Bearing on a great circle

Average Error: 13.5 → 0.2

Time: 33.8s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r76662 = lambda1;
        double r76663 = lambda2;
        double r76664 = r76662 - r76663;
        double r76665 = sin(r76664);
        double r76666 = phi2;
        double r76667 = cos(r76666);
        double r76668 = r76665 * r76667;
        double r76669 = phi1;
        double r76670 = cos(r76669);
        double r76671 = sin(r76666);
        double r76672 = r76670 * r76671;
        double r76673 = sin(r76669);
        double r76674 = r76673 * r76667;
        double r76675 = cos(r76664);
        double r76676 = r76674 * r76675;
        double r76677 = r76672 - r76676;
        double r76678 = atan2(r76668, r76677);
        return r76678;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r76679 = lambda1;
        double r76680 = sin(r76679);
        double r76681 = lambda2;
        double r76682 = cos(r76681);
        double r76683 = r76680 * r76682;
        double r76684 = cos(r76679);
        double r76685 = -r76681;
        double r76686 = sin(r76685);
        double r76687 = r76684 * r76686;
        double r76688 = r76683 + r76687;
        double r76689 = phi2;
        double r76690 = cos(r76689);
        double r76691 = r76688 * r76690;
        double r76692 = phi1;
        double r76693 = cos(r76692);
        double r76694 = sin(r76689);
        double r76695 = r76693 * r76694;
        double r76696 = sin(r76692);
        double r76697 = r76696 * r76690;
        double r76698 = r76682 * r76684;
        double r76699 = 2.0;
        double r76700 = sin(r76681);
        double r76701 = r76700 * r76680;
        double r76702 = exp(r76701);
        double r76703 = cbrt(r76702);
        double r76704 = log(r76703);
        double r76705 = r76699 * r76704;
        double r76706 = r76705 + r76704;
        double r76707 = r76698 + r76706;
        double r76708 = r76697 * r76707;
        double r76709 = r76695 - r76708;
        double r76710 = atan2(r76691, r76709);
        return r76710;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.5

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

3. Applied sub-neg 13.5

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

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 cos-diff 0.2

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

8. Simplified 0.2

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

9. Simplified 0.2

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

11. Applied add-log-exp 0.2

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

13. Applied add-cube-cbrt 0.2

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

14. Applied log-prod 0.2

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

15. Simplified 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019235 
(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))))))