Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 1.1m

Precision: 64

• unused variable phi2

Math

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

↓

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

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r2107765 = lambda1;
        double r2107766 = theta;
        double r2107767 = sin(r2107766);
        double r2107768 = delta;
        double r2107769 = sin(r2107768);
        double r2107770 = r2107767 * r2107769;
        double r2107771 = phi1;
        double r2107772 = cos(r2107771);
        double r2107773 = r2107770 * r2107772;
        double r2107774 = cos(r2107768);
        double r2107775 = sin(r2107771);
        double r2107776 = r2107775 * r2107774;
        double r2107777 = r2107772 * r2107769;
        double r2107778 = cos(r2107766);
        double r2107779 = r2107777 * r2107778;
        double r2107780 = r2107776 + r2107779;
        double r2107781 = asin(r2107780);
        double r2107782 = sin(r2107781);
        double r2107783 = r2107775 * r2107782;
        double r2107784 = r2107774 - r2107783;
        double r2107785 = atan2(r2107773, r2107784);
        double r2107786 = r2107765 + r2107785;
        return r2107786;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r2107787 = phi1;
        double r2107788 = cos(r2107787);
        double r2107789 = delta;
        double r2107790 = sin(r2107789);
        double r2107791 = theta;
        double r2107792 = sin(r2107791);
        double r2107793 = r2107790 * r2107792;
        double r2107794 = r2107788 * r2107793;
        double r2107795 = cos(r2107789);
        double r2107796 = 3.0;
        double r2107797 = pow(r2107795, r2107796);
        double r2107798 = sin(r2107787);
        double r2107799 = r2107788 * r2107790;
        double r2107800 = cos(r2107791);
        double r2107801 = r2107799 * r2107800;
        double r2107802 = fma(r2107798, r2107795, r2107801);
        double r2107803 = asin(r2107802);
        double r2107804 = sin(r2107803);
        double r2107805 = r2107804 * r2107798;
        double r2107806 = expm1(r2107805);
        double r2107807 = log1p(r2107806);
        double r2107808 = pow(r2107807, r2107796);
        double r2107809 = r2107797 - r2107808;
        double r2107810 = r2107795 * r2107795;
        double r2107811 = r2107807 * r2107807;
        double r2107812 = exp(r2107803);
        double r2107813 = log(r2107812);
        double r2107814 = sin(r2107813);
        double r2107815 = r2107798 * r2107814;
        double r2107816 = expm1(r2107815);
        double r2107817 = log1p(r2107816);
        double r2107818 = r2107817 * r2107795;
        double r2107819 = r2107811 + r2107818;
        double r2107820 = r2107810 + r2107819;
        double r2107821 = r2107809 / r2107820;
        double r2107822 = atan2(r2107794, r2107821);
        double r2107823 = lambda1;
        double r2107824 = r2107822 + r2107823;
        return r2107824;
}

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 log1p-expm1-u 0.2

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

4. Simplified 0.2

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

6. Applied flip3-- 0.2

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

8. Applied add-log-exp 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))))))))))