Bearing on a great circle

Average Error: 13.3 → 0.2

Time: 48.3s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r119360 = lambda1;
        double r119361 = lambda2;
        double r119362 = r119360 - r119361;
        double r119363 = sin(r119362);
        double r119364 = phi2;
        double r119365 = cos(r119364);
        double r119366 = r119363 * r119365;
        double r119367 = phi1;
        double r119368 = cos(r119367);
        double r119369 = sin(r119364);
        double r119370 = r119368 * r119369;
        double r119371 = sin(r119367);
        double r119372 = r119371 * r119365;
        double r119373 = cos(r119362);
        double r119374 = r119372 * r119373;
        double r119375 = r119370 - r119374;
        double r119376 = atan2(r119366, r119375);
        return r119376;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r119377 = lambda1;
        double r119378 = sin(r119377);
        double r119379 = lambda2;
        double r119380 = cos(r119379);
        double r119381 = r119378 * r119380;
        double r119382 = cos(r119377);
        double r119383 = -r119379;
        double r119384 = sin(r119383);
        double r119385 = r119382 * r119384;
        double r119386 = r119381 + r119385;
        double r119387 = phi2;
        double r119388 = cos(r119387);
        double r119389 = r119386 * r119388;
        double r119390 = phi1;
        double r119391 = cos(r119390);
        double r119392 = sin(r119387);
        double r119393 = r119391 * r119392;
        double r119394 = r119380 * r119382;
        double r119395 = 3.0;
        double r119396 = pow(r119394, r119395);
        double r119397 = r119384 * r119378;
        double r119398 = pow(r119397, r119395);
        double r119399 = r119396 - r119398;
        double r119400 = sin(r119390);
        double r119401 = r119400 * r119388;
        double r119402 = r119399 * r119401;
        double r119403 = r119394 * r119394;
        double r119404 = exp(r119397);
        double r119405 = log(r119404);
        double r119406 = r119405 * r119405;
        double r119407 = r119394 * r119405;
        double r119408 = r119406 + r119407;
        double r119409 = r119403 + r119408;
        double r119410 = r119402 / r119409;
        double r119411 = r119393 - r119410;
        double r119412 = atan2(r119389, r119411);
        return r119412;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.3

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

3. Applied sub-neg 13.3

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

4. Applied sin-sum 6.7

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

5. Simplified 6.7

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

7. Applied sub-neg 6.7

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

8. Applied cos-sum 0.2

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

9. Simplified 0.2

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

10. Simplified 0.2

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

12. Applied add-log-exp 0.2

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

14. Applied flip3-- 0.2

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

15. Applied associate-*r/ 0.2

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

16. Simplified 0.2

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

17. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019199 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))