Average Error: 12.9 → 0.2
Time: 22.6s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \left(\left(\sin \phi_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) + \sqrt[3]{{\left(\sin \lambda_2 \cdot \sin \lambda_1\right)}^{3}} \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]
\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 - \left(\cos \lambda_1 \cdot \left(\left(\sin \phi_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) + \sqrt[3]{{\left(\sin \lambda_2 \cdot \sin \lambda_1\right)}^{3}} \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r136759 = lambda1;
        double r136760 = lambda2;
        double r136761 = r136759 - r136760;
        double r136762 = sin(r136761);
        double r136763 = phi2;
        double r136764 = cos(r136763);
        double r136765 = r136762 * r136764;
        double r136766 = phi1;
        double r136767 = cos(r136766);
        double r136768 = sin(r136763);
        double r136769 = r136767 * r136768;
        double r136770 = sin(r136766);
        double r136771 = r136770 * r136764;
        double r136772 = cos(r136761);
        double r136773 = r136771 * r136772;
        double r136774 = r136769 - r136773;
        double r136775 = atan2(r136765, r136774);
        return r136775;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r136776 = lambda1;
        double r136777 = sin(r136776);
        double r136778 = lambda2;
        double r136779 = cos(r136778);
        double r136780 = r136777 * r136779;
        double r136781 = cos(r136776);
        double r136782 = sin(r136778);
        double r136783 = r136781 * r136782;
        double r136784 = r136780 - r136783;
        double r136785 = phi2;
        double r136786 = cos(r136785);
        double r136787 = r136784 * r136786;
        double r136788 = phi1;
        double r136789 = cos(r136788);
        double r136790 = sin(r136785);
        double r136791 = r136789 * r136790;
        double r136792 = sin(r136788);
        double r136793 = r136792 * r136779;
        double r136794 = r136793 * r136786;
        double r136795 = r136781 * r136794;
        double r136796 = r136782 * r136777;
        double r136797 = 3.0;
        double r136798 = pow(r136796, r136797);
        double r136799 = cbrt(r136798);
        double r136800 = r136792 * r136786;
        double r136801 = r136799 * r136800;
        double r136802 = r136795 + r136801;
        double r136803 = r136791 - r136802;
        double r136804 = atan2(r136787, r136803);
        return r136804;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.9

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

  3. Applied sin-diff6.7

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

  5. Applied cos-diff0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Simplified0.2

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

  8. Simplified0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.2

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

  13. Applied add-cbrt-cube0.2

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

  14. Applied add-cbrt-cube0.2

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

  15. Applied cbrt-unprod0.2

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019351 
(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))))))