Average Error: 16.9 → 3.9
Time: 15.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)\right)\right)} \cdot R\]
\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
↓
e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\sin \phi_1 \cdot \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 \left(-\lambda_2\right)\right)\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25842 = phi1;
        double r25843 = sin(r25842);
        double r25844 = phi2;
        double r25845 = sin(r25844);
        double r25846 = r25843 * r25845;
        double r25847 = cos(r25842);
        double r25848 = cos(r25844);
        double r25849 = r25847 * r25848;
        double r25850 = lambda1;
        double r25851 = lambda2;
        double r25852 = r25850 - r25851;
        double r25853 = cos(r25852);
        double r25854 = r25849 * r25853;
        double r25855 = r25846 + r25854;
        double r25856 = acos(r25855);
        double r25857 = R;
        double r25858 = r25856 * r25857;
        return r25858;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r25859 = phi1;
        double r25860 = sin(r25859);
        double r25861 = phi2;
        double r25862 = sin(r25861);
        double r25863 = r25860 * r25862;
        double r25864 = cos(r25859);
        double r25865 = cos(r25861);
        double r25866 = r25864 * r25865;
        double r25867 = lambda1;
        double r25868 = cos(r25867);
        double r25869 = lambda2;
        double r25870 = cos(r25869);
        double r25871 = r25868 * r25870;
        double r25872 = sin(r25867);
        double r25873 = -r25869;
        double r25874 = sin(r25873);
        double r25875 = r25872 * r25874;
        double r25876 = r25871 - r25875;
        double r25877 = r25866 * r25876;
        double r25878 = r25863 + r25877;
        double r25879 = acos(r25878);
        double r25880 = log(r25879);
        double r25881 = expm1(r25880);
        double r25882 = log1p(r25881);
        double r25883 = exp(r25882);
        double r25884 = R;
        double r25885 = r25883 * r25884;
        return r25885;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied sub-neg16.9

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

  4. Applied cos-sum3.9

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

  5. Simplified3.9

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

  7. Applied add-exp-log3.9

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

  9. Applied log1p-expm1-u3.9

    \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)\right)\right)}} \cdot R\]

  10. Final simplification3.9

    \[\leadsto e^{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)\right)\right)} \cdot R\]

Reproduce

herbie shell --seed 2020083 +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))