Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 3.8m

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r5357751 = lambda1;
        double r5357752 = theta;
        double r5357753 = sin(r5357752);
        double r5357754 = delta;
        double r5357755 = sin(r5357754);
        double r5357756 = r5357753 * r5357755;
        double r5357757 = phi1;
        double r5357758 = cos(r5357757);
        double r5357759 = r5357756 * r5357758;
        double r5357760 = cos(r5357754);
        double r5357761 = sin(r5357757);
        double r5357762 = r5357761 * r5357760;
        double r5357763 = r5357758 * r5357755;
        double r5357764 = cos(r5357752);
        double r5357765 = r5357763 * r5357764;
        double r5357766 = r5357762 + r5357765;
        double r5357767 = asin(r5357766);
        double r5357768 = sin(r5357767);
        double r5357769 = r5357761 * r5357768;
        double r5357770 = r5357760 - r5357769;
        double r5357771 = atan2(r5357759, r5357770);
        double r5357772 = r5357751 + r5357771;
        return r5357772;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r5357773 = phi1;
        double r5357774 = cos(r5357773);
        double r5357775 = delta;
        double r5357776 = sin(r5357775);
        double r5357777 = theta;
        double r5357778 = sin(r5357777);
        double r5357779 = r5357776 * r5357778;
        double r5357780 = r5357774 * r5357779;
        double r5357781 = cos(r5357775);
        double r5357782 = r5357781 * r5357781;
        double r5357783 = cos(r5357777);
        double r5357784 = r5357774 * r5357776;
        double r5357785 = r5357784 * r5357781;
        double r5357786 = sin(r5357773);
        double r5357787 = r5357786 * r5357786;
        double r5357788 = r5357786 * r5357787;
        double r5357789 = r5357785 * r5357788;
        double r5357790 = r5357783 * r5357789;
        double r5357791 = r5357781 * r5357786;
        double r5357792 = r5357787 * r5357791;
        double r5357793 = r5357776 * r5357792;
        double r5357794 = r5357783 * r5357793;
        double r5357795 = r5357794 * r5357774;
        double r5357796 = r5357790 + r5357795;
        double r5357797 = r5357787 * r5357781;
        double r5357798 = r5357797 * r5357797;
        double r5357799 = log(r5357798);
        double r5357800 = exp(r5357799);
        double r5357801 = r5357796 + r5357800;
        double r5357802 = r5357782 - r5357801;
        double r5357803 = r5357784 * r5357783;
        double r5357804 = r5357791 + r5357803;
        double r5357805 = asin(r5357804);
        double r5357806 = sin(r5357805);
        double r5357807 = r5357786 * r5357806;
        double r5357808 = r5357781 + r5357807;
        double r5357809 = r5357802 / r5357808;
        double r5357810 = r5357784 * r5357784;
        double r5357811 = r5357783 * r5357786;
        double r5357812 = r5357811 * r5357811;
        double r5357813 = r5357810 * r5357812;
        double r5357814 = r5357813 / r5357808;
        double r5357815 = r5357809 - r5357814;
        double r5357816 = atan2(r5357780, r5357815);
        double r5357817 = lambda1;
        double r5357818 = r5357816 + r5357817;
        return r5357818;
}

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

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

5. Simplified 0.2

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

7. Applied div-sub 0.2

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

9. Applied add-exp-log 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019174 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))