Average Error: 16.4 → 3.7
Time: 2.2m
Precision: 64
Internal Precision: 2112
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)\right) \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 16.4

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]

  2. Applied simplify16.4

    \[\leadsto \color{blue}{\cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R}\]
  3. Using strategy rm

  4. Applied sub-neg16.4

    \[\leadsto \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]

  5. Applied cos-sum3.7

    \[\leadsto \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]

  6. Applied simplify3.7

    \[\leadsto \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]
  7. Using strategy rm

  8. Applied add-cube-cbrt3.7

    \[\leadsto \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right) \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)}\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]

  9. Applied associate-*r*3.7

    \[\leadsto \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}\right)\right) \cdot \sqrt[3]{\sin \left(-\lambda_2\right)}}\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right) \cdot R\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))