Average Error: 17.3 → 3.8
Time: 14.2s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)\right)\right)} \cdot R\]
\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
↓
e^{\log \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\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 \left(-\lambda_2\right)\right)\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22820 = phi1;
        double r22821 = sin(r22820);
        double r22822 = phi2;
        double r22823 = sin(r22822);
        double r22824 = r22821 * r22823;
        double r22825 = cos(r22820);
        double r22826 = cos(r22822);
        double r22827 = r22825 * r22826;
        double r22828 = lambda1;
        double r22829 = lambda2;
        double r22830 = r22828 - r22829;
        double r22831 = cos(r22830);
        double r22832 = r22827 * r22831;
        double r22833 = r22824 + r22832;
        double r22834 = acos(r22833);
        double r22835 = R;
        double r22836 = r22834 * r22835;
        return r22836;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22837 = phi1;
        double r22838 = sin(r22837);
        double r22839 = phi2;
        double r22840 = sin(r22839);
        double r22841 = r22838 * r22840;
        double r22842 = cos(r22837);
        double r22843 = cos(r22839);
        double r22844 = r22842 * r22843;
        double r22845 = lambda1;
        double r22846 = cos(r22845);
        double r22847 = lambda2;
        double r22848 = cos(r22847);
        double r22849 = r22846 * r22848;
        double r22850 = sin(r22845);
        double r22851 = -r22847;
        double r22852 = sin(r22851);
        double r22853 = r22850 * r22852;
        double r22854 = r22849 - r22853;
        double r22855 = r22844 * r22854;
        double r22856 = r22841 + r22855;
        double r22857 = acos(r22856);
        double r22858 = expm1(r22857);
        double r22859 = log1p(r22858);
        double r22860 = log(r22859);
        double r22861 = exp(r22860);
        double r22862 = R;
        double r22863 = r22861 * r22862;
        return r22863;
}

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 17.3

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

  3. Applied sub-neg17.3

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

  4. Applied cos-sum3.8

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

  5. Simplified3.8

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

  7. Applied add-exp-log3.8

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

  9. Applied add-log-exp3.8

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

  11. Applied log1p-expm1-u3.8

    \[\leadsto e^{\log \color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right)\right)\right)\right)}} \cdot R\]

  12. Simplified3.8

    \[\leadsto e^{\log \left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)}\right)\right)} \cdot R\]

  13. Final simplification3.8

    \[\leadsto e^{\log \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right)\right)\right)} \cdot R\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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))