Average Error: 16.9 → 4.0
Time: 15.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left({\left(\cos \lambda_1\right)}^{3} \cdot {\left(\cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}\right)}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R\]
\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
↓
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left({\left(\cos \lambda_1\right)}^{3} \cdot {\left(\cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}\right)}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r26848 = phi1;
        double r26849 = sin(r26848);
        double r26850 = phi2;
        double r26851 = sin(r26850);
        double r26852 = r26849 * r26851;
        double r26853 = cos(r26848);
        double r26854 = cos(r26850);
        double r26855 = r26853 * r26854;
        double r26856 = lambda1;
        double r26857 = lambda2;
        double r26858 = r26856 - r26857;
        double r26859 = cos(r26858);
        double r26860 = r26855 * r26859;
        double r26861 = r26852 + r26860;
        double r26862 = acos(r26861);
        double r26863 = R;
        double r26864 = r26862 * r26863;
        return r26864;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r26865 = phi1;
        double r26866 = sin(r26865);
        double r26867 = phi2;
        double r26868 = sin(r26867);
        double r26869 = r26866 * r26868;
        double r26870 = cos(r26865);
        double r26871 = cos(r26867);
        double r26872 = r26870 * r26871;
        double r26873 = lambda1;
        double r26874 = cos(r26873);
        double r26875 = 3.0;
        double r26876 = pow(r26874, r26875);
        double r26877 = lambda2;
        double r26878 = cos(r26877);
        double r26879 = pow(r26878, r26875);
        double r26880 = r26876 * r26879;
        double r26881 = sin(r26873);
        double r26882 = sin(r26877);
        double r26883 = r26881 * r26882;
        double r26884 = pow(r26883, r26875);
        double r26885 = r26880 + r26884;
        double r26886 = r26872 * r26885;
        double r26887 = r26874 * r26878;
        double r26888 = r26887 * r26887;
        double r26889 = r26883 * r26883;
        double r26890 = r26887 * r26883;
        double r26891 = r26889 - r26890;
        double r26892 = r26888 + r26891;
        double r26893 = r26886 / r26892;
        double r26894 = r26869 + r26893;
        double r26895 = acos(r26894);
        double r26896 = R;
        double r26897 = r26895 * r26896;
        return r26897;
}

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 cos-diff3.9

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

  5. Applied flip3-+4.0

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

  6. Applied associate-*r/4.0

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

  8. Applied unpow-prod-down4.0

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

  9. Final simplification4.0

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

Reproduce

herbie shell --seed 2020083 
(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))