Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 19.2s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left(\log \left(e^{{\left(\sin \phi_1\right)}^{6}}\right) \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sqrt[3]{{\left({\left(\sin \phi_1\right)}^{5}\right)}^{3}} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r123886 = lambda1;
        double r123887 = theta;
        double r123888 = sin(r123887);
        double r123889 = delta;
        double r123890 = sin(r123889);
        double r123891 = r123888 * r123890;
        double r123892 = phi1;
        double r123893 = cos(r123892);
        double r123894 = r123891 * r123893;
        double r123895 = cos(r123889);
        double r123896 = sin(r123892);
        double r123897 = r123896 * r123895;
        double r123898 = r123893 * r123890;
        double r123899 = cos(r123887);
        double r123900 = r123898 * r123899;
        double r123901 = r123897 + r123900;
        double r123902 = asin(r123901);
        double r123903 = sin(r123902);
        double r123904 = r123896 * r123903;
        double r123905 = r123895 - r123904;
        double r123906 = atan2(r123894, r123905);
        double r123907 = r123886 + r123906;
        return r123907;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r123908 = lambda1;
        double r123909 = theta;
        double r123910 = sin(r123909);
        double r123911 = delta;
        double r123912 = sin(r123911);
        double r123913 = r123910 * r123912;
        double r123914 = phi1;
        double r123915 = cos(r123914);
        double r123916 = r123913 * r123915;
        double r123917 = cos(r123911);
        double r123918 = 3.0;
        double r123919 = pow(r123917, r123918);
        double r123920 = pow(r123912, r123918);
        double r123921 = pow(r123915, r123918);
        double r123922 = cos(r123909);
        double r123923 = pow(r123922, r123918);
        double r123924 = sin(r123914);
        double r123925 = pow(r123924, r123918);
        double r123926 = r123923 * r123925;
        double r123927 = r123921 * r123926;
        double r123928 = r123920 * r123927;
        double r123929 = 6.0;
        double r123930 = pow(r123924, r123929);
        double r123931 = exp(r123930);
        double r123932 = log(r123931);
        double r123933 = r123932 * r123919;
        double r123934 = 5.0;
        double r123935 = pow(r123924, r123934);
        double r123936 = pow(r123935, r123918);
        double r123937 = cbrt(r123936);
        double r123938 = 2.0;
        double r123939 = pow(r123917, r123938);
        double r123940 = r123939 * r123922;
        double r123941 = r123937 * r123940;
        double r123942 = r123915 * r123941;
        double r123943 = r123912 * r123942;
        double r123944 = r123918 * r123943;
        double r123945 = pow(r123912, r123938);
        double r123946 = pow(r123915, r123938);
        double r123947 = 4.0;
        double r123948 = pow(r123924, r123947);
        double r123949 = pow(r123922, r123938);
        double r123950 = r123917 * r123949;
        double r123951 = r123948 * r123950;
        double r123952 = r123946 * r123951;
        double r123953 = r123945 * r123952;
        double r123954 = r123918 * r123953;
        double r123955 = r123944 + r123954;
        double r123956 = r123933 + r123955;
        double r123957 = r123928 + r123956;
        double r123958 = r123919 - r123957;
        double r123959 = r123917 * r123922;
        double r123960 = r123924 * r123959;
        double r123961 = r123915 * r123960;
        double r123962 = r123912 * r123961;
        double r123963 = pow(r123924, r123938);
        double r123964 = r123949 * r123963;
        double r123965 = r123946 * r123964;
        double r123966 = r123945 * r123965;
        double r123967 = r123963 * r123939;
        double r123968 = r123925 * r123959;
        double r123969 = r123915 * r123968;
        double r123970 = r123912 * r123969;
        double r123971 = r123938 * r123970;
        double r123972 = r123948 * r123939;
        double r123973 = r123971 + r123972;
        double r123974 = r123967 + r123973;
        double r123975 = r123966 + r123974;
        double r123976 = r123962 + r123975;
        double r123977 = r123939 + r123976;
        double r123978 = r123958 / r123977;
        double r123979 = atan2(r123916, r123978);
        double r123980 = r123908 + r123979;
        return r123980;
}

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 0.1

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

3. Applied flip3-- 0.2

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

4. Simplified 0.2

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

5. Taylor expanded around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left({\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)}}}\]
6. Using strategy rm

7. Applied add-log-exp 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left(\color{blue}{\log \left(e^{{\left(\sin \phi_1\right)}^{6}}\right)} \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)}}\]
8. Using strategy rm

9. Applied add-cbrt-cube 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left(\log \left(e^{{\left(\sin \phi_1\right)}^{6}}\right) \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\color{blue}{\sqrt[3]{\left({\left(\sin \phi_1\right)}^{5} \cdot {\left(\sin \phi_1\right)}^{5}\right) \cdot {\left(\sin \phi_1\right)}^{5}}} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)}}\]

10. Simplified 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left(\log \left(e^{{\left(\sin \phi_1\right)}^{6}}\right) \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sqrt[3]{\color{blue}{{\left({\left(\sin \phi_1\right)}^{5}\right)}^{3}}} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)}}\]

11. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left(\log \left(e^{{\left(\sin \phi_1\right)}^{6}}\right) \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sqrt[3]{{\left({\left(\sin \phi_1\right)}^{5}\right)}^{3}} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}{{\left(\cos delta\right)}^{2} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\sin \phi_1 \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos delta\right)}^{2} + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2020003 +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))))))))))