Average Error: 16.5 → 3.8
Time: 49.3s
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 r24826 = phi1;
        double r24827 = sin(r24826);
        double r24828 = phi2;
        double r24829 = sin(r24828);
        double r24830 = r24827 * r24829;
        double r24831 = cos(r24826);
        double r24832 = cos(r24828);
        double r24833 = r24831 * r24832;
        double r24834 = lambda1;
        double r24835 = lambda2;
        double r24836 = r24834 - r24835;
        double r24837 = cos(r24836);
        double r24838 = r24833 * r24837;
        double r24839 = r24830 + r24838;
        double r24840 = acos(r24839);
        double r24841 = R;
        double r24842 = r24840 * r24841;
        return r24842;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24843 = phi1;
        double r24844 = sin(r24843);
        double r24845 = phi2;
        double r24846 = sin(r24845);
        double r24847 = cos(r24843);
        double r24848 = cos(r24845);
        double r24849 = r24847 * r24848;
        double r24850 = lambda1;
        double r24851 = cos(r24850);
        double r24852 = lambda2;
        double r24853 = cos(r24852);
        double r24854 = r24851 * r24853;
        double r24855 = r24849 * r24854;
        double r24856 = sin(r24850);
        double r24857 = sin(r24852);
        double r24858 = r24856 * r24857;
        double r24859 = r24849 * r24858;
        double r24860 = r24855 + r24859;
        double r24861 = fma(r24844, r24846, r24860);
        double r24862 = acos(r24861);
        double r24863 = R;
        double r24864 = r24862 * r24863;
        return r24864;
}

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))