Average Error: 17.2 → 3.9
Time: 44.9s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right)\]
\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 \log \left(e^{\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 \lambda_2\right)\right)}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22805 = phi1;
        double r22806 = sin(r22805);
        double r22807 = phi2;
        double r22808 = sin(r22807);
        double r22809 = r22806 * r22808;
        double r22810 = cos(r22805);
        double r22811 = cos(r22807);
        double r22812 = r22810 * r22811;
        double r22813 = lambda1;
        double r22814 = lambda2;
        double r22815 = r22813 - r22814;
        double r22816 = cos(r22815);
        double r22817 = r22812 * r22816;
        double r22818 = r22809 + r22817;
        double r22819 = acos(r22818);
        double r22820 = R;
        double r22821 = r22819 * r22820;
        return r22821;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22822 = R;
        double r22823 = phi1;
        double r22824 = sin(r22823);
        double r22825 = phi2;
        double r22826 = sin(r22825);
        double r22827 = r22824 * r22826;
        double r22828 = cos(r22823);
        double r22829 = cos(r22825);
        double r22830 = r22828 * r22829;
        double r22831 = lambda1;
        double r22832 = cos(r22831);
        double r22833 = lambda2;
        double r22834 = cos(r22833);
        double r22835 = r22832 * r22834;
        double r22836 = sin(r22831);
        double r22837 = sin(r22833);
        double r22838 = r22836 * r22837;
        double r22839 = r22835 + r22838;
        double r22840 = r22830 * r22839;
        double r22841 = r22827 + r22840;
        double r22842 = acos(r22841);
        double r22843 = exp(r22842);
        double r22844 = log(r22843);
        double r22845 = r22822 * r22844;
        return r22845;
}

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 17.2

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
  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-log-exp3.9

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

  7. Applied acos-asin3.9

    \[\leadsto \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \cdot R\]
  8. Using strategy rm

  9. Applied asin-acos3.9

    \[\leadsto \log \left(e^{\frac{\pi}{2} - \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}\right) \cdot R\]

  10. Applied associate--r-3.9

    \[\leadsto \log \left(e^{\color{blue}{\left(\frac{\pi}{2} - \frac{\pi}{2}\right) + \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \cdot R\]

  11. Simplified3.9

    \[\leadsto \log \left(e^{\color{blue}{0} + \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R\]

  12. Final simplification3.9

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

Reproduce

herbie shell --seed 2019306 
(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))