Average Error: 17.1 → 4.1
Time: 1.4m
Precision: 64
Internal Precision: 2112
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\left(\left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right) \cdot \sin \lambda_1\right) \cdot \sqrt[3]{\sin \lambda_2} + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) + \sin \phi_2 \cdot \sin \phi_1\right)\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied cos-diff4.0

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

  4. Taylor expanded around -inf 4.0

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

  6. Applied add-cube-cbrt4.1

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

  7. Applied associate-*r*4.1

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

  8. Final simplification4.1

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))