Bearing on a great circle

Average Error: 13.1 → 0.2

Time: 19.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r101651 = lambda1;
        double r101652 = lambda2;
        double r101653 = r101651 - r101652;
        double r101654 = sin(r101653);
        double r101655 = phi2;
        double r101656 = cos(r101655);
        double r101657 = r101654 * r101656;
        double r101658 = phi1;
        double r101659 = cos(r101658);
        double r101660 = sin(r101655);
        double r101661 = r101659 * r101660;
        double r101662 = sin(r101658);
        double r101663 = r101662 * r101656;
        double r101664 = cos(r101653);
        double r101665 = r101663 * r101664;
        double r101666 = r101661 - r101665;
        double r101667 = atan2(r101657, r101666);
        return r101667;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r101668 = lambda1;
        double r101669 = sin(r101668);
        double r101670 = lambda2;
        double r101671 = cos(r101670);
        double r101672 = r101669 * r101671;
        double r101673 = cos(r101668);
        double r101674 = -r101670;
        double r101675 = sin(r101674);
        double r101676 = r101673 * r101675;
        double r101677 = r101672 + r101676;
        double r101678 = phi2;
        double r101679 = cos(r101678);
        double r101680 = r101677 * r101679;
        double r101681 = phi1;
        double r101682 = cos(r101681);
        double r101683 = sin(r101678);
        double r101684 = r101682 * r101683;
        double r101685 = sin(r101681);
        double r101686 = r101685 * r101679;
        double r101687 = r101671 * r101673;
        double r101688 = 3.0;
        double r101689 = pow(r101687, r101688);
        double r101690 = pow(r101689, r101688);
        double r101691 = cbrt(r101690);
        double r101692 = r101669 * r101675;
        double r101693 = pow(r101692, r101688);
        double r101694 = r101691 - r101693;
        double r101695 = r101686 * r101694;
        double r101696 = r101687 * r101687;
        double r101697 = r101692 * r101692;
        double r101698 = r101687 * r101692;
        double r101699 = r101697 + r101698;
        double r101700 = r101696 + r101699;
        double r101701 = r101695 / r101700;
        double r101702 = r101684 - r101701;
        double r101703 = atan2(r101680, r101702);
        return r101703;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.1

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

3. Applied sub-neg 13.1

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

4. Applied sin-sum 6.8

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

5. Simplified 6.8

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

7. Applied sub-neg 6.8

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

8. Applied cos-sum 0.2

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

9. Simplified 0.2

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

11. Applied flip3-- 0.2

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

12. Applied associate-*r/ 0.2

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

14. Applied add-cbrt-cube 0.2

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

15. Simplified 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020047 +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))))))