Average Error: 17.0 → 3.9
Time: 14.6s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \lambda_1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right) + \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R\]
\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(\sin \lambda_1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right) + \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R);
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (acos(((sin(lambda1) * (cos(phi1) * (cos(phi2) * sin(lambda2)))) + ((sin(phi1) * sin(phi2)) + (cos(lambda1) * (cos(phi1) * (cos(phi2) * cos(lambda2))))))) * R);
}

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 17.0

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

  3. Applied cos-diff3.9

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

  4. Applied distribute-lft-in3.9

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

  6. Applied associate-*l*3.9

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

  8. Applied add-log-exp3.9

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

  10. Applied pow13.9

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

  11. Applied pow13.9

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

  12. Applied pow-prod-down3.9

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

  13. Simplified3.9

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

  14. Final simplification3.9

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))