Average Error: 16.5 → 3.8
Time: 49.6s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24886 = phi1;
        double r24887 = sin(r24886);
        double r24888 = phi2;
        double r24889 = sin(r24888);
        double r24890 = r24887 * r24889;
        double r24891 = cos(r24886);
        double r24892 = cos(r24888);
        double r24893 = r24891 * r24892;
        double r24894 = lambda1;
        double r24895 = lambda2;
        double r24896 = r24894 - r24895;
        double r24897 = cos(r24896);
        double r24898 = r24893 * r24897;
        double r24899 = r24890 + r24898;
        double r24900 = acos(r24899);
        double r24901 = R;
        double r24902 = r24900 * r24901;
        return r24902;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24903 = phi1;
        double r24904 = sin(r24903);
        double r24905 = phi2;
        double r24906 = sin(r24905);
        double r24907 = cos(r24903);
        double r24908 = cos(r24905);
        double r24909 = r24907 * r24908;
        double r24910 = lambda1;
        double r24911 = cos(r24910);
        double r24912 = lambda2;
        double r24913 = cos(r24912);
        double r24914 = r24911 * r24913;
        double r24915 = r24909 * r24914;
        double r24916 = sin(r24910);
        double r24917 = sin(r24912);
        double r24918 = r24916 * r24917;
        double r24919 = r24909 * r24918;
        double r24920 = r24915 + r24919;
        double r24921 = fma(r24904, r24906, r24920);
        double r24922 = acos(r24921);
        double r24923 = R;
        double r24924 = r24922 * r24923;
        return r24924;
}

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 16.5

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

  2. Simplified16.5

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

  4. Applied cos-diff3.8

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

  5. Applied distribute-lft-in3.8

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

  6. Final simplification3.8

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))