Average Error: 17.0 → 3.7
Time: 45.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sqrt[3]{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2\right)\right)\right) \cdot R\]
\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
↓
\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sqrt[3]{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1240063 = phi1;
        double r1240064 = sin(r1240063);
        double r1240065 = phi2;
        double r1240066 = sin(r1240065);
        double r1240067 = r1240064 * r1240066;
        double r1240068 = cos(r1240063);
        double r1240069 = cos(r1240065);
        double r1240070 = r1240068 * r1240069;
        double r1240071 = lambda1;
        double r1240072 = lambda2;
        double r1240073 = r1240071 - r1240072;
        double r1240074 = cos(r1240073);
        double r1240075 = r1240070 * r1240074;
        double r1240076 = r1240067 + r1240075;
        double r1240077 = acos(r1240076);
        double r1240078 = R;
        double r1240079 = r1240077 * r1240078;
        return r1240079;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1240080 = phi2;
        double r1240081 = sin(r1240080);
        double r1240082 = phi1;
        double r1240083 = sin(r1240082);
        double r1240084 = cos(r1240082);
        double r1240085 = lambda2;
        double r1240086 = cos(r1240085);
        double r1240087 = lambda1;
        double r1240088 = cos(r1240087);
        double r1240089 = sin(r1240087);
        double r1240090 = sin(r1240085);
        double r1240091 = r1240089 * r1240090;
        double r1240092 = r1240091 * r1240091;
        double r1240093 = r1240091 * r1240092;
        double r1240094 = cbrt(r1240093);
        double r1240095 = fma(r1240086, r1240088, r1240094);
        double r1240096 = cos(r1240080);
        double r1240097 = r1240095 * r1240096;
        double r1240098 = r1240084 * r1240097;
        double r1240099 = fma(r1240081, r1240083, r1240098);
        double r1240100 = acos(r1240099);
        double r1240101 = R;
        double r1240102 = r1240100 * r1240101;
        return r1240102;
}

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 17.0

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

  3. Applied cos-diff3.7

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

  4. Applied distribute-rgt-in3.7

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

  6. Applied add-exp-log3.7

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

  7. Simplified3.7

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

  8. Taylor expanded around 0 3.6

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

  10. Applied add-cbrt-cube3.7

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

  11. Applied add-cbrt-cube3.7

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

  12. Applied cbrt-unprod3.7

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

  13. Simplified3.7

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

  14. Final simplification3.7

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sqrt[3]{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot \cos \phi_2\right)\right)\right) \cdot R\]

Reproduce

herbie shell --seed 2019168 +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))