Average Error: 13.2 → 0.2
Time: 21.8s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2 + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_1}\right) \cdot \left(\sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2\right)\right)\right)}\]
\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(\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(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2 + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_1}\right) \cdot \left(\sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2\right)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r95921 = lambda1;
        double r95922 = lambda2;
        double r95923 = r95921 - r95922;
        double r95924 = sin(r95923);
        double r95925 = phi2;
        double r95926 = cos(r95925);
        double r95927 = r95924 * r95926;
        double r95928 = phi1;
        double r95929 = cos(r95928);
        double r95930 = sin(r95925);
        double r95931 = r95929 * r95930;
        double r95932 = sin(r95928);
        double r95933 = r95932 * r95926;
        double r95934 = cos(r95923);
        double r95935 = r95933 * r95934;
        double r95936 = r95931 - r95935;
        double r95937 = atan2(r95927, r95936);
        return r95937;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r95938 = lambda1;
        double r95939 = sin(r95938);
        double r95940 = lambda2;
        double r95941 = cos(r95940);
        double r95942 = r95939 * r95941;
        double r95943 = cos(r95938);
        double r95944 = sin(r95940);
        double r95945 = r95943 * r95944;
        double r95946 = r95942 - r95945;
        double r95947 = phi2;
        double r95948 = cos(r95947);
        double r95949 = r95946 * r95948;
        double r95950 = phi1;
        double r95951 = cos(r95950);
        double r95952 = sin(r95947);
        double r95953 = r95951 * r95952;
        double r95954 = sin(r95950);
        double r95955 = r95954 * r95948;
        double r95956 = r95955 * r95943;
        double r95957 = r95956 * r95941;
        double r95958 = cbrt(r95939);
        double r95959 = r95958 * r95958;
        double r95960 = r95958 * r95944;
        double r95961 = r95959 * r95960;
        double r95962 = r95955 * r95961;
        double r95963 = r95957 + r95962;
        double r95964 = r95953 - r95963;
        double r95965 = atan2(r95949, r95964);
        return r95965;
}

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 13.2

    \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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. 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 associate-*r*0.2

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

  10. Applied add-cube-cbrt0.2

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

  11. Applied associate-*l*0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2020042 
(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))))))