Average Error: 16.5 → 3.8
Time: 33.3s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\]
\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 \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\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)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21885 = phi1;
        double r21886 = sin(r21885);
        double r21887 = phi2;
        double r21888 = sin(r21887);
        double r21889 = r21886 * r21888;
        double r21890 = cos(r21885);
        double r21891 = cos(r21887);
        double r21892 = r21890 * r21891;
        double r21893 = lambda1;
        double r21894 = lambda2;
        double r21895 = r21893 - r21894;
        double r21896 = cos(r21895);
        double r21897 = r21892 * r21896;
        double r21898 = r21889 + r21897;
        double r21899 = acos(r21898);
        double r21900 = R;
        double r21901 = r21899 * r21900;
        return r21901;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21902 = R;
        double r21903 = phi1;
        double r21904 = sin(r21903);
        double r21905 = phi2;
        double r21906 = sin(r21905);
        double r21907 = r21904 * r21906;
        double r21908 = cos(r21903);
        double r21909 = cos(r21905);
        double r21910 = r21908 * r21909;
        double r21911 = lambda1;
        double r21912 = cos(r21911);
        double r21913 = lambda2;
        double r21914 = cos(r21913);
        double r21915 = r21912 * r21914;
        double r21916 = r21910 * r21915;
        double r21917 = sin(r21911);
        double r21918 = sin(r21913);
        double r21919 = r21917 * r21918;
        double r21920 = r21910 * r21919;
        double r21921 = r21916 + r21920;
        double r21922 = r21907 + r21921;
        double r21923 = acos(r21922);
        double r21924 = r21902 * r21923;
        return r21924;
}

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-diff3.8

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

  4. Applied distribute-lft-in3.8

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

  6. Applied add-log-exp3.8

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

  8. Applied add-exp-log3.8

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

  9. Final simplification3.8

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

Reproduce

herbie shell --seed 2019298 
(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))