Average Error: 16.9 → 3.6
Time: 46.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)\]
\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 \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r822844 = phi1;
        double r822845 = sin(r822844);
        double r822846 = phi2;
        double r822847 = sin(r822846);
        double r822848 = r822845 * r822847;
        double r822849 = cos(r822844);
        double r822850 = cos(r822846);
        double r822851 = r822849 * r822850;
        double r822852 = lambda1;
        double r822853 = lambda2;
        double r822854 = r822852 - r822853;
        double r822855 = cos(r822854);
        double r822856 = r822851 * r822855;
        double r822857 = r822848 + r822856;
        double r822858 = acos(r822857);
        double r822859 = R;
        double r822860 = r822858 * r822859;
        return r822860;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r822861 = R;
        double r822862 = phi1;
        double r822863 = sin(r822862);
        double r822864 = phi2;
        double r822865 = sin(r822864);
        double r822866 = cos(r822864);
        double r822867 = cos(r822862);
        double r822868 = r822866 * r822867;
        double r822869 = lambda2;
        double r822870 = cos(r822869);
        double r822871 = lambda1;
        double r822872 = cos(r822871);
        double r822873 = sin(r822871);
        double r822874 = sin(r822869);
        double r822875 = r822873 * r822874;
        double r822876 = fma(r822870, r822872, r822875);
        double r822877 = r822868 * r822876;
        double r822878 = fma(r822863, r822865, r822877);
        double r822879 = acos(r822878);
        double r822880 = expm1(r822879);
        double r822881 = log1p(r822880);
        double r822882 = r822861 * r822881;
        return r822882;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

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. Simplified16.9

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

  4. Applied cos-diff3.6

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

  6. Applied add-log-exp3.7

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

  7. Simplified3.7

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

  9. Applied log1p-expm1-u3.7

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

  10. Simplified3.6

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

  11. Final simplification3.6

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))