Bearing on a great circle

Average Error: 13.6 → 0.2

Time: 15.4s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r109590 = lambda1;
        double r109591 = lambda2;
        double r109592 = r109590 - r109591;
        double r109593 = sin(r109592);
        double r109594 = phi2;
        double r109595 = cos(r109594);
        double r109596 = r109593 * r109595;
        double r109597 = phi1;
        double r109598 = cos(r109597);
        double r109599 = sin(r109594);
        double r109600 = r109598 * r109599;
        double r109601 = sin(r109597);
        double r109602 = r109601 * r109595;
        double r109603 = cos(r109592);
        double r109604 = r109602 * r109603;
        double r109605 = r109600 - r109604;
        double r109606 = atan2(r109596, r109605);
        return r109606;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r109607 = lambda1;
        double r109608 = sin(r109607);
        double r109609 = lambda2;
        double r109610 = cos(r109609);
        double r109611 = r109608 * r109610;
        double r109612 = cos(r109607);
        double r109613 = sin(r109609);
        double r109614 = r109612 * r109613;
        double r109615 = r109611 - r109614;
        double r109616 = phi2;
        double r109617 = cos(r109616);
        double r109618 = r109615 * r109617;
        double r109619 = phi1;
        double r109620 = cos(r109619);
        double r109621 = sin(r109616);
        double r109622 = r109620 * r109621;
        double r109623 = sin(r109619);
        double r109624 = r109623 * r109617;
        double r109625 = r109612 * r109610;
        double r109626 = 3.0;
        double r109627 = pow(r109625, r109626);
        double r109628 = -r109609;
        double r109629 = sin(r109628);
        double r109630 = r109608 * r109629;
        double r109631 = pow(r109630, r109626);
        double r109632 = r109627 - r109631;
        double r109633 = r109624 * r109632;
        double r109634 = -r109633;
        double r109635 = -r109630;
        double r109636 = fma(r109608, r109629, r109625);
        double r109637 = r109635 * r109636;
        double r109638 = 2.0;
        double r109639 = pow(r109612, r109638);
        double r109640 = pow(r109610, r109638);
        double r109641 = r109639 * r109640;
        double r109642 = -r109641;
        double r109643 = r109637 + r109642;
        double r109644 = r109634 / r109643;
        double r109645 = r109622 - r109644;
        double r109646 = atan2(r109618, r109645);
        return r109646;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.6

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

3. Applied sin-diff 6.9

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

5. Applied sub-neg 6.9

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

6. Applied cos-sum 0.2

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

7. Simplified 0.2

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

9. Applied flip3-- 0.2

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

10. Applied associate-*r/ 0.2

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

12. Applied frac-2neg 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))))