Average Error: 0.1 → 0.2
Time: 2.3m
Precision: 64
Internal Precision: 384
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + \left(3 \cdot \left(\cos delta \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{6} + 3 \cdot \left({\left(\cos delta\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)\right)\right)\right)\right)}{(\left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right) \cdot \sin \phi_1\right) \cdot \left((\left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right) \cdot \left(\sin \phi_1\right) + \left(\cos delta\right))_*\right) + \left(\cos delta \cdot \cos delta\right))_*}}\]

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 flip3--0.2

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

  4. Applied simplify0.2

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

  5. Taylor expanded around inf 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{3} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + \left(3 \cdot \left(\cos delta \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{6} + 3 \cdot \left({\left(\cos delta\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)\right)\right)\right)\right)}}{(\left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right) \cdot \sin \phi_1\right) \cdot \left((\left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right) \cdot \left(\sin \phi_1\right) + \left(\cos delta\right))_*\right) + \left(\cos delta \cdot \cos delta\right))_*}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +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))))))))))