Average Error: 13.3 → 0.2
Time: 51.3s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right) \cdot \sin \lambda_1\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right)}\]
\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(\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(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right) \cdot \sin \lambda_1\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4374647 = lambda1;
        double r4374648 = lambda2;
        double r4374649 = r4374647 - r4374648;
        double r4374650 = sin(r4374649);
        double r4374651 = phi2;
        double r4374652 = cos(r4374651);
        double r4374653 = r4374650 * r4374652;
        double r4374654 = phi1;
        double r4374655 = cos(r4374654);
        double r4374656 = sin(r4374651);
        double r4374657 = r4374655 * r4374656;
        double r4374658 = sin(r4374654);
        double r4374659 = r4374658 * r4374652;
        double r4374660 = cos(r4374649);
        double r4374661 = r4374659 * r4374660;
        double r4374662 = r4374657 - r4374661;
        double r4374663 = atan2(r4374653, r4374662);
        return r4374663;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4374664 = lambda2;
        double r4374665 = cos(r4374664);
        double r4374666 = lambda1;
        double r4374667 = sin(r4374666);
        double r4374668 = r4374665 * r4374667;
        double r4374669 = cos(r4374666);
        double r4374670 = sin(r4374664);
        double r4374671 = r4374669 * r4374670;
        double r4374672 = r4374668 - r4374671;
        double r4374673 = phi2;
        double r4374674 = cos(r4374673);
        double r4374675 = r4374672 * r4374674;
        double r4374676 = sin(r4374673);
        double r4374677 = phi1;
        double r4374678 = cos(r4374677);
        double r4374679 = r4374676 * r4374678;
        double r4374680 = r4374665 * r4374669;
        double r4374681 = sin(r4374677);
        double r4374682 = r4374681 * r4374674;
        double r4374683 = r4374680 * r4374682;
        double r4374684 = cbrt(r4374670);
        double r4374685 = r4374684 * r4374684;
        double r4374686 = r4374685 * r4374667;
        double r4374687 = r4374686 * r4374684;
        double r4374688 = r4374682 * r4374687;
        double r4374689 = r4374683 + r4374688;
        double r4374690 = r4374679 - r4374689;
        double r4374691 = atan2(r4374675, r4374690);
        return r4374691;
}

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 13.3

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied associate-*r*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019149 +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))))))