Average Error: 13.5 → 0.2
Time: 40.6s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \frac{\sin \phi_1 \cdot \cos \phi_2}{\frac{\frac{\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_2 \cdot \sin \lambda_1}{\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1}}{\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_2 \cdot \sin \lambda_1}}}\]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \frac{\sin \phi_1 \cdot \cos \phi_2}{\frac{\frac{\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_2 \cdot \sin \lambda_1}{\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1}}{\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_2 \cdot \sin \lambda_1}}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r5343850 = lambda1;
        double r5343851 = lambda2;
        double r5343852 = r5343850 - r5343851;
        double r5343853 = sin(r5343852);
        double r5343854 = phi2;
        double r5343855 = cos(r5343854);
        double r5343856 = r5343853 * r5343855;
        double r5343857 = phi1;
        double r5343858 = cos(r5343857);
        double r5343859 = sin(r5343854);
        double r5343860 = r5343858 * r5343859;
        double r5343861 = sin(r5343857);
        double r5343862 = r5343861 * r5343855;
        double r5343863 = cos(r5343852);
        double r5343864 = r5343862 * r5343863;
        double r5343865 = r5343860 - r5343864;
        double r5343866 = atan2(r5343856, r5343865);
        return r5343866;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r5343867 = lambda2;
        double r5343868 = cos(r5343867);
        double r5343869 = lambda1;
        double r5343870 = sin(r5343869);
        double r5343871 = r5343868 * r5343870;
        double r5343872 = cos(r5343869);
        double r5343873 = sin(r5343867);
        double r5343874 = r5343872 * r5343873;
        double r5343875 = r5343871 - r5343874;
        double r5343876 = phi2;
        double r5343877 = cos(r5343876);
        double r5343878 = r5343875 * r5343877;
        double r5343879 = sin(r5343876);
        double r5343880 = phi1;
        double r5343881 = cos(r5343880);
        double r5343882 = r5343879 * r5343881;
        double r5343883 = sin(r5343880);
        double r5343884 = r5343883 * r5343877;
        double r5343885 = r5343868 * r5343872;
        double r5343886 = r5343873 * r5343870;
        double r5343887 = r5343885 - r5343886;
        double r5343888 = r5343886 + r5343885;
        double r5343889 = r5343887 / r5343888;
        double r5343890 = r5343889 / r5343887;
        double r5343891 = r5343884 / r5343890;
        double r5343892 = r5343882 - r5343891;
        double r5343893 = atan2(r5343878, r5343892);
        return r5343893;
}

Error

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 13.5

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

  3. Applied sin-diff6.7

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

  5. Applied cos-diff0.2

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

  7. Applied flip-+0.2

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

  8. Applied associate-*r/0.2

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

  10. Applied associate-/l*0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019169 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))