Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 40.5s

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r69404 = lambda1;
        double r69405 = theta;
        double r69406 = sin(r69405);
        double r69407 = delta;
        double r69408 = sin(r69407);
        double r69409 = r69406 * r69408;
        double r69410 = phi1;
        double r69411 = cos(r69410);
        double r69412 = r69409 * r69411;
        double r69413 = cos(r69407);
        double r69414 = sin(r69410);
        double r69415 = r69414 * r69413;
        double r69416 = r69411 * r69408;
        double r69417 = cos(r69405);
        double r69418 = r69416 * r69417;
        double r69419 = r69415 + r69418;
        double r69420 = asin(r69419);
        double r69421 = sin(r69420);
        double r69422 = r69414 * r69421;
        double r69423 = r69413 - r69422;
        double r69424 = atan2(r69412, r69423);
        double r69425 = r69404 + r69424;
        return r69425;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r69426 = lambda1;
        double r69427 = theta;
        double r69428 = sin(r69427);
        double r69429 = delta;
        double r69430 = sin(r69429);
        double r69431 = r69428 * r69430;
        double r69432 = phi1;
        double r69433 = cos(r69432);
        double r69434 = r69431 * r69433;
        double r69435 = cos(r69429);
        double r69436 = 2.0;
        double r69437 = pow(r69435, r69436);
        double r69438 = sin(r69432);
        double r69439 = 4.0;
        double r69440 = pow(r69438, r69439);
        double r69441 = 3.0;
        double r69442 = pow(r69438, r69441);
        double r69443 = pow(r69442, r69441);
        double r69444 = cbrt(r69443);
        double r69445 = cos(r69427);
        double r69446 = r69435 * r69445;
        double r69447 = r69444 * r69446;
        double r69448 = r69433 * r69447;
        double r69449 = r69448 * r69436;
        double r69450 = pow(r69433, r69436);
        double r69451 = pow(r69445, r69436);
        double r69452 = pow(r69438, r69436);
        double r69453 = r69451 * r69452;
        double r69454 = r69450 * r69453;
        double r69455 = r69430 * r69454;
        double r69456 = r69449 + r69455;
        double r69457 = r69430 * r69456;
        double r69458 = fma(r69440, r69437, r69457);
        double r69459 = r69437 - r69458;
        double r69460 = r69433 * r69430;
        double r69461 = r69460 * r69445;
        double r69462 = fma(r69438, r69435, r69461);
        double r69463 = asin(r69462);
        double r69464 = sin(r69463);
        double r69465 = fma(r69438, r69464, r69435);
        double r69466 = r69459 / r69465;
        double r69467 = atan2(r69434, r69466);
        double r69468 = r69426 + r69467;
        return r69468;
}

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. Simplified 0.2

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

4. Applied flip-- 0.2

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

5. Simplified 0.2

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

6. Simplified 0.2

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

7. Taylor expanded around inf 0.2

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

8. Simplified 0.2

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

10. Applied add-cbrt-cube 0.2

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

11. Simplified 0.2

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

12. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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))))))))))