Average Error: 17.2 → 3.9
Time: 1.1m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\sin \lambda_1\right), \left(\sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right)\]
\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 \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\sin \lambda_1\right), \left(\sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1377738 = phi1;
        double r1377739 = sin(r1377738);
        double r1377740 = phi2;
        double r1377741 = sin(r1377740);
        double r1377742 = r1377739 * r1377741;
        double r1377743 = cos(r1377738);
        double r1377744 = cos(r1377740);
        double r1377745 = r1377743 * r1377744;
        double r1377746 = lambda1;
        double r1377747 = lambda2;
        double r1377748 = r1377746 - r1377747;
        double r1377749 = cos(r1377748);
        double r1377750 = r1377745 * r1377749;
        double r1377751 = r1377742 + r1377750;
        double r1377752 = acos(r1377751);
        double r1377753 = R;
        double r1377754 = r1377752 * r1377753;
        return r1377754;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1377755 = R;
        double r1377756 = atan2(1.0, 0.0);
        double r1377757 = 2.0;
        double r1377758 = r1377756 / r1377757;
        double r1377759 = phi2;
        double r1377760 = cos(r1377759);
        double r1377761 = lambda1;
        double r1377762 = sin(r1377761);
        double r1377763 = lambda2;
        double r1377764 = sin(r1377763);
        double r1377765 = cos(r1377761);
        double r1377766 = cos(r1377763);
        double r1377767 = r1377765 * r1377766;
        double r1377768 = fma(r1377762, r1377764, r1377767);
        double r1377769 = r1377760 * r1377768;
        double r1377770 = phi1;
        double r1377771 = cos(r1377770);
        double r1377772 = sin(r1377770);
        double r1377773 = sin(r1377759);
        double r1377774 = r1377772 * r1377773;
        double r1377775 = fma(r1377769, r1377771, r1377774);
        double r1377776 = asin(r1377775);
        double r1377777 = r1377758 - r1377776;
        double r1377778 = exp(r1377777);
        double r1377779 = log(r1377778);
        double r1377780 = r1377755 * r1377779;
        return r1377780;
}

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 17.2

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

  2. Simplified17.2

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

  4. Applied cos-diff3.9

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

  5. Taylor expanded around inf 3.9

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

  6. Simplified3.9

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

  8. Applied add-log-exp3.9

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

  10. Applied acos-asin3.9

    \[\leadsto R \cdot \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\sin \lambda_1\right), \left(\sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}}\right)\]

  11. Final simplification3.9

    \[\leadsto R \cdot \log \left(e^{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\sin \lambda_1\right), \left(\sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019120 +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))