Average Error: 17.1 → 4.0
Time: 52.0s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\cos^{-1} \left((\left(\cos \phi_2 \cdot (\left(\sin \lambda_2\right) \cdot \left(\sin \lambda_1\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right))_*\right) \cdot \left(\cos \phi_1\right) + \left(\log_* (1 + (e^{\sin \phi_1 \cdot \sin \phi_2} - 1)^*)\right))_*\right)}\right)\]

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 17.1

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

  2. Simplified17.1

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

  4. Applied cos-diff3.9

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

  5. Taylor expanded around -inf 3.9

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

  6. Simplified3.9

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

  8. Applied add-log-exp4.0

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

  10. Applied log1p-expm1-u4.0

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

  11. Final simplification4.0

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

Reproduce

herbie shell --seed 2019053 +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))