Average Error: 0.2 → 0.2
Time: 44.8s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \cdot \sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right)\right)}\right)}\]
\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{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \cdot \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right) \cdot \left(\sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \cdot \sqrt[3]{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right)\right)}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4061767 = lambda1;
        double r4061768 = theta;
        double r4061769 = sin(r4061768);
        double r4061770 = delta;
        double r4061771 = sin(r4061770);
        double r4061772 = r4061769 * r4061771;
        double r4061773 = phi1;
        double r4061774 = cos(r4061773);
        double r4061775 = r4061772 * r4061774;
        double r4061776 = cos(r4061770);
        double r4061777 = sin(r4061773);
        double r4061778 = r4061777 * r4061776;
        double r4061779 = r4061774 * r4061771;
        double r4061780 = cos(r4061768);
        double r4061781 = r4061779 * r4061780;
        double r4061782 = r4061778 + r4061781;
        double r4061783 = asin(r4061782);
        double r4061784 = sin(r4061783);
        double r4061785 = r4061777 * r4061784;
        double r4061786 = r4061776 - r4061785;
        double r4061787 = atan2(r4061775, r4061786);
        double r4061788 = r4061767 + r4061787;
        return r4061788;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r4061789 = lambda1;
        double r4061790 = phi1;
        double r4061791 = cos(r4061790);
        double r4061792 = delta;
        double r4061793 = sin(r4061792);
        double r4061794 = theta;
        double r4061795 = sin(r4061794);
        double r4061796 = r4061793 * r4061795;
        double r4061797 = r4061791 * r4061796;
        double r4061798 = cos(r4061792);
        double r4061799 = sin(r4061790);
        double r4061800 = r4061798 * r4061799;
        double r4061801 = cos(r4061794);
        double r4061802 = r4061791 * r4061793;
        double r4061803 = r4061801 * r4061802;
        double r4061804 = r4061800 + r4061803;
        double r4061805 = asin(r4061804);
        double r4061806 = cbrt(r4061805);
        double r4061807 = r4061806 * r4061806;
        double r4061808 = r4061805 * r4061807;
        double r4061809 = r4061806 * r4061808;
        double r4061810 = r4061805 * r4061809;
        double r4061811 = cbrt(r4061810);
        double r4061812 = sin(r4061811);
        double r4061813 = r4061799 * r4061812;
        double r4061814 = r4061798 - r4061813;
        double r4061815 = atan2(r4061797, r4061814);
        double r4061816 = r4061789 + r4061815;
        return r4061816;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-cube0.2

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

  5. Applied add-cube-cbrt0.2

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

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019172 
(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))))))))))