Midpoint on a great circle

Average Error: 0.9 → 0.9

Time: 26.7s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r480629 = lambda1;
        double r480630 = phi2;
        double r480631 = cos(r480630);
        double r480632 = lambda2;
        double r480633 = r480629 - r480632;
        double r480634 = sin(r480633);
        double r480635 = r480631 * r480634;
        double r480636 = phi1;
        double r480637 = cos(r480636);
        double r480638 = cos(r480633);
        double r480639 = r480631 * r480638;
        double r480640 = r480637 + r480639;
        double r480641 = atan2(r480635, r480640);
        double r480642 = r480629 + r480641;
        return r480642;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r480643 = lambda1;
        double r480644 = phi2;
        double r480645 = cos(r480644);
        double r480646 = lambda2;
        double r480647 = r480643 - r480646;
        double r480648 = sin(r480647);
        double r480649 = r480645 * r480648;
        double r480650 = cos(r480647);
        double r480651 = r480650 * r480645;
        double r480652 = sin(r480643);
        double r480653 = sin(r480646);
        double r480654 = r480652 * r480653;
        double r480655 = cos(r480646);
        double r480656 = cos(r480643);
        double r480657 = r480655 * r480656;
        double r480658 = r480654 + r480657;
        double r480659 = r480658 * r480645;
        double r480660 = r480650 * r480650;
        double r480661 = r480650 * r480660;
        double r480662 = cbrt(r480661);
        double r480663 = r480662 * r480645;
        double r480664 = r480659 * r480663;
        double r480665 = r480651 * r480664;
        double r480666 = phi1;
        double r480667 = cos(r480666);
        double r480668 = r480667 * r480667;
        double r480669 = r480668 * r480667;
        double r480670 = r480665 + r480669;
        double r480671 = r480651 * r480651;
        double r480672 = r480667 - r480659;
        double r480673 = r480667 * r480672;
        double r480674 = r480671 + r480673;
        double r480675 = r480670 / r480674;
        double r480676 = atan2(r480649, r480675);
        double r480677 = r480643 + r480676;
        return r480677;
}

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 flip3-+ 0.9

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

4. Simplified 0.9

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

5. Simplified 0.9

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

7. Applied cos-diff 0.9

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

9. Applied cos-diff 0.9

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

11. Applied add-cbrt-cube 0.9

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

12. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019156 
(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)))))))