Spherical law of cosines

Average Error: 16.7 → 3.8

Time: 48.5s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1302284 = phi1;
        double r1302285 = sin(r1302284);
        double r1302286 = phi2;
        double r1302287 = sin(r1302286);
        double r1302288 = r1302285 * r1302287;
        double r1302289 = cos(r1302284);
        double r1302290 = cos(r1302286);
        double r1302291 = r1302289 * r1302290;
        double r1302292 = lambda1;
        double r1302293 = lambda2;
        double r1302294 = r1302292 - r1302293;
        double r1302295 = cos(r1302294);
        double r1302296 = r1302291 * r1302295;
        double r1302297 = r1302288 + r1302296;
        double r1302298 = acos(r1302297);
        double r1302299 = R;
        double r1302300 = r1302298 * r1302299;
        return r1302300;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1302301 = R;
        double r1302302 = lambda2;
        double r1302303 = sin(r1302302);
        double r1302304 = lambda1;
        double r1302305 = sin(r1302304);
        double r1302306 = cos(r1302304);
        double r1302307 = cos(r1302302);
        double r1302308 = r1302306 * r1302307;
        double r1302309 = fma(r1302303, r1302305, r1302308);
        double r1302310 = phi2;
        double r1302311 = cos(r1302310);
        double r1302312 = phi1;
        double r1302313 = cos(r1302312);
        double r1302314 = r1302311 * r1302313;
        double r1302315 = sin(r1302312);
        double r1302316 = sin(r1302310);
        double r1302317 = r1302315 * r1302316;
        double r1302318 = fma(r1302309, r1302314, r1302317);
        double r1302319 = acos(r1302318);
        double r1302320 = exp(r1302319);
        double r1302321 = 1.0;
        double r1302322 = r1302320 - r1302321;
        double r1302323 = log1p(r1302322);
        double r1302324 = r1302301 * r1302323;
        return r1302324;
}

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. Simplified 16.7

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

4. Applied cos-diff 3.8

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

6. Applied log1p-expm1-u 3.8

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

7. Simplified 3.8

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

9. Applied add-log-exp 3.8

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

11. Applied expm1-udef 3.8

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

12. Applied exp-diff 3.8

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

13. Applied log-div 3.8

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

14. Simplified 3.8

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

15. Simplified 3.8

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

16. Final simplification 3.8

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

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))