Average Error: 16.2 → 3.9
Time: 42.0s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)\right)} \cdot R\]
\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(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1255909 = phi1;
        double r1255910 = sin(r1255909);
        double r1255911 = phi2;
        double r1255912 = sin(r1255911);
        double r1255913 = r1255910 * r1255912;
        double r1255914 = cos(r1255909);
        double r1255915 = cos(r1255911);
        double r1255916 = r1255914 * r1255915;
        double r1255917 = lambda1;
        double r1255918 = lambda2;
        double r1255919 = r1255917 - r1255918;
        double r1255920 = cos(r1255919);
        double r1255921 = r1255916 * r1255920;
        double r1255922 = r1255913 + r1255921;
        double r1255923 = acos(r1255922);
        double r1255924 = R;
        double r1255925 = r1255923 * r1255924;
        return r1255925;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1255926 = phi1;
        double r1255927 = sin(r1255926);
        double r1255928 = phi2;
        double r1255929 = sin(r1255928);
        double r1255930 = lambda1;
        double r1255931 = cos(r1255930);
        double r1255932 = lambda2;
        double r1255933 = cos(r1255932);
        double r1255934 = sin(r1255932);
        double r1255935 = sin(r1255930);
        double r1255936 = r1255934 * r1255935;
        double r1255937 = fma(r1255931, r1255933, r1255936);
        double r1255938 = cos(r1255928);
        double r1255939 = cos(r1255926);
        double r1255940 = r1255938 * r1255939;
        double r1255941 = r1255937 * r1255940;
        double r1255942 = fma(r1255927, r1255929, r1255941);
        double r1255943 = acos(r1255942);
        double r1255944 = log(r1255943);
        double r1255945 = exp(r1255944);
        double r1255946 = R;
        double r1255947 = r1255945 * r1255946;
        return r1255947;
}

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 16.2

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

  3. Applied cos-diff3.9

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

  5. Applied add-log-exp3.9

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

  7. Applied add-log-exp3.9

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

  8. Simplified3.9

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

  10. Applied add-exp-log3.9

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

  11. Simplified3.9

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

  12. Final simplification3.9

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

Reproduce

herbie shell --seed 2019164 +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))