Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 31.9s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2327328 = lambda1;
        double r2327329 = phi2;
        double r2327330 = cos(r2327329);
        double r2327331 = lambda2;
        double r2327332 = r2327328 - r2327331;
        double r2327333 = sin(r2327332);
        double r2327334 = r2327330 * r2327333;
        double r2327335 = phi1;
        double r2327336 = cos(r2327335);
        double r2327337 = cos(r2327332);
        double r2327338 = r2327330 * r2327337;
        double r2327339 = r2327336 + r2327338;
        double r2327340 = atan2(r2327334, r2327339);
        double r2327341 = r2327328 + r2327340;
        return r2327341;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2327342 = lambda1;
        double r2327343 = phi2;
        double r2327344 = cos(r2327343);
        double r2327345 = sin(r2327342);
        double r2327346 = lambda2;
        double r2327347 = cos(r2327346);
        double r2327348 = r2327345 * r2327347;
        double r2327349 = cos(r2327342);
        double r2327350 = sin(r2327346);
        double r2327351 = r2327349 * r2327350;
        double r2327352 = r2327348 - r2327351;
        double r2327353 = r2327344 * r2327352;
        double r2327354 = r2327350 * r2327345;
        double r2327355 = r2327344 * r2327354;
        double r2327356 = phi1;
        double r2327357 = cos(r2327356);
        double r2327358 = r2327357 * r2327357;
        double r2327359 = r2327358 * r2327357;
        double r2327360 = r2327349 * r2327344;
        double r2327361 = r2327360 * r2327347;
        double r2327362 = r2327361 * r2327361;
        double r2327363 = r2327362 * r2327361;
        double r2327364 = r2327359 + r2327363;
        double r2327365 = r2327358 - r2327362;
        double r2327366 = r2327357 + r2327361;
        double r2327367 = r2327365 / r2327366;
        double r2327368 = r2327357 * r2327367;
        double r2327369 = r2327362 + r2327368;
        double r2327370 = r2327364 / r2327369;
        double r2327371 = r2327355 + r2327370;
        double r2327372 = atan2(r2327353, r2327371);
        double r2327373 = r2327342 + r2327372;
        return r2327373;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 0.9

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

3. Applied sin-diff 0.8

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
4. Using strategy rm

5. Applied cos-diff 0.2

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

6. Applied distribute-lft-in 0.2

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

7. Applied associate-+r+ 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\left(\cos \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}}\]
8. Using strategy rm

9. Applied flip3-+ 0.3

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\cos \phi_1 \cdot \cos \phi_1 + \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

10. Simplified 0.3

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

11. Simplified 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{\left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \left(\left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)}{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_1 - \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) + \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
12. Using strategy rm

13. Applied flip-- 0.3

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

14. Final simplification 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) + \frac{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos \phi_1 + \left(\left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)}{\left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) + \cos \phi_1 \cdot \frac{\cos \phi_1 \cdot \cos \phi_1 - \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \left(\left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)}{\cos \phi_1 + \left(\cos \lambda_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}}}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))