Average Error: 0.2 → 0.2
Time: 3.4m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left((\left(\cos \phi_1\right) \cdot \left(\sin delta \cdot \cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left((\left(\cos \phi_1\right) \cdot \left(\sin delta \cdot \cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left((\left(\cos theta \cdot \cos \phi_1\right) \cdot \left(\sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}{(\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1\right) \cdot \left(\cos theta \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right) \cdot \left((\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1\right) \cdot \left(\cos theta \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right) + \left(\cos delta \cdot \cos delta\right))_*}}\]

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. Using strategy rm

  3. Applied log1p-expm1-u0.2

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

  4. Applied simplify0.2

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

  6. Applied flip3--0.2

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

  7. Applied simplify0.2

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

  9. Applied add-cube-cbrt0.2

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

  10. Applied simplify0.2

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

  11. Applied simplify0.2

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018296 +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))))))))))