Average Error: 0.2 → 0.2
Time: 40.0s
Precision: 64
Internal Precision: 384
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) \cdot \sin \phi_1}\right)} + \lambda_1\]

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 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

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

  4. Applied add-log-exp0.2

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

  5. Applied add-log-exp0.2

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

  6. Applied diff-log0.2

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

  7. Applied simplify0.2

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

Runtime

Time bar (total: 40.0s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))))))))))