Bearing on a great circle

Average Error: 13.3 → 0.3

Time: 34.1s

Precision: 64

Math

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

↓

\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left({\left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_2\right)}^{2} \cdot \left({\left(\cos \lambda_1\right)}^{2} \cdot {\left(\cos \lambda_2\right)}^{2}\right)\right)\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)} + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r120851 = lambda1;
        double r120852 = lambda2;
        double r120853 = r120851 - r120852;
        double r120854 = sin(r120853);
        double r120855 = phi2;
        double r120856 = cos(r120855);
        double r120857 = r120854 * r120856;
        double r120858 = phi1;
        double r120859 = cos(r120858);
        double r120860 = sin(r120855);
        double r120861 = r120859 * r120860;
        double r120862 = sin(r120858);
        double r120863 = r120862 * r120856;
        double r120864 = cos(r120853);
        double r120865 = r120863 * r120864;
        double r120866 = r120861 - r120865;
        double r120867 = atan2(r120857, r120866);
        return r120867;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r120868 = lambda1;
        double r120869 = sin(r120868);
        double r120870 = lambda2;
        double r120871 = cos(r120870);
        double r120872 = r120869 * r120871;
        double r120873 = cos(r120868);
        double r120874 = -r120870;
        double r120875 = sin(r120874);
        double r120876 = r120873 * r120875;
        double r120877 = r120872 + r120876;
        double r120878 = phi2;
        double r120879 = cos(r120878);
        double r120880 = r120877 * r120879;
        double r120881 = phi1;
        double r120882 = cos(r120881);
        double r120883 = sin(r120878);
        double r120884 = r120882 * r120883;
        double r120885 = sin(r120881);
        double r120886 = 2.0;
        double r120887 = pow(r120885, r120886);
        double r120888 = pow(r120879, r120886);
        double r120889 = pow(r120873, r120886);
        double r120890 = pow(r120871, r120886);
        double r120891 = r120889 * r120890;
        double r120892 = r120888 * r120891;
        double r120893 = r120887 * r120892;
        double r120894 = 0.3333333333333333;
        double r120895 = pow(r120893, r120894);
        double r120896 = r120885 * r120879;
        double r120897 = r120873 * r120871;
        double r120898 = r120896 * r120897;
        double r120899 = cbrt(r120898);
        double r120900 = r120895 * r120899;
        double r120901 = sin(r120870);
        double r120902 = r120869 * r120901;
        double r120903 = r120896 * r120902;
        double r120904 = r120900 + r120903;
        double r120905 = r120884 - r120904;
        double r120906 = atan2(r120880, r120905);
        return r120906;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.3

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

3. Applied sub-neg 13.3

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

4. Applied sin-sum 6.8

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

5. Simplified 6.8

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

7. Applied cos-diff 0.2

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

8. Applied distribute-lft-in 0.2

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

10. Applied add-cube-cbrt 0.3

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

11. Taylor expanded around inf 0.3

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{{\left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_2\right)}^{2} \cdot \left({\left(\cos \lambda_1\right)}^{2} \cdot {\left(\cos \lambda_2\right)}^{2}\right)\right)\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)} + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]

12. Final simplification 0.3

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left({\left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_2\right)}^{2} \cdot \left({\left(\cos \lambda_1\right)}^{2} \cdot {\left(\cos \lambda_2\right)}^{2}\right)\right)\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)} + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]

Reproduce

herbie shell --seed 2019297 
(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))))))