Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 48.9s

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({\left(\cos delta\right)}^{2}\right)}^{3} - {\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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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)}^{3}}{\left(\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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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({\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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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}\right) + {\left(\cos delta\right)}^{4}\right) \cdot \left(\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)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r95527 = lambda1;
        double r95528 = theta;
        double r95529 = sin(r95528);
        double r95530 = delta;
        double r95531 = sin(r95530);
        double r95532 = r95529 * r95531;
        double r95533 = phi1;
        double r95534 = cos(r95533);
        double r95535 = r95532 * r95534;
        double r95536 = cos(r95530);
        double r95537 = sin(r95533);
        double r95538 = r95537 * r95536;
        double r95539 = r95534 * r95531;
        double r95540 = cos(r95528);
        double r95541 = r95539 * r95540;
        double r95542 = r95538 + r95541;
        double r95543 = asin(r95542);
        double r95544 = sin(r95543);
        double r95545 = r95537 * r95544;
        double r95546 = r95536 - r95545;
        double r95547 = atan2(r95535, r95546);
        double r95548 = r95527 + r95547;
        return r95548;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r95549 = lambda1;
        double r95550 = theta;
        double r95551 = sin(r95550);
        double r95552 = delta;
        double r95553 = sin(r95552);
        double r95554 = r95551 * r95553;
        double r95555 = phi1;
        double r95556 = cos(r95555);
        double r95557 = r95554 * r95556;
        double r95558 = cos(r95552);
        double r95559 = 2.0;
        double r95560 = pow(r95558, r95559);
        double r95561 = 3.0;
        double r95562 = pow(r95560, r95561);
        double r95563 = sin(r95555);
        double r95564 = pow(r95563, r95561);
        double r95565 = cos(r95550);
        double r95566 = r95553 * r95565;
        double r95567 = r95558 * r95566;
        double r95568 = r95556 * r95567;
        double r95569 = r95564 * r95568;
        double r95570 = pow(r95563, r95559);
        double r95571 = pow(r95556, r95559);
        double r95572 = pow(r95565, r95559);
        double r95573 = pow(r95553, r95559);
        double r95574 = r95572 * r95573;
        double r95575 = r95571 * r95574;
        double r95576 = r95570 * r95575;
        double r95577 = r95570 * r95570;
        double r95578 = r95577 * r95560;
        double r95579 = r95558 * r95565;
        double r95580 = r95553 * r95579;
        double r95581 = r95556 * r95580;
        double r95582 = r95564 * r95581;
        double r95583 = r95578 + r95582;
        double r95584 = r95576 + r95583;
        double r95585 = r95569 + r95584;
        double r95586 = pow(r95585, r95561);
        double r95587 = r95562 - r95586;
        double r95588 = r95585 + r95560;
        double r95589 = r95585 * r95588;
        double r95590 = 4.0;
        double r95591 = pow(r95558, r95590);
        double r95592 = r95589 + r95591;
        double r95593 = r95563 * r95558;
        double r95594 = r95556 * r95553;
        double r95595 = r95594 * r95565;
        double r95596 = r95593 + r95595;
        double r95597 = asin(r95596);
        double r95598 = sin(r95597);
        double r95599 = r95563 * r95598;
        double r95600 = r95558 + r95599;
        double r95601 = r95592 * r95600;
        double r95602 = r95587 / r95601;
        double r95603 = atan2(r95557, r95602);
        double r95604 = r95549 + r95603;
        return r95604;
}

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

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

6. Applied sqr-pow 0.2

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

7. Simplified 0.2

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

8. Simplified 0.2

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

10. Applied flip3-- 0.2

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

11. 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({\left(\cos delta\right)}^{2}\right)}^{3} - {\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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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)}^{3}}{\left(\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)\right) \cdot \left({\left(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(\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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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} \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({\left(\sin \phi_1\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right) \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)\right)}}}\]

12. Simplified 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019298 
(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))))))))))