Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 17.2s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left(\left({\left(\cos delta\right)}^{3} - {\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right)\right) - \left({\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot {\left(\cos delta\right)}^{3}\right) - 3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right) + {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left(\left(\sqrt{{\left(\sin \phi_1\right)}^{4}} \cdot \sqrt{{\left(\sin \phi_1\right)}^{4}}\right) \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r124545 = lambda1;
        double r124546 = theta;
        double r124547 = sin(r124546);
        double r124548 = delta;
        double r124549 = sin(r124548);
        double r124550 = r124547 * r124549;
        double r124551 = phi1;
        double r124552 = cos(r124551);
        double r124553 = r124550 * r124552;
        double r124554 = cos(r124548);
        double r124555 = sin(r124551);
        double r124556 = r124555 * r124554;
        double r124557 = r124552 * r124549;
        double r124558 = cos(r124546);
        double r124559 = r124557 * r124558;
        double r124560 = r124556 + r124559;
        double r124561 = asin(r124560);
        double r124562 = sin(r124561);
        double r124563 = r124555 * r124562;
        double r124564 = r124554 - r124563;
        double r124565 = atan2(r124553, r124564);
        double r124566 = r124545 + r124565;
        return r124566;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r124567 = lambda1;
        double r124568 = theta;
        double r124569 = sin(r124568);
        double r124570 = delta;
        double r124571 = sin(r124570);
        double r124572 = r124569 * r124571;
        double r124573 = phi1;
        double r124574 = cos(r124573);
        double r124575 = r124572 * r124574;
        double r124576 = cos(r124570);
        double r124577 = 3.0;
        double r124578 = pow(r124576, r124577);
        double r124579 = pow(r124571, r124577);
        double r124580 = pow(r124574, r124577);
        double r124581 = cos(r124568);
        double r124582 = pow(r124581, r124577);
        double r124583 = sin(r124573);
        double r124584 = pow(r124583, r124577);
        double r124585 = r124582 * r124584;
        double r124586 = r124580 * r124585;
        double r124587 = r124579 * r124586;
        double r124588 = r124578 - r124587;
        double r124589 = r124584 * r124584;
        double r124590 = r124589 * r124578;
        double r124591 = r124588 - r124590;
        double r124592 = 5.0;
        double r124593 = pow(r124583, r124592);
        double r124594 = 2.0;
        double r124595 = pow(r124576, r124594);
        double r124596 = r124595 * r124581;
        double r124597 = r124593 * r124596;
        double r124598 = r124574 * r124597;
        double r124599 = r124571 * r124598;
        double r124600 = pow(r124571, r124594);
        double r124601 = pow(r124574, r124594);
        double r124602 = 4.0;
        double r124603 = pow(r124583, r124602);
        double r124604 = sqrt(r124603);
        double r124605 = r124604 * r124604;
        double r124606 = pow(r124581, r124594);
        double r124607 = r124576 * r124606;
        double r124608 = r124605 * r124607;
        double r124609 = r124601 * r124608;
        double r124610 = r124600 * r124609;
        double r124611 = r124599 + r124610;
        double r124612 = r124577 * r124611;
        double r124613 = r124591 - r124612;
        double r124614 = r124583 * r124576;
        double r124615 = r124574 * r124571;
        double r124616 = r124615 * r124581;
        double r124617 = r124614 + r124616;
        double r124618 = asin(r124617);
        double r124619 = sin(r124618);
        double r124620 = r124583 * r124619;
        double r124621 = r124620 + r124576;
        double r124622 = r124620 * r124621;
        double r124623 = r124576 * r124576;
        double r124624 = r124622 + r124623;
        double r124625 = r124613 / r124624;
        double r124626 = atan2(r124575, r124625);
        double r124627 = r124567 + r124626;
        return r124627;
}

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 flip3-- 0.2

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

4. Simplified 0.2

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

5. Taylor expanded around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left({\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

6. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\left(\left({\left(\cos delta\right)}^{3} - {\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right)\right) - {\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3}\right) - 3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right) + {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 32.3

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left(\left({\left(\cos delta\right)}^{3} - {\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right)\right) - {\color{blue}{\left(\sqrt{\sin \phi_1} \cdot \sqrt{\sin \phi_1}\right)}}^{6} \cdot {\left(\cos delta\right)}^{3}\right) - 3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right) + {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

9. Applied unpow-prod-down 32.3

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left(\left({\left(\cos delta\right)}^{3} - {\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right)\right) - \color{blue}{\left({\left(\sqrt{\sin \phi_1}\right)}^{6} \cdot {\left(\sqrt{\sin \phi_1}\right)}^{6}\right)} \cdot {\left(\cos delta\right)}^{3}\right) - 3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right) + {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

10. Simplified 32.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left(\left({\left(\cos delta\right)}^{3} - {\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right)\right) - \left(\color{blue}{{\left(\sin \phi_1\right)}^{3}} \cdot {\left(\sqrt{\sin \phi_1}\right)}^{6}\right) \cdot {\left(\cos delta\right)}^{3}\right) - 3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right) + {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) + \cos delta\right) + \cos delta \cdot \cos delta}}\]

11. Simplified 0.2

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

13. Applied add-sqr-sqrt 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020057 
(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))))))))))