Average Error: 0.9 → 0.9
Time: 10.2s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) - \cos \phi_1, \cos \phi_1 \cdot \cos \phi_1\right)}}\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) - \cos \phi_1, \cos \phi_1 \cdot \cos \phi_1\right)}}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r47745 = lambda1;
        double r47746 = phi2;
        double r47747 = cos(r47746);
        double r47748 = lambda2;
        double r47749 = r47745 - r47748;
        double r47750 = sin(r47749);
        double r47751 = r47747 * r47750;
        double r47752 = phi1;
        double r47753 = cos(r47752);
        double r47754 = cos(r47749);
        double r47755 = r47747 * r47754;
        double r47756 = r47753 + r47755;
        double r47757 = atan2(r47751, r47756);
        double r47758 = r47745 + r47757;
        return r47758;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r47759 = lambda1;
        double r47760 = phi2;
        double r47761 = cos(r47760);
        double r47762 = lambda2;
        double r47763 = r47759 - r47762;
        double r47764 = sin(r47763);
        double r47765 = r47761 * r47764;
        double r47766 = phi1;
        double r47767 = cos(r47766);
        double r47768 = 3.0;
        double r47769 = pow(r47767, r47768);
        double r47770 = cos(r47759);
        double r47771 = cos(r47762);
        double r47772 = r47770 * r47771;
        double r47773 = sin(r47759);
        double r47774 = sin(r47762);
        double r47775 = r47773 * r47774;
        double r47776 = r47772 + r47775;
        double r47777 = r47761 * r47776;
        double r47778 = pow(r47777, r47768);
        double r47779 = r47769 + r47778;
        double r47780 = cos(r47763);
        double r47781 = r47761 * r47780;
        double r47782 = r47781 - r47767;
        double r47783 = r47767 * r47767;
        double r47784 = fma(r47781, r47782, r47783);
        double r47785 = r47779 / r47784;
        double r47786 = atan2(r47765, r47785);
        double r47787 = r47759 + r47786;
        return r47787;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

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

  3. Applied flip3-+0.9

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

  4. Simplified0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) - \cos \phi_1, \cos \phi_1 \cdot \cos \phi_1\right)}}}\]
  5. Using strategy rm

  6. Applied cos-diff0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))