin \phi_1\right)}^{2}\right)}}\]

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 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. Using strategy rm

  3. Applied flip--0.2

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

  4. Taylor expanded around inf 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\cos delta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{4} + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}\right)\right) + 2 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)\right)\right)\right)}{\cos delta + \left(\sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right) + \cos delta \cdot {\left(\sin \phi_1\right)}^{2}\right)}}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))))))))))