Average Error: 0.2 → 0.2
Time: 48.2s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \log \left(e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}\right)}\]
\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{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\log \left(e^{\cos delta - \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)}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r100709 = lambda1;
        double r100710 = theta;
        double r100711 = sin(r100710);
        double r100712 = delta;
        double r100713 = sin(r100712);
        double r100714 = r100711 * r100713;
        double r100715 = phi1;
        double r100716 = cos(r100715);
        double r100717 = r100714 * r100716;
        double r100718 = cos(r100712);
        double r100719 = sin(r100715);
        double r100720 = r100719 * r100718;
        double r100721 = r100716 * r100713;
        double r100722 = cos(r100710);
        double r100723 = r100721 * r100722;
        double r100724 = r100720 + r100723;
        double r100725 = asin(r100724);
        double r100726 = sin(r100725);
        double r100727 = r100719 * r100726;
        double r100728 = r100718 - r100727;
        double r100729 = atan2(r100717, r100728);
        double r100730 = r100709 + r100729;
        return r100730;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r100731 = lambda1;
        double r100732 = theta;
        double r100733 = sin(r100732);
        double r100734 = delta;
        double r100735 = sin(r100734);
        double r100736 = r100733 * r100735;
        double r100737 = phi1;
        double r100738 = cos(r100737);
        double r100739 = r100736 * r100738;
        double r100740 = cos(r100734);
        double r100741 = sin(r100737);
        double r100742 = r100741 * r100740;
        double r100743 = r100738 * r100735;
        double r100744 = cos(r100732);
        double r100745 = r100743 * r100744;
        double r100746 = r100742 + r100745;
        double r100747 = asin(r100746);
        double r100748 = sin(r100747);
        double r100749 = r100741 * r100748;
        double r100750 = exp(r100749);
        double r100751 = log(r100750);
        double r100752 = r100740 - r100751;
        double r100753 = exp(r100752);
        double r100754 = log(r100753);
        double r100755 = atan2(r100739, r100754);
        double r100756 = r100731 + r100755;
        return r100756;
}

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-log-exp0.2

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

  4. Applied add-log-exp0.2

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

  5. Applied diff-log0.2

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

  6. Simplified0.2

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

  8. Applied add-log-exp0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 
(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))))))))))