Average Error: 16.7 → 3.8
Time: 41.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right)\right)}\right)\right)\]
\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(\log \left(e^{\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)\right)\right)}\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1096163 = phi1;
        double r1096164 = sin(r1096163);
        double r1096165 = phi2;
        double r1096166 = sin(r1096165);
        double r1096167 = r1096164 * r1096166;
        double r1096168 = cos(r1096163);
        double r1096169 = cos(r1096165);
        double r1096170 = r1096168 * r1096169;
        double r1096171 = lambda1;
        double r1096172 = lambda2;
        double r1096173 = r1096171 - r1096172;
        double r1096174 = cos(r1096173);
        double r1096175 = r1096170 * r1096174;
        double r1096176 = r1096167 + r1096175;
        double r1096177 = acos(r1096176);
        double r1096178 = R;
        double r1096179 = r1096177 * r1096178;
        return r1096179;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1096180 = R;
        double r1096181 = phi2;
        double r1096182 = sin(r1096181);
        double r1096183 = phi1;
        double r1096184 = sin(r1096183);
        double r1096185 = lambda2;
        double r1096186 = sin(r1096185);
        double r1096187 = lambda1;
        double r1096188 = sin(r1096187);
        double r1096189 = cos(r1096187);
        double r1096190 = cos(r1096185);
        double r1096191 = r1096189 * r1096190;
        double r1096192 = fma(r1096186, r1096188, r1096191);
        double r1096193 = cos(r1096181);
        double r1096194 = r1096192 * r1096193;
        double r1096195 = cos(r1096183);
        double r1096196 = r1096194 * r1096195;
        double r1096197 = fma(r1096182, r1096184, r1096196);
        double r1096198 = acos(r1096197);
        double r1096199 = expm1(r1096198);
        double r1096200 = exp(r1096199);
        double r1096201 = log(r1096200);
        double r1096202 = log1p(r1096201);
        double r1096203 = r1096180 * r1096202;
        return r1096203;
}

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 16.7

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

  2. Simplified16.7

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

  4. Applied cos-diff3.8

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

  6. Applied log1p-expm1-u3.8

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

  7. Simplified3.8

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

  9. Applied expm1-log1p-u3.8

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

  10. Simplified3.8

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

  12. Applied add-log-exp3.8

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

  13. Final simplification3.8

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

Reproduce

herbie shell --seed 2019172 +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))