Bearing on a great circle

Average Error: 13.0 → 0.2

Time: 35.7s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r124687 = lambda1;
        double r124688 = lambda2;
        double r124689 = r124687 - r124688;
        double r124690 = sin(r124689);
        double r124691 = phi2;
        double r124692 = cos(r124691);
        double r124693 = r124690 * r124692;
        double r124694 = phi1;
        double r124695 = cos(r124694);
        double r124696 = sin(r124691);
        double r124697 = r124695 * r124696;
        double r124698 = sin(r124694);
        double r124699 = r124698 * r124692;
        double r124700 = cos(r124689);
        double r124701 = r124699 * r124700;
        double r124702 = r124697 - r124701;
        double r124703 = atan2(r124693, r124702);
        return r124703;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r124704 = phi2;
        double r124705 = cos(r124704);
        double r124706 = lambda2;
        double r124707 = sin(r124706);
        double r124708 = -r124707;
        double r124709 = lambda1;
        double r124710 = cos(r124709);
        double r124711 = r124708 * r124710;
        double r124712 = sin(r124709);
        double r124713 = cos(r124706);
        double r124714 = r124712 * r124713;
        double r124715 = r124711 + r124714;
        double r124716 = r124705 * r124715;
        double r124717 = phi1;
        double r124718 = sin(r124717);
        double r124719 = r124718 * r124712;
        double r124720 = r124707 * r124719;
        double r124721 = cbrt(r124720);
        double r124722 = r124721 * r124721;
        double r124723 = r124721 * r124722;
        double r124724 = r124718 * r124710;
        double r124725 = r124724 * r124713;
        double r124726 = r124723 + r124725;
        double r124727 = -r124705;
        double r124728 = sin(r124704);
        double r124729 = cos(r124717);
        double r124730 = r124728 * r124729;
        double r124731 = fma(r124726, r124727, r124730);
        double r124732 = atan2(r124716, r124731);
        return r124732;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.0

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

2. Simplified 13.0

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

4. Applied sub-neg 13.0

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

5. Applied sin-sum 6.4

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

6. Simplified 6.4

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

7. Simplified 6.4

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

9. Applied cos-diff 0.2

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

10. Applied distribute-lft-in 0.2

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

11. Simplified 0.2

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

12. Simplified 0.2

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

14. Applied add-cube-cbrt 0.2

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

15. Simplified 0.2

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

16. Simplified 0.2

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

17. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))