Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 45.7s

Precision: 64

Math

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

↓

\[\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(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\left(\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)\right) \cdot \cos \phi_1\right) + \left(\left(\left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot 2}{\left(\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2 - \cos \phi_1\right)\right) \cdot \left(\left(\left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot 2\right)}} + \lambda_1\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1139794 = lambda1;
        double r1139795 = phi2;
        double r1139796 = cos(r1139795);
        double r1139797 = lambda2;
        double r1139798 = r1139794 - r1139797;
        double r1139799 = sin(r1139798);
        double r1139800 = r1139796 * r1139799;
        double r1139801 = phi1;
        double r1139802 = cos(r1139801);
        double r1139803 = cos(r1139798);
        double r1139804 = r1139796 * r1139803;
        double r1139805 = r1139802 + r1139804;
        double r1139806 = atan2(r1139800, r1139805);
        double r1139807 = r1139794 + r1139806;
        return r1139807;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1139808 = phi2;
        double r1139809 = cos(r1139808);
        double r1139810 = lambda1;
        double r1139811 = sin(r1139810);
        double r1139812 = lambda2;
        double r1139813 = cos(r1139812);
        double r1139814 = r1139811 * r1139813;
        double r1139815 = cos(r1139810);
        double r1139816 = sin(r1139812);
        double r1139817 = r1139815 * r1139816;
        double r1139818 = r1139814 - r1139817;
        double r1139819 = r1139809 * r1139818;
        double r1139820 = r1139816 * r1139811;
        double r1139821 = r1139815 * r1139813;
        double r1139822 = r1139820 - r1139821;
        double r1139823 = r1139822 * r1139822;
        double r1139824 = r1139823 * r1139822;
        double r1139825 = phi1;
        double r1139826 = r1139825 + r1139825;
        double r1139827 = cos(r1139826);
        double r1139828 = r1139825 - r1139825;
        double r1139829 = cos(r1139828);
        double r1139830 = r1139827 + r1139829;
        double r1139831 = cos(r1139825);
        double r1139832 = r1139830 * r1139831;
        double r1139833 = r1139824 * r1139832;
        double r1139834 = r1139820 * r1139820;
        double r1139835 = r1139821 * r1139821;
        double r1139836 = r1139834 - r1139835;
        double r1139837 = r1139809 * r1139836;
        double r1139838 = r1139837 * r1139837;
        double r1139839 = r1139838 * r1139837;
        double r1139840 = 2.0;
        double r1139841 = r1139839 * r1139840;
        double r1139842 = r1139833 + r1139841;
        double r1139843 = r1139831 * r1139831;
        double r1139844 = r1139821 + r1139820;
        double r1139845 = r1139844 * r1139809;
        double r1139846 = r1139845 - r1139831;
        double r1139847 = r1139845 * r1139846;
        double r1139848 = r1139843 + r1139847;
        double r1139849 = r1139824 * r1139840;
        double r1139850 = r1139848 * r1139849;
        double r1139851 = r1139842 / r1139850;
        double r1139852 = atan2(r1139819, r1139851);
        double r1139853 = r1139852 + r1139810;
        return r1139853;
}

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. Using strategy rm

7. Applied flip3-+ 0.3

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

8. Simplified 0.3

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

9. Simplified 0.3

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

11. Applied cos-mult 0.3

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

12. Applied associate-*r/ 0.3

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

13. Applied flip-+ 0.3

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

14. Applied associate-*r/ 0.3

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

15. Applied flip-+ 0.3

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

16. Applied associate-*r/ 0.3

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

17. Applied frac-times 0.3

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

18. Applied flip-+ 0.3

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

19. Applied associate-*r/ 0.3

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

20. Applied frac-times 0.3

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

21. Applied frac-add 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}{\frac{\left(\left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right) \cdot 2 + \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \left(\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)\right)\right)}{\left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot 2}}}{\left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1}}\]

22. Applied associate-/l/ 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(\left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right) \cdot 2 + \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \left(\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)\right)\right)}{\left(\left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right) - \cos \phi_1\right) + \cos \phi_1 \cdot \cos \phi_1\right) \cdot \left(\left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot 2\right)}}}\]

23. Final simplification 0.3

    \[\leadsto \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(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\left(\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)\right) \cdot \cos \phi_1\right) + \left(\left(\left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot 2}{\left(\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2 - \cos \phi_1\right)\right) \cdot \left(\left(\left(\left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot 2\right)}} + \lambda_1\]

Reproduce

herbie shell --seed 2019151 
(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)))))))