Midpoint on a great circle

Average Error: 0.9 → 0.3

Time: 26.0s

Precision: 64

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right) \cdot \left(\cos \left(\phi_1 + \phi_1\right) + 1\right) + \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left({\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{2} - {\left(\cos \phi_1\right)}^{2}\right)} \cdot \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right)\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r51355 = lambda1;
        double r51356 = phi2;
        double r51357 = cos(r51356);
        double r51358 = lambda2;
        double r51359 = r51355 - r51358;
        double r51360 = sin(r51359);
        double r51361 = r51357 * r51360;
        double r51362 = phi1;
        double r51363 = cos(r51362);
        double r51364 = cos(r51359);
        double r51365 = r51357 * r51364;
        double r51366 = r51363 + r51365;
        double r51367 = atan2(r51361, r51366);
        double r51368 = r51355 + r51367;
        return r51368;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r51369 = lambda1;
        double r51370 = phi2;
        double r51371 = cos(r51370);
        double r51372 = lambda2;
        double r51373 = cos(r51372);
        double r51374 = sin(r51369);
        double r51375 = r51373 * r51374;
        double r51376 = cos(r51369);
        double r51377 = sin(r51372);
        double r51378 = r51376 * r51377;
        double r51379 = -r51378;
        double r51380 = r51375 + r51379;
        double r51381 = r51371 * r51380;
        double r51382 = phi1;
        double r51383 = cos(r51382);
        double r51384 = 3.0;
        double r51385 = pow(r51383, r51384);
        double r51386 = r51376 * r51373;
        double r51387 = r51371 * r51386;
        double r51388 = pow(r51387, r51384);
        double r51389 = r51385 + r51388;
        double r51390 = r51387 + r51383;
        double r51391 = r51382 + r51382;
        double r51392 = cos(r51391);
        double r51393 = 1.0;
        double r51394 = r51392 + r51393;
        double r51395 = r51390 * r51394;
        double r51396 = 2.0;
        double r51397 = r51396 * r51387;
        double r51398 = pow(r51387, r51396);
        double r51399 = pow(r51383, r51396);
        double r51400 = r51398 - r51399;
        double r51401 = r51397 * r51400;
        double r51402 = r51395 + r51401;
        double r51403 = r51389 / r51402;
        double r51404 = r51396 * r51390;
        double r51405 = r51403 * r51404;
        double r51406 = r51374 * r51377;
        double r51407 = r51371 * r51406;
        double r51408 = r51405 + r51407;
        double r51409 = atan2(r51381, r51408);
        double r51410 = r51369 + r51409;
        return r51410;
}

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 cos-diff 0.8

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

4. Applied distribute-lft-in 0.8

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

5. Applied associate-+r+ 0.8

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

7. Applied sub-neg 0.8

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

8. Applied sin-sum 0.2

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

9. Simplified 0.2

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

10. Simplified 0.2

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

12. Applied flip3-+ 0.3

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

13. Simplified 0.3

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

15. Applied flip-- 0.3

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

16. Applied associate-*r/ 0.3

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

17. Applied cos-mult 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\color{blue}{\frac{\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)}{2}} + \frac{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \phi_1 \cdot \cos \phi_1\right)}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

18. Applied frac-add 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right) + 2 \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \phi_1 \cdot \cos \phi_1\right)\right)}{2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right)}}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

19. Applied associate-/r/ 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\color{blue}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\left(\cos \left(\phi_1 + \phi_1\right) + \cos \left(\phi_1 - \phi_1\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right) + 2 \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \phi_1 \cdot \cos \phi_1\right)\right)} \cdot \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right)\right)} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

20. Simplified 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\color{blue}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right) \cdot \left(\cos \left(\phi_1 + \phi_1\right) + 1\right) + \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left({\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{2} - {\left(\cos \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right)\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

21. Final simplification 0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 + \left(-\cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{{\left(\cos \phi_1\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right) \cdot \left(\cos \left(\phi_1 + \phi_1\right) + 1\right) + \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right) \cdot \left({\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{2} - {\left(\cos \phi_1\right)}^{2}\right)} \cdot \left(2 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1\right)\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))