Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 18.4s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right) \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{\left(\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)\right) \cdot \left({\left(\cos delta\right)}^{2} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r93335 = lambda1;
        double r93336 = theta;
        double r93337 = sin(r93336);
        double r93338 = delta;
        double r93339 = sin(r93338);
        double r93340 = r93337 * r93339;
        double r93341 = phi1;
        double r93342 = cos(r93341);
        double r93343 = r93340 * r93342;
        double r93344 = cos(r93338);
        double r93345 = sin(r93341);
        double r93346 = r93345 * r93344;
        double r93347 = r93342 * r93339;
        double r93348 = cos(r93336);
        double r93349 = r93347 * r93348;
        double r93350 = r93346 + r93349;
        double r93351 = asin(r93350);
        double r93352 = sin(r93351);
        double r93353 = r93345 * r93352;
        double r93354 = r93344 - r93353;
        double r93355 = atan2(r93343, r93354);
        double r93356 = r93335 + r93355;
        return r93356;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r93357 = lambda1;
        double r93358 = theta;
        double r93359 = sin(r93358);
        double r93360 = delta;
        double r93361 = sin(r93360);
        double r93362 = r93359 * r93361;
        double r93363 = phi1;
        double r93364 = cos(r93363);
        double r93365 = r93362 * r93364;
        double r93366 = cos(r93360);
        double r93367 = 2.0;
        double r93368 = pow(r93366, r93367);
        double r93369 = r93368 * r93368;
        double r93370 = sin(r93363);
        double r93371 = 3.0;
        double r93372 = pow(r93370, r93371);
        double r93373 = cos(r93358);
        double r93374 = r93361 * r93373;
        double r93375 = r93366 * r93374;
        double r93376 = r93364 * r93375;
        double r93377 = r93372 * r93376;
        double r93378 = pow(r93370, r93367);
        double r93379 = pow(r93364, r93367);
        double r93380 = pow(r93373, r93367);
        double r93381 = pow(r93361, r93367);
        double r93382 = r93380 * r93381;
        double r93383 = r93379 * r93382;
        double r93384 = r93378 * r93383;
        double r93385 = 4.0;
        double r93386 = pow(r93370, r93385);
        double r93387 = r93386 * r93368;
        double r93388 = r93366 * r93373;
        double r93389 = r93361 * r93388;
        double r93390 = r93364 * r93389;
        double r93391 = r93372 * r93390;
        double r93392 = r93387 + r93391;
        double r93393 = r93384 + r93392;
        double r93394 = r93377 + r93393;
        double r93395 = r93394 * r93394;
        double r93396 = r93369 - r93395;
        double r93397 = r93378 * r93366;
        double r93398 = r93373 * r93361;
        double r93399 = r93364 * r93398;
        double r93400 = r93370 * r93399;
        double r93401 = r93397 + r93400;
        double r93402 = r93366 + r93401;
        double r93403 = r93368 + r93394;
        double r93404 = r93402 * r93403;
        double r93405 = r93396 / r93404;
        double r93406 = atan2(r93365, r93405);
        double r93407 = r93357 + r93406;
        return r93407;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

1. Initial program 0.2

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

3. Applied flip-- 0.2

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

4. Taylor expanded around -inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)}}}\]
5. Using strategy rm

6. Applied flip-- 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\frac{{\left(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right) \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}}}{\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)}}\]

7. Applied associate-/l/ 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right) \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{\left(\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)\right) \cdot \left({\left(\cos delta\right)}^{2} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)\right)}}}\]

8. Final simplification 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right) \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{\left(\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)\right) \cdot \left({\left(\cos delta\right)}^{2} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))