Average Error: 16.5 → 3.6
Time: 1.2m
Precision: 64
Internal Precision: 1856
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \left(\log \left(e^{\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) + \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.5

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

  3. Applied sub-neg16.5

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

  4. Applied cos-sum3.6

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

  5. Simplified3.6

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

  7. Applied sub-neg3.6

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

  8. Applied distribute-lft-in3.6

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

  9. Simplified3.6

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

  11. Applied add-log-exp3.6

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

  12. Final simplification3.6

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018220 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))