Average Error: 16.5 → 3.6
Time: 42.5s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot R\]
\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(\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 + \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21876 = phi1;
        double r21877 = sin(r21876);
        double r21878 = phi2;
        double r21879 = sin(r21878);
        double r21880 = r21877 * r21879;
        double r21881 = cos(r21876);
        double r21882 = cos(r21878);
        double r21883 = r21881 * r21882;
        double r21884 = lambda1;
        double r21885 = lambda2;
        double r21886 = r21884 - r21885;
        double r21887 = cos(r21886);
        double r21888 = r21883 * r21887;
        double r21889 = r21880 + r21888;
        double r21890 = acos(r21889);
        double r21891 = R;
        double r21892 = r21890 * r21891;
        return r21892;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21893 = phi1;
        double r21894 = sin(r21893);
        double r21895 = phi2;
        double r21896 = sin(r21895);
        double r21897 = r21894 * r21896;
        double r21898 = cos(r21893);
        double r21899 = cos(r21895);
        double r21900 = r21898 * r21899;
        double r21901 = lambda1;
        double r21902 = cos(r21901);
        double r21903 = lambda2;
        double r21904 = cos(r21903);
        double r21905 = r21902 * r21904;
        double r21906 = sin(r21901);
        double r21907 = sin(r21903);
        double r21908 = cbrt(r21907);
        double r21909 = r21908 * r21908;
        double r21910 = r21906 * r21909;
        double r21911 = r21910 * r21908;
        double r21912 = r21905 + r21911;
        double r21913 = r21900 * r21912;
        double r21914 = r21897 + r21913;
        double r21915 = acos(r21914);
        double r21916 = R;
        double r21917 = r21915 * r21916;
        return r21917;
}

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 16.5

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

  3. Applied cos-diff3.6

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

  5. Applied add-cube-cbrt3.6

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right) \cdot \sqrt[3]{\sin \lambda_2}\right)}\right)\right) \cdot R\]

  6. Applied associate-*r*3.6

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}}\right)\right) \cdot R\]

  7. Final simplification3.6

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot R\]

Reproduce

herbie shell --seed 2019303 
(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))