Average Error: 13.4 → 0.2
Time: 47.9s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right) \cdot \sin \lambda_1\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right)}\]
\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(\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 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \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}\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4003954 = lambda1;
        double r4003955 = lambda2;
        double r4003956 = r4003954 - r4003955;
        double r4003957 = sin(r4003956);
        double r4003958 = phi2;
        double r4003959 = cos(r4003958);
        double r4003960 = r4003957 * r4003959;
        double r4003961 = phi1;
        double r4003962 = cos(r4003961);
        double r4003963 = sin(r4003958);
        double r4003964 = r4003962 * r4003963;
        double r4003965 = sin(r4003961);
        double r4003966 = r4003965 * r4003959;
        double r4003967 = cos(r4003956);
        double r4003968 = r4003966 * r4003967;
        double r4003969 = r4003964 - r4003968;
        double r4003970 = atan2(r4003960, r4003969);
        return r4003970;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4003971 = lambda2;
        double r4003972 = cos(r4003971);
        double r4003973 = lambda1;
        double r4003974 = sin(r4003973);
        double r4003975 = r4003972 * r4003974;
        double r4003976 = cos(r4003973);
        double r4003977 = sin(r4003971);
        double r4003978 = r4003976 * r4003977;
        double r4003979 = r4003975 - r4003978;
        double r4003980 = phi2;
        double r4003981 = cos(r4003980);
        double r4003982 = r4003979 * r4003981;
        double r4003983 = sin(r4003980);
        double r4003984 = phi1;
        double r4003985 = cos(r4003984);
        double r4003986 = r4003983 * r4003985;
        double r4003987 = r4003972 * r4003976;
        double r4003988 = sin(r4003984);
        double r4003989 = r4003988 * r4003981;
        double r4003990 = r4003987 * r4003989;
        double r4003991 = cbrt(r4003977);
        double r4003992 = r4003991 * r4003991;
        double r4003993 = r4003992 * r4003974;
        double r4003994 = r4003993 * r4003991;
        double r4003995 = r4003989 * r4003994;
        double r4003996 = r4003990 + r4003995;
        double r4003997 = r4003986 - r4003996;
        double r4003998 = atan2(r4003982, r4003997);
        return r4003998;
}

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 13.4

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

  3. Applied sin-diff6.8

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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. Applied distribute-lft-in0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied associate-*r*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(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))))))