Average Error: 16.7 → 3.9
Time: 58.4s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[e^{\log \left(\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right)\right)\right)} \cdot R\]
\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(\mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)}\right)\right)\right)} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22202 = phi1;
        double r22203 = sin(r22202);
        double r22204 = phi2;
        double r22205 = sin(r22204);
        double r22206 = r22203 * r22205;
        double r22207 = cos(r22202);
        double r22208 = cos(r22204);
        double r22209 = r22207 * r22208;
        double r22210 = lambda1;
        double r22211 = lambda2;
        double r22212 = r22210 - r22211;
        double r22213 = cos(r22212);
        double r22214 = r22209 * r22213;
        double r22215 = r22206 + r22214;
        double r22216 = acos(r22215);
        double r22217 = R;
        double r22218 = r22216 * r22217;
        return r22218;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r22219 = phi1;
        double r22220 = cos(r22219);
        double r22221 = phi2;
        double r22222 = cos(r22221);
        double r22223 = r22220 * r22222;
        double r22224 = lambda2;
        double r22225 = cos(r22224);
        double r22226 = lambda1;
        double r22227 = cos(r22226);
        double r22228 = sin(r22224);
        double r22229 = sin(r22226);
        double r22230 = r22228 * r22229;
        double r22231 = fma(r22225, r22227, r22230);
        double r22232 = sin(r22219);
        double r22233 = sin(r22221);
        double r22234 = r22232 * r22233;
        double r22235 = fma(r22223, r22231, r22234);
        double r22236 = acos(r22235);
        double r22237 = expm1(r22236);
        double r22238 = exp(r22237);
        double r22239 = log(r22238);
        double r22240 = log1p(r22239);
        double r22241 = log(r22240);
        double r22242 = exp(r22241);
        double r22243 = R;
        double r22244 = r22242 * r22243;
        return r22244;
}

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

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

  4. Applied cos-diff3.8

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

  5. Applied distribute-lft-in3.8

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

  6. Simplified3.8

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

  8. Applied add-exp-log3.9

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

  9. Simplified3.8

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

  11. Applied log1p-expm1-u3.9

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

  13. Applied add-log-exp3.9

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

  14. Final simplification3.9

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

Reproduce

herbie shell --seed 2019199 +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))