Average Error: 0.2 → 0.2
Time: 1.6m
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(\sqrt[3]{\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 {\left(\sin \phi_1\right)}^{3}\right) \cdot \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 \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)}\right)}^{3}}{(\left(\cos delta\right) \cdot \left(\cos delta + \sqrt[3]{\left({\left(\sin \phi_1\right)}^{3} \cdot \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) \cdot \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right)}\right) + \left(\sqrt[3]{\left({\left(\sin \phi_1\right)}^{3} \cdot \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) \cdot \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right)} \cdot \sqrt[3]{\left({\left(\sin \phi_1\right)}^{3} \cdot \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) \cdot \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right)}\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 0.2

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

  3. Applied add-cbrt-cube0.2

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

  4. Applied add-cbrt-cube0.2

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

  5. Applied cbrt-unprod0.2

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

  6. Applied simplify0.2

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

  8. 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(\sqrt[3]{\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 {\left(\sin \phi_1\right)}^{3}\right) \cdot \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 \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)}\right)}^{3}}{\cos delta \cdot \cos delta + \left(\sqrt[3]{\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 {\left(\sin \phi_1\right)}^{3}\right) \cdot \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 \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 \sqrt[3]{\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 {\left(\sin \phi_1\right)}^{3}\right) \cdot \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 \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)} + \cos delta \cdot \sqrt[3]{\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 {\left(\sin \phi_1\right)}^{3}\right) \cdot \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 \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)}\right)}}}\]

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))))))))))