Average Error: 16.9 → 3.9
Time: 1.2m
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \left(\frac{\pi}{2} - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \cos \phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right)\right)\]
\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 \left(\frac{\pi}{2} - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \cos \phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right)\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1584321 = phi1;
        double r1584322 = sin(r1584321);
        double r1584323 = phi2;
        double r1584324 = sin(r1584323);
        double r1584325 = r1584322 * r1584324;
        double r1584326 = cos(r1584321);
        double r1584327 = cos(r1584323);
        double r1584328 = r1584326 * r1584327;
        double r1584329 = lambda1;
        double r1584330 = lambda2;
        double r1584331 = r1584329 - r1584330;
        double r1584332 = cos(r1584331);
        double r1584333 = r1584328 * r1584332;
        double r1584334 = r1584325 + r1584333;
        double r1584335 = acos(r1584334);
        double r1584336 = R;
        double r1584337 = r1584335 * r1584336;
        return r1584337;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1584338 = R;
        double r1584339 = atan2(1.0, 0.0);
        double r1584340 = 2.0;
        double r1584341 = r1584339 / r1584340;
        double r1584342 = phi2;
        double r1584343 = cos(r1584342);
        double r1584344 = phi1;
        double r1584345 = cos(r1584344);
        double r1584346 = r1584343 * r1584345;
        double r1584347 = lambda2;
        double r1584348 = cos(r1584347);
        double r1584349 = lambda1;
        double r1584350 = cos(r1584349);
        double r1584351 = r1584348 * r1584350;
        double r1584352 = sin(r1584347);
        double r1584353 = sin(r1584349);
        double r1584354 = r1584352 * r1584353;
        double r1584355 = r1584351 + r1584354;
        double r1584356 = sin(r1584342);
        double r1584357 = sin(r1584344);
        double r1584358 = r1584356 * r1584357;
        double r1584359 = fma(r1584346, r1584355, r1584358);
        double r1584360 = asin(r1584359);
        double r1584361 = exp(r1584360);
        double r1584362 = log(r1584361);
        double r1584363 = r1584341 - r1584362;
        double r1584364 = r1584338 * r1584363;
        return r1584364;
}

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 16.9

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

  2. Simplified16.9

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

  4. Applied cos-diff3.8

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

  6. Applied add-log-exp3.9

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

  8. Applied acos-asin3.9

    \[\leadsto R \cdot \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}}\right)\]

  9. Applied exp-diff3.9

    \[\leadsto R \cdot \log \color{blue}{\left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}}\right)}\]

  10. Applied log-div3.9

    \[\leadsto R \cdot \color{blue}{\left(\log \left(e^{\frac{\pi}{2}}\right) - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right)\right)}\]

  11. Simplified3.9

    \[\leadsto R \cdot \left(\color{blue}{\frac{\pi}{2}} - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right)\right)\]

  12. Final simplification3.9

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\left(\cos \phi_2 \cdot \cos \phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), \left(\sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019121 +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))