Average Error: 3.6 → 3.7
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\sqrt[3]{\cos \phi_1} \cdot \left(\left(\sqrt[3]{\cos \phi_1} \cdot \sqrt[3]{\cos \phi_1}\right) \cdot \left(\sin delta \cdot \sin theta\right)\right)}{(\left(-\sin \phi_1\right) \cdot \left((\left(\sin delta\right) \cdot \left(\cos theta \cdot \cos \phi_1\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right) + \left(\cos delta\right))_*} + \lambda_1\]

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 3.6

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

  2. Taylor expanded around inf 3.6

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

  3. Simplified3.6

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

  5. Applied add-cube-cbrt3.7

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

  6. Applied associate-*r*3.7

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

  7. Final simplification3.7

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018285 +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))))))))))