Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 27.4s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r36621 = lambda1;
        double r36622 = phi2;
        double r36623 = cos(r36622);
        double r36624 = lambda2;
        double r36625 = r36621 - r36624;
        double r36626 = sin(r36625);
        double r36627 = r36623 * r36626;
        double r36628 = phi1;
        double r36629 = cos(r36628);
        double r36630 = cos(r36625);
        double r36631 = r36623 * r36630;
        double r36632 = r36629 + r36631;
        double r36633 = atan2(r36627, r36632);
        double r36634 = r36621 + r36633;
        return r36634;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r36635 = lambda1;
        double r36636 = phi2;
        double r36637 = cos(r36636);
        double r36638 = sin(r36635);
        double r36639 = lambda2;
        double r36640 = cos(r36639);
        double r36641 = r36638 * r36640;
        double r36642 = cos(r36635);
        double r36643 = -r36639;
        double r36644 = sin(r36643);
        double r36645 = r36642 * r36644;
        double r36646 = r36641 + r36645;
        double r36647 = r36637 * r36646;
        double r36648 = phi1;
        double r36649 = cos(r36648);
        double r36650 = 3.0;
        double r36651 = pow(r36649, r36650);
        double r36652 = r36642 * r36640;
        double r36653 = r36637 * r36652;
        double r36654 = pow(r36653, r36650);
        double r36655 = r36651 + r36654;
        double r36656 = r36649 * r36649;
        double r36657 = r36637 * r36640;
        double r36658 = r36642 * r36657;
        double r36659 = 2.0;
        double r36660 = pow(r36658, r36659);
        double r36661 = pow(r36649, r36659);
        double r36662 = r36660 - r36661;
        double r36663 = r36649 + r36658;
        double r36664 = r36662 / r36663;
        double r36665 = r36658 * r36664;
        double r36666 = r36656 + r36665;
        double r36667 = r36655 / r36666;
        double r36668 = sin(r36639);
        double r36669 = r36638 * r36668;
        double r36670 = r36669 * r36637;
        double r36671 = r36667 + r36670;
        double r36672 = atan2(r36647, r36671);
        double r36673 = r36635 + r36672;
        return r36673;
}

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 sub-neg 0.9

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

4. Applied sin-sum 0.8

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

5. Simplified 0.8

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

7. Applied cos-diff 0.2

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

8. Applied distribute-rgt-in 0.2

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

9. Applied associate-+r+ 0.2

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

10. Simplified 0.2

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

12. Applied flip3-+ 0.3

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

13. Simplified 0.3

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

15. Applied flip-- 0.3

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

16. Simplified 0.3

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019305 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))