Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 35.5s

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r64590 = lambda1;
        double r64591 = theta;
        double r64592 = sin(r64591);
        double r64593 = delta;
        double r64594 = sin(r64593);
        double r64595 = r64592 * r64594;
        double r64596 = phi1;
        double r64597 = cos(r64596);
        double r64598 = r64595 * r64597;
        double r64599 = cos(r64593);
        double r64600 = sin(r64596);
        double r64601 = r64600 * r64599;
        double r64602 = r64597 * r64594;
        double r64603 = cos(r64591);
        double r64604 = r64602 * r64603;
        double r64605 = r64601 + r64604;
        double r64606 = asin(r64605);
        double r64607 = sin(r64606);
        double r64608 = r64600 * r64607;
        double r64609 = r64599 - r64608;
        double r64610 = atan2(r64598, r64609);
        double r64611 = r64590 + r64610;
        return r64611;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r64612 = lambda1;
        double r64613 = theta;
        double r64614 = sin(r64613);
        double r64615 = delta;
        double r64616 = sin(r64615);
        double r64617 = r64614 * r64616;
        double r64618 = phi1;
        double r64619 = cos(r64618);
        double r64620 = r64617 * r64619;
        double r64621 = cos(r64615);
        double r64622 = 2.0;
        double r64623 = pow(r64621, r64622);
        double r64624 = sin(r64618);
        double r64625 = 3.0;
        double r64626 = pow(r64624, r64625);
        double r64627 = cos(r64613);
        double r64628 = r64616 * r64627;
        double r64629 = r64621 * r64628;
        double r64630 = r64619 * r64629;
        double r64631 = r64626 * r64630;
        double r64632 = pow(r64624, r64622);
        double r64633 = pow(r64619, r64622);
        double r64634 = pow(r64627, r64622);
        double r64635 = pow(r64616, r64622);
        double r64636 = r64634 * r64635;
        double r64637 = r64633 * r64636;
        double r64638 = r64632 * r64637;
        double r64639 = 4.0;
        double r64640 = pow(r64624, r64639);
        double r64641 = sqrt(r64640);
        double r64642 = r64641 * r64641;
        double r64643 = r64642 * r64623;
        double r64644 = r64621 * r64627;
        double r64645 = r64616 * r64644;
        double r64646 = r64619 * r64645;
        double r64647 = r64626 * r64646;
        double r64648 = r64643 + r64647;
        double r64649 = r64638 + r64648;
        double r64650 = r64631 + r64649;
        double r64651 = r64623 - r64650;
        double r64652 = 1.0;
        double r64653 = r64632 + r64652;
        double r64654 = r64653 * r64621;
        double r64655 = r64627 * r64616;
        double r64656 = r64619 * r64655;
        double r64657 = r64624 * r64656;
        double r64658 = r64654 + r64657;
        double r64659 = r64651 / r64658;
        double r64660 = atan2(r64620, r64659);
        double r64661 = r64612 + r64660;
        return r64661;
}

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-log-exp 0.2

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

4. Applied add-log-exp 0.2

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

5. Applied diff-log 0.2

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

6. Simplified 0.2

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

8. Applied flip-- 0.2

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

9. Simplified 0.2

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

10. Simplified 0.2

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

11. Taylor expanded around -inf 0.2

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

12. Simplified 0.2

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

14. Applied add-sqr-sqrt 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019209 
(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))))))))))