Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 31.0s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2171856 = lambda1;
        double r2171857 = phi2;
        double r2171858 = cos(r2171857);
        double r2171859 = lambda2;
        double r2171860 = r2171856 - r2171859;
        double r2171861 = sin(r2171860);
        double r2171862 = r2171858 * r2171861;
        double r2171863 = phi1;
        double r2171864 = cos(r2171863);
        double r2171865 = cos(r2171860);
        double r2171866 = r2171858 * r2171865;
        double r2171867 = r2171864 + r2171866;
        double r2171868 = atan2(r2171862, r2171867);
        double r2171869 = r2171856 + r2171868;
        return r2171869;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2171870 = lambda1;
        double r2171871 = phi2;
        double r2171872 = cos(r2171871);
        double r2171873 = sin(r2171870);
        double r2171874 = lambda2;
        double r2171875 = cos(r2171874);
        double r2171876 = r2171873 * r2171875;
        double r2171877 = cos(r2171870);
        double r2171878 = sin(r2171874);
        double r2171879 = r2171877 * r2171878;
        double r2171880 = r2171876 - r2171879;
        double r2171881 = r2171872 * r2171880;
        double r2171882 = phi1;
        double r2171883 = cos(r2171882);
        double r2171884 = r2171883 * r2171883;
        double r2171885 = r2171884 * r2171883;
        double r2171886 = r2171877 * r2171872;
        double r2171887 = r2171886 * r2171875;
        double r2171888 = r2171887 * r2171887;
        double r2171889 = r2171888 * r2171887;
        double r2171890 = r2171885 + r2171889;
        double r2171891 = r2171888 - r2171884;
        double r2171892 = r2171891 * r2171887;
        double r2171893 = r2171883 + r2171887;
        double r2171894 = r2171892 / r2171893;
        double r2171895 = r2171884 + r2171894;
        double r2171896 = r2171890 / r2171895;
        double r2171897 = r2171878 * r2171873;
        double r2171898 = r2171897 * r2171872;
        double r2171899 = r2171896 + r2171898;
        double r2171900 = atan2(r2171881, r2171899);
        double r2171901 = r2171870 + r2171900;
        return r2171901;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 0.9

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

3. Applied sin-diff 0.8

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

5. Applied cos-diff 0.2

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

6. Applied distribute-rgt-in 0.2

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

7. Applied associate-+r+ 0.2

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

9. Applied flip3-+ 0.3

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

10. Simplified 0.3

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

11. Simplified 0.3

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

13. Applied flip-- 0.3

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

14. Applied associate-*r/ 0.3

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

15. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))