Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 42.6s

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r89540 = lambda1;
        double r89541 = theta;
        double r89542 = sin(r89541);
        double r89543 = delta;
        double r89544 = sin(r89543);
        double r89545 = r89542 * r89544;
        double r89546 = phi1;
        double r89547 = cos(r89546);
        double r89548 = r89545 * r89547;
        double r89549 = cos(r89543);
        double r89550 = sin(r89546);
        double r89551 = r89550 * r89549;
        double r89552 = r89547 * r89544;
        double r89553 = cos(r89541);
        double r89554 = r89552 * r89553;
        double r89555 = r89551 + r89554;
        double r89556 = asin(r89555);
        double r89557 = sin(r89556);
        double r89558 = r89550 * r89557;
        double r89559 = r89549 - r89558;
        double r89560 = atan2(r89548, r89559);
        double r89561 = r89540 + r89560;
        return r89561;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r89562 = lambda1;
        double r89563 = theta;
        double r89564 = sin(r89563);
        double r89565 = delta;
        double r89566 = sin(r89565);
        double r89567 = r89564 * r89566;
        double r89568 = phi1;
        double r89569 = cos(r89568);
        double r89570 = r89567 * r89569;
        double r89571 = sin(r89568);
        double r89572 = cos(r89565);
        double r89573 = r89571 * r89572;
        double r89574 = r89569 * r89566;
        double r89575 = cos(r89563);
        double r89576 = r89574 * r89575;
        double r89577 = r89573 + r89576;
        double r89578 = asin(r89577);
        double r89579 = sin(r89578);
        double r89580 = 2.0;
        double r89581 = r89579 / r89580;
        double r89582 = -r89581;
        double r89583 = r89582 * r89571;
        double r89584 = r89583 + r89572;
        double r89585 = atan2(1.0, 0.0);
        double r89586 = r89585 / r89580;
        double r89587 = sin(r89586);
        double r89588 = r89587 * r89571;
        double r89589 = acos(r89577);
        double r89590 = cos(r89589);
        double r89591 = r89580 / r89590;
        double r89592 = r89588 / r89591;
        double r89593 = cos(r89586);
        double r89594 = r89593 * r89571;
        double r89595 = sin(r89589);
        double r89596 = r89580 / r89595;
        double r89597 = r89594 / r89596;
        double r89598 = r89592 - r89597;
        double r89599 = r89584 - r89598;
        double r89600 = atan2(r89570, r89599);
        double r89601 = r89562 + r89600;
        return r89601;
}

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 add-cbrt-cube 0.2

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

4. Applied add-cbrt-cube 0.2

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

5. Applied cbrt-unprod 0.2

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

6. Simplified 0.2

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

8. Applied add-log-exp 0.2

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

9. Simplified 0.2

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

11. Applied sqr-pow 0.2

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

12. Applied log-prod 0.2

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

13. Applied associate--r+ 0.2

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

14. Simplified 0.2

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

16. Applied asin-acos 0.2

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

17. Applied sin-diff 0.2

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

18. Applied div-sub 0.2

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

19. Applied pow-sub 0.2

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

20. Applied log-div 0.2

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

21. Simplified 0.2

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

22. Simplified 0.2

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

23. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019297 
(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))))))))))