Average Error: 0.2 → 0.2
Time: 14.5s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right), \sin \phi_1, \sqrt[3]{{\left(\cos delta\right)}^{3}}\right) + \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \left(\left(-\sin \phi_1\right) + \sin \phi_1\right)}\]
\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}{\mathsf{fma}\left(-\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\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), \sin \phi_1, \sqrt[3]{{\left(\cos delta\right)}^{3}}\right) + \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 \left(\left(-\sin \phi_1\right) + \sin \phi_1\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r93762 = lambda1;
        double r93763 = theta;
        double r93764 = sin(r93763);
        double r93765 = delta;
        double r93766 = sin(r93765);
        double r93767 = r93764 * r93766;
        double r93768 = phi1;
        double r93769 = cos(r93768);
        double r93770 = r93767 * r93769;
        double r93771 = cos(r93765);
        double r93772 = sin(r93768);
        double r93773 = r93772 * r93771;
        double r93774 = r93769 * r93766;
        double r93775 = cos(r93763);
        double r93776 = r93774 * r93775;
        double r93777 = r93773 + r93776;
        double r93778 = asin(r93777);
        double r93779 = sin(r93778);
        double r93780 = r93772 * r93779;
        double r93781 = r93771 - r93780;
        double r93782 = atan2(r93770, r93781);
        double r93783 = r93762 + r93782;
        return r93783;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r93784 = lambda1;
        double r93785 = theta;
        double r93786 = sin(r93785);
        double r93787 = delta;
        double r93788 = sin(r93787);
        double r93789 = r93786 * r93788;
        double r93790 = phi1;
        double r93791 = cos(r93790);
        double r93792 = r93789 * r93791;
        double r93793 = sin(r93790);
        double r93794 = cos(r93787);
        double r93795 = r93793 * r93794;
        double r93796 = r93791 * r93788;
        double r93797 = cos(r93785);
        double r93798 = r93796 * r93797;
        double r93799 = r93795 + r93798;
        double r93800 = asin(r93799);
        double r93801 = expm1(r93800);
        double r93802 = log1p(r93801);
        double r93803 = sin(r93802);
        double r93804 = -r93803;
        double r93805 = 3.0;
        double r93806 = pow(r93794, r93805);
        double r93807 = cbrt(r93806);
        double r93808 = fma(r93804, r93793, r93807);
        double r93809 = sin(r93800);
        double r93810 = -r93793;
        double r93811 = r93810 + r93793;
        double r93812 = r93809 * r93811;
        double r93813 = r93808 + r93812;
        double r93814 = atan2(r93792, r93813);
        double r93815 = r93784 + r93814;
        return r93815;
}

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-cube-cbrt0.3

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

  4. Applied prod-diff0.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  8. Applied add-cbrt-cube0.3

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

  9. Simplified0.2

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

  11. Applied log1p-expm1-u0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}, \sin \phi_1, \sqrt[3]{{\left(\cos delta\right)}^{3}}\right) + \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \left(\left(-\sin \phi_1\right) + \sin \phi_1\right)}\]

  12. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right), \sin \phi_1, \sqrt[3]{{\left(\cos delta\right)}^{3}}\right) + \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \left(\left(-\sin \phi_1\right) + \sin \phi_1\right)}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))))))))))