Average Error: 16.5 → 3.8
Time: 42.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}\right)\right) \cdot R\]
\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 + \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22878 = phi1;
        double r22879 = sin(r22878);
        double r22880 = phi2;
        double r22881 = sin(r22880);
        double r22882 = r22879 * r22881;
        double r22883 = cos(r22878);
        double r22884 = cos(r22880);
        double r22885 = r22883 * r22884;
        double r22886 = lambda1;
        double r22887 = lambda2;
        double r22888 = r22886 - r22887;
        double r22889 = cos(r22888);
        double r22890 = r22885 * r22889;
        double r22891 = r22882 + r22890;
        double r22892 = acos(r22891);
        double r22893 = R;
        double r22894 = r22892 * r22893;
        return r22894;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22895 = phi1;
        double r22896 = sin(r22895);
        double r22897 = phi2;
        double r22898 = sin(r22897);
        double r22899 = r22896 * r22898;
        double r22900 = cos(r22895);
        double r22901 = cos(r22897);
        double r22902 = r22900 * r22901;
        double r22903 = lambda1;
        double r22904 = cos(r22903);
        double r22905 = lambda2;
        double r22906 = cos(r22905);
        double r22907 = r22904 * r22906;
        double r22908 = sin(r22903);
        double r22909 = sin(r22905);
        double r22910 = r22908 * r22909;
        double r22911 = 3.0;
        double r22912 = pow(r22910, r22911);
        double r22913 = cbrt(r22912);
        double r22914 = r22907 + r22913;
        double r22915 = r22902 * r22914;
        double r22916 = r22899 + r22915;
        double r22917 = acos(r22916);
        double r22918 = R;
        double r22919 = r22917 * r22918;
        return r22919;
}

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. Using strategy rm

  5. Applied add-cbrt-cube3.8

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

  6. Applied add-cbrt-cube3.8

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

  7. Applied cbrt-unprod3.8

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

  8. Simplified3.8

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

  9. Final simplification3.8

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

Reproduce

herbie shell --seed 2019323 
(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))