Average Error: 0.1 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1\right)\right)}^{3}}{\cos delta \cdot \cos delta + \left(\left(\cos delta \cdot \sin \phi_1\right) \cdot \left(\cos delta \cdot \sin \phi_1\right)\right) \cdot \left(1 + \sin \phi_1 \cdot \sin \phi_1\right)} - \left(\cos theta \cdot \sin \phi_1\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right)}\]

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 add-log-exp0.2

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

  4. Taylor expanded around inf 0.2

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

  5. Applied simplify0.1

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

  7. Applied flip3--0.1

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

  8. Applied simplify0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(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))))))))))