Average Error: 13.2 → 0.2
Time: 47.9s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)}\]
\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(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4682676 = lambda1;
        double r4682677 = lambda2;
        double r4682678 = r4682676 - r4682677;
        double r4682679 = sin(r4682678);
        double r4682680 = phi2;
        double r4682681 = cos(r4682680);
        double r4682682 = r4682679 * r4682681;
        double r4682683 = phi1;
        double r4682684 = cos(r4682683);
        double r4682685 = sin(r4682680);
        double r4682686 = r4682684 * r4682685;
        double r4682687 = sin(r4682683);
        double r4682688 = r4682687 * r4682681;
        double r4682689 = cos(r4682678);
        double r4682690 = r4682688 * r4682689;
        double r4682691 = r4682686 - r4682690;
        double r4682692 = atan2(r4682682, r4682691);
        return r4682692;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4682693 = lambda2;
        double r4682694 = cos(r4682693);
        double r4682695 = lambda1;
        double r4682696 = sin(r4682695);
        double r4682697 = r4682694 * r4682696;
        double r4682698 = cos(r4682695);
        double r4682699 = sin(r4682693);
        double r4682700 = r4682698 * r4682699;
        double r4682701 = r4682697 - r4682700;
        double r4682702 = phi2;
        double r4682703 = cos(r4682702);
        double r4682704 = r4682701 * r4682703;
        double r4682705 = sin(r4682702);
        double r4682706 = phi1;
        double r4682707 = cos(r4682706);
        double r4682708 = r4682705 * r4682707;
        double r4682709 = r4682694 * r4682698;
        double r4682710 = sin(r4682706);
        double r4682711 = r4682710 * r4682703;
        double r4682712 = r4682709 * r4682711;
        double r4682713 = expm1(r4682712);
        double r4682714 = log1p(r4682713);
        double r4682715 = log1p(r4682714);
        double r4682716 = expm1(r4682715);
        double r4682717 = r4682699 * r4682696;
        double r4682718 = r4682711 * r4682717;
        double r4682719 = r4682716 + r4682718;
        double r4682720 = r4682708 - r4682719;
        double r4682721 = atan2(r4682704, r4682720);
        return r4682721;
}

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 13.2

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

  8. Applied expm1-log1p-u0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)} + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]
  9. Using strategy rm

  10. Applied log1p-expm1-u0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]

  11. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)}\]

Reproduce

herbie shell --seed 2019151 +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))))))