Average Error: 0.9 → 0.3
Time: 31.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\log \left(e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}\right)}\]
\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{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\log \left(e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1\right)}\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2356849 = lambda1;
        double r2356850 = phi2;
        double r2356851 = cos(r2356850);
        double r2356852 = lambda2;
        double r2356853 = r2356849 - r2356852;
        double r2356854 = sin(r2356853);
        double r2356855 = r2356851 * r2356854;
        double r2356856 = phi1;
        double r2356857 = cos(r2356856);
        double r2356858 = cos(r2356853);
        double r2356859 = r2356851 * r2356858;
        double r2356860 = r2356857 + r2356859;
        double r2356861 = atan2(r2356855, r2356860);
        double r2356862 = r2356849 + r2356861;
        return r2356862;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2356863 = lambda1;
        double r2356864 = sin(r2356863);
        double r2356865 = lambda2;
        double r2356866 = cos(r2356865);
        double r2356867 = r2356864 * r2356866;
        double r2356868 = cos(r2356863);
        double r2356869 = sin(r2356865);
        double r2356870 = r2356868 * r2356869;
        double r2356871 = r2356867 - r2356870;
        double r2356872 = phi2;
        double r2356873 = cos(r2356872);
        double r2356874 = r2356871 * r2356873;
        double r2356875 = r2356868 * r2356866;
        double r2356876 = fma(r2356864, r2356869, r2356875);
        double r2356877 = phi1;
        double r2356878 = cos(r2356877);
        double r2356879 = fma(r2356873, r2356876, r2356878);
        double r2356880 = exp(r2356879);
        double r2356881 = log(r2356880);
        double r2356882 = atan2(r2356874, r2356881);
        double r2356883 = r2356863 + r2356882;
        return r2356883;
}

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. Simplified0.8

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

  4. Applied cos-diff0.8

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

  6. Applied sin-diff0.2

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

  8. Applied add-log-exp0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))