Average Error: 12.8 → 0.2
Time: 49.2s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot \sin \phi_1\right)\right)}\]
\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 - \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot \sin \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4588686 = lambda1;
        double r4588687 = lambda2;
        double r4588688 = r4588686 - r4588687;
        double r4588689 = sin(r4588688);
        double r4588690 = phi2;
        double r4588691 = cos(r4588690);
        double r4588692 = r4588689 * r4588691;
        double r4588693 = phi1;
        double r4588694 = cos(r4588693);
        double r4588695 = sin(r4588690);
        double r4588696 = r4588694 * r4588695;
        double r4588697 = sin(r4588693);
        double r4588698 = r4588697 * r4588691;
        double r4588699 = cos(r4588688);
        double r4588700 = r4588698 * r4588699;
        double r4588701 = r4588696 - r4588700;
        double r4588702 = atan2(r4588692, r4588701);
        return r4588702;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4588703 = lambda2;
        double r4588704 = cos(r4588703);
        double r4588705 = lambda1;
        double r4588706 = sin(r4588705);
        double r4588707 = r4588704 * r4588706;
        double r4588708 = cos(r4588705);
        double r4588709 = sin(r4588703);
        double r4588710 = r4588708 * r4588709;
        double r4588711 = r4588707 - r4588710;
        double r4588712 = phi2;
        double r4588713 = cos(r4588712);
        double r4588714 = r4588711 * r4588713;
        double r4588715 = sin(r4588712);
        double r4588716 = phi1;
        double r4588717 = cos(r4588716);
        double r4588718 = r4588715 * r4588717;
        double r4588719 = r4588704 * r4588708;
        double r4588720 = fma(r4588706, r4588709, r4588719);
        double r4588721 = r4588713 * r4588720;
        double r4588722 = sin(r4588716);
        double r4588723 = r4588721 * r4588722;
        double r4588724 = expm1(r4588723);
        double r4588725 = log1p(r4588724);
        double r4588726 = r4588718 - r4588725;
        double r4588727 = atan2(r4588714, r4588726);
        return r4588727;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 12.8

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

  7. Applied log1p-expm1-u0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +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))))))