Average Error: 16.9 → 3.7
Time: 47.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)\right)\right)}\right) \cdot R\]
\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
↓
\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)\right)\right)}\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2328703 = phi1;
        double r2328704 = sin(r2328703);
        double r2328705 = phi2;
        double r2328706 = sin(r2328705);
        double r2328707 = r2328704 * r2328706;
        double r2328708 = cos(r2328703);
        double r2328709 = cos(r2328705);
        double r2328710 = r2328708 * r2328709;
        double r2328711 = lambda1;
        double r2328712 = lambda2;
        double r2328713 = r2328711 - r2328712;
        double r2328714 = cos(r2328713);
        double r2328715 = r2328710 * r2328714;
        double r2328716 = r2328707 + r2328715;
        double r2328717 = acos(r2328716);
        double r2328718 = R;
        double r2328719 = r2328717 * r2328718;
        return r2328719;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2328720 = phi1;
        double r2328721 = sin(r2328720);
        double r2328722 = phi2;
        double r2328723 = sin(r2328722);
        double r2328724 = cos(r2328720);
        double r2328725 = lambda1;
        double r2328726 = sin(r2328725);
        double r2328727 = lambda2;
        double r2328728 = sin(r2328727);
        double r2328729 = cos(r2328725);
        double r2328730 = cos(r2328727);
        double r2328731 = r2328729 * r2328730;
        double r2328732 = fma(r2328726, r2328728, r2328731);
        double r2328733 = cos(r2328722);
        double r2328734 = r2328732 * r2328733;
        double r2328735 = r2328724 * r2328734;
        double r2328736 = fma(r2328721, r2328723, r2328735);
        double r2328737 = acos(r2328736);
        double r2328738 = exp(r2328737);
        double r2328739 = log(r2328738);
        double r2328740 = R;
        double r2328741 = r2328739 * r2328740;
        return r2328741;
}

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 16.9

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

  2. Simplified16.9

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

  4. Applied cos-diff3.6

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

  6. Applied add-log-exp3.7

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

  7. Simplified3.7

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

  9. Applied *-commutative3.7

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

  10. Final simplification3.7

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

Reproduce

herbie shell --seed 2019174 +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))