Average Error: 0.2 → 0.2
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\log_* (1 + (e^{\cos \phi_1 \cdot \sin delta} - 1)^*) \cdot \sin theta}{(\left(-\sin \phi_1\right) \cdot \left((\left(\cos \phi_1\right) \cdot \left(\cos theta \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right) + \left(\cos delta\right))_*}\]

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 0.2

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

  2. Applied simplify0.2

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

  3. Taylor expanded around -inf 0.2

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

  4. Applied simplify0.2

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

  6. Applied log1p-expm1-u0.2

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018193 +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))))))))))