Average Error: 13.1 → 0.2
Time: 42.2s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right) + \left(\left(\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_1}\right) \cdot \left(\sin \lambda_2 \cdot \sqrt[3]{\sin \lambda_1}\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\]
\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(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right) + \left(\left(\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_1}\right) \cdot \left(\sin \lambda_2 \cdot \sqrt[3]{\sin \lambda_1}\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r5086810 = lambda1;
        double r5086811 = lambda2;
        double r5086812 = r5086810 - r5086811;
        double r5086813 = sin(r5086812);
        double r5086814 = phi2;
        double r5086815 = cos(r5086814);
        double r5086816 = r5086813 * r5086815;
        double r5086817 = phi1;
        double r5086818 = cos(r5086817);
        double r5086819 = sin(r5086814);
        double r5086820 = r5086818 * r5086819;
        double r5086821 = sin(r5086817);
        double r5086822 = r5086821 * r5086815;
        double r5086823 = cos(r5086812);
        double r5086824 = r5086822 * r5086823;
        double r5086825 = r5086820 - r5086824;
        double r5086826 = atan2(r5086816, r5086825);
        return r5086826;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r5086827 = lambda2;
        double r5086828 = cos(r5086827);
        double r5086829 = lambda1;
        double r5086830 = sin(r5086829);
        double r5086831 = r5086828 * r5086830;
        double r5086832 = cos(r5086829);
        double r5086833 = sin(r5086827);
        double r5086834 = r5086832 * r5086833;
        double r5086835 = r5086831 - r5086834;
        double r5086836 = phi2;
        double r5086837 = cos(r5086836);
        double r5086838 = r5086835 * r5086837;
        double r5086839 = sin(r5086836);
        double r5086840 = phi1;
        double r5086841 = cos(r5086840);
        double r5086842 = r5086839 * r5086841;
        double r5086843 = r5086828 * r5086832;
        double r5086844 = sin(r5086840);
        double r5086845 = r5086837 * r5086844;
        double r5086846 = r5086843 * r5086845;
        double r5086847 = cbrt(r5086830);
        double r5086848 = r5086847 * r5086847;
        double r5086849 = r5086833 * r5086847;
        double r5086850 = r5086848 * r5086849;
        double r5086851 = r5086850 * r5086845;
        double r5086852 = r5086846 + r5086851;
        double r5086853 = r5086842 - r5086852;
        double r5086854 = atan2(r5086838, r5086853);
        return r5086854;
}

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 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 sin-diff6.6

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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-rgt-in0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied associate-*l*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 
(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))))))