Average Error: 17.2 → 3.7
Time: 1.1m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\sin \lambda_2\right), \left(\sin \lambda_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\log \left(e^{\sin \phi_1 \cdot \sin \phi_2}\right)\right)\right)\right)\]
\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 \cos^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \mathsf{fma}\left(\left(\sin \lambda_2\right), \left(\sin \lambda_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), \left(\cos \phi_1\right), \left(\log \left(e^{\sin \phi_1 \cdot \sin \phi_2}\right)\right)\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1436868 = phi1;
        double r1436869 = sin(r1436868);
        double r1436870 = phi2;
        double r1436871 = sin(r1436870);
        double r1436872 = r1436869 * r1436871;
        double r1436873 = cos(r1436868);
        double r1436874 = cos(r1436870);
        double r1436875 = r1436873 * r1436874;
        double r1436876 = lambda1;
        double r1436877 = lambda2;
        double r1436878 = r1436876 - r1436877;
        double r1436879 = cos(r1436878);
        double r1436880 = r1436875 * r1436879;
        double r1436881 = r1436872 + r1436880;
        double r1436882 = acos(r1436881);
        double r1436883 = R;
        double r1436884 = r1436882 * r1436883;
        return r1436884;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1436885 = R;
        double r1436886 = phi2;
        double r1436887 = cos(r1436886);
        double r1436888 = lambda2;
        double r1436889 = sin(r1436888);
        double r1436890 = lambda1;
        double r1436891 = sin(r1436890);
        double r1436892 = cos(r1436890);
        double r1436893 = cos(r1436888);
        double r1436894 = r1436892 * r1436893;
        double r1436895 = fma(r1436889, r1436891, r1436894);
        double r1436896 = r1436887 * r1436895;
        double r1436897 = phi1;
        double r1436898 = cos(r1436897);
        double r1436899 = sin(r1436897);
        double r1436900 = sin(r1436886);
        double r1436901 = r1436899 * r1436900;
        double r1436902 = exp(r1436901);
        double r1436903 = log(r1436902);
        double r1436904 = fma(r1436896, r1436898, r1436903);
        double r1436905 = acos(r1436904);
        double r1436906 = r1436885 * r1436905;
        return r1436906;
}

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 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. Simplified17.2

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

  4. Applied cos-diff3.7

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

  5. Taylor expanded around -inf 3.7

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

  6. Simplified3.7

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

  8. Applied add-log-exp3.7

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

  9. Final simplification3.7

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

Reproduce

herbie shell --seed 2019124 +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))