Average Error: 12.6 → 0.2
Time: 41.2s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) - \sin \lambda_1 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)}}\]
\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(\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{\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) - \sin \lambda_1 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_2\right) \cdot \sin \lambda_1\right)\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r120882 = lambda1;
        double r120883 = lambda2;
        double r120884 = r120882 - r120883;
        double r120885 = sin(r120884);
        double r120886 = phi2;
        double r120887 = cos(r120886);
        double r120888 = r120885 * r120887;
        double r120889 = phi1;
        double r120890 = cos(r120889);
        double r120891 = sin(r120886);
        double r120892 = r120890 * r120891;
        double r120893 = sin(r120889);
        double r120894 = r120893 * r120887;
        double r120895 = cos(r120884);
        double r120896 = r120894 * r120895;
        double r120897 = r120892 - r120896;
        double r120898 = atan2(r120888, r120897);
        return r120898;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r120899 = lambda1;
        double r120900 = sin(r120899);
        double r120901 = lambda2;
        double r120902 = cos(r120901);
        double r120903 = r120900 * r120902;
        double r120904 = cos(r120899);
        double r120905 = sin(r120901);
        double r120906 = r120904 * r120905;
        double r120907 = r120903 - r120906;
        double r120908 = phi2;
        double r120909 = cos(r120908);
        double r120910 = r120907 * r120909;
        double r120911 = phi1;
        double r120912 = cos(r120911);
        double r120913 = sin(r120908);
        double r120914 = r120912 * r120913;
        double r120915 = r120902 * r120904;
        double r120916 = r120915 * r120915;
        double r120917 = r120905 * r120905;
        double r120918 = r120917 * r120900;
        double r120919 = r120900 * r120918;
        double r120920 = r120916 - r120919;
        double r120921 = sin(r120911);
        double r120922 = r120921 * r120909;
        double r120923 = r120920 * r120922;
        double r120924 = -r120901;
        double r120925 = sin(r120924);
        double r120926 = r120900 * r120925;
        double r120927 = r120915 + r120926;
        double r120928 = r120923 / r120927;
        double r120929 = r120914 - r120928;
        double r120930 = atan2(r120910, r120929);
        return r120930;
}

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 12.6

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

  3. Applied sin-diff6.6

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

  5. Applied sub-neg6.6

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

  6. Applied cos-sum0.2

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

  7. Simplified0.2

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

  9. Applied flip--0.2

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

  10. Applied associate-*r/0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))