Average Error: 16.4 → 3.7
Time: 43.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right)\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1086887 = phi1;
        double r1086888 = sin(r1086887);
        double r1086889 = phi2;
        double r1086890 = sin(r1086889);
        double r1086891 = r1086888 * r1086890;
        double r1086892 = cos(r1086887);
        double r1086893 = cos(r1086889);
        double r1086894 = r1086892 * r1086893;
        double r1086895 = lambda1;
        double r1086896 = lambda2;
        double r1086897 = r1086895 - r1086896;
        double r1086898 = cos(r1086897);
        double r1086899 = r1086894 * r1086898;
        double r1086900 = r1086891 + r1086899;
        double r1086901 = acos(r1086900);
        double r1086902 = R;
        double r1086903 = r1086901 * r1086902;
        return r1086903;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1086904 = R;
        double r1086905 = phi1;
        double r1086906 = sin(r1086905);
        double r1086907 = phi2;
        double r1086908 = sin(r1086907);
        double r1086909 = lambda2;
        double r1086910 = cos(r1086909);
        double r1086911 = lambda1;
        double r1086912 = cos(r1086911);
        double r1086913 = sin(r1086911);
        double r1086914 = sin(r1086909);
        double r1086915 = r1086913 * r1086914;
        double r1086916 = fma(r1086910, r1086912, r1086915);
        double r1086917 = cos(r1086907);
        double r1086918 = r1086916 * r1086917;
        double r1086919 = cos(r1086905);
        double r1086920 = r1086918 * r1086919;
        double r1086921 = fma(r1086906, r1086908, r1086920);
        double r1086922 = acos(r1086921);
        double r1086923 = r1086904 * r1086922;
        return r1086923;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 16.4

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]

  2. Simplified16.4

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

  4. Applied cos-diff3.7

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

  6. Applied add-log-exp3.7

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

  7. Taylor expanded around 0 3.7

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

  8. Simplified3.7

    \[\leadsto R \cdot \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)}\]

  9. Final simplification3.7

    \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))