Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 30.2s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2411434 = lambda1;
        double r2411435 = phi2;
        double r2411436 = cos(r2411435);
        double r2411437 = lambda2;
        double r2411438 = r2411434 - r2411437;
        double r2411439 = sin(r2411438);
        double r2411440 = r2411436 * r2411439;
        double r2411441 = phi1;
        double r2411442 = cos(r2411441);
        double r2411443 = cos(r2411438);
        double r2411444 = r2411436 * r2411443;
        double r2411445 = r2411442 + r2411444;
        double r2411446 = atan2(r2411440, r2411445);
        double r2411447 = r2411434 + r2411446;
        return r2411447;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2411448 = phi2;
        double r2411449 = cos(r2411448);
        double r2411450 = lambda1;
        double r2411451 = sin(r2411450);
        double r2411452 = lambda2;
        double r2411453 = cos(r2411452);
        double r2411454 = r2411451 * r2411453;
        double r2411455 = cos(r2411450);
        double r2411456 = sin(r2411452);
        double r2411457 = r2411455 * r2411456;
        double r2411458 = r2411454 - r2411457;
        double r2411459 = r2411449 * r2411458;
        double r2411460 = r2411455 * r2411449;
        double r2411461 = r2411460 * r2411453;
        double r2411462 = r2411461 * r2411461;
        double r2411463 = r2411462 * r2411461;
        double r2411464 = r2411463 * r2411463;
        double r2411465 = cbrt(r2411464);
        double r2411466 = r2411465 * r2411461;
        double r2411467 = phi1;
        double r2411468 = cos(r2411467);
        double r2411469 = r2411468 * r2411468;
        double r2411470 = r2411468 * r2411469;
        double r2411471 = r2411466 + r2411470;
        double r2411472 = r2411461 - r2411468;
        double r2411473 = r2411461 * r2411472;
        double r2411474 = r2411469 + r2411473;
        double r2411475 = r2411471 / r2411474;
        double r2411476 = r2411456 * r2411451;
        double r2411477 = r2411449 * r2411476;
        double r2411478 = r2411475 + r2411477;
        double r2411479 = atan2(r2411459, r2411478);
        double r2411480 = r2411479 + r2411450;
        return r2411480;
}

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 0.9

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
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-rgt-in 0.2

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

7. Applied associate-+r+ 0.2

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

9. Applied flip3-+ 0.2

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

10. Simplified 0.2

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

11. Simplified 0.2

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

13. Applied add-cbrt-cube 0.3

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

14. Applied add-cbrt-cube 0.3

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

15. Applied cbrt-unprod 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019192 
(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)))))))