Equirectangular approximation to distance on a great circle

Average Error: 39.2 → 39.0

Time: 1.5m

Precision: 64

Math

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \le -2.6219847008147082 \cdot 10^{208}:\\ \;\;\;\;R \cdot \left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \le -5.5068609279832755 \cdot 10^{47}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le -343.899314599184834:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le -1.39901496078204538 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 1.1505150553105537 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}{\lambda_1 + \lambda_2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 3.62734158549870918 \cdot 10^{-162}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.7202990652582576 \cdot 10^{32}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 1.8914867016819411 \cdot 10^{68}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.83361654007986611 \cdot 10^{153}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r392858 = R;
        double r392859 = lambda1;
        double r392860 = lambda2;
        double r392861 = r392859 - r392860;
        double r392862 = phi1;
        double r392863 = phi2;
        double r392864 = r392862 + r392863;
        double r392865 = 2.0;
        double r392866 = r392864 / r392865;
        double r392867 = cos(r392866);
        double r392868 = r392861 * r392867;
        double r392869 = r392868 * r392868;
        double r392870 = r392862 - r392863;
        double r392871 = r392870 * r392870;
        double r392872 = r392869 + r392871;
        double r392873 = sqrt(r392872);
        double r392874 = r392858 * r392873;
        return r392874;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r392875 = lambda2;
        double r392876 = -2.621984700814708e+208;
        bool r392877 = r392875 <= r392876;
        double r392878 = R;
        double r392879 = phi2;
        double r392880 = phi1;
        double r392881 = r392875 * r392880;
        double r392882 = r392879 * r392881;
        double r392883 = 0.5;
        double r392884 = r392879 + r392880;
        double r392885 = r392883 * r392884;
        double r392886 = cos(r392885);
        double r392887 = lambda1;
        double r392888 = 2.0;
        double r392889 = pow(r392887, r392888);
        double r392890 = r392886 * r392889;
        double r392891 = r392882 / r392890;
        double r392892 = r392879 * r392880;
        double r392893 = r392886 * r392887;
        double r392894 = r392892 / r392893;
        double r392895 = r392875 * r392886;
        double r392896 = r392894 + r392895;
        double r392897 = r392891 + r392896;
        double r392898 = -r392897;
        double r392899 = r392878 * r392898;
        double r392900 = -5.5068609279832755e+47;
        bool r392901 = r392875 <= r392900;
        double r392902 = r392879 - r392880;
        double r392903 = r392878 * r392902;
        double r392904 = -343.89931459918483;
        bool r392905 = r392875 <= r392904;
        double r392906 = r392887 - r392875;
        double r392907 = r392880 + r392879;
        double r392908 = 2.0;
        double r392909 = r392907 / r392908;
        double r392910 = cos(r392909);
        double r392911 = r392906 * r392910;
        double r392912 = r392910 * r392911;
        double r392913 = r392906 * r392912;
        double r392914 = r392880 - r392879;
        double r392915 = r392914 * r392914;
        double r392916 = r392913 + r392915;
        double r392917 = sqrt(r392916);
        double r392918 = r392878 * r392917;
        double r392919 = -1.3990149607820454e-33;
        bool r392920 = r392875 <= r392919;
        double r392921 = 1.1505150553105537e-291;
        bool r392922 = r392875 <= r392921;
        double r392923 = r392887 * r392887;
        double r392924 = r392875 * r392875;
        double r392925 = r392923 - r392924;
        double r392926 = r392925 * r392910;
        double r392927 = r392910 * r392926;
        double r392928 = r392887 + r392875;
        double r392929 = r392927 / r392928;
        double r392930 = r392906 * r392929;
        double r392931 = r392930 + r392915;
        double r392932 = sqrt(r392931);
        double r392933 = r392878 * r392932;
        double r392934 = 3.627341585498709e-162;
        bool r392935 = r392875 <= r392934;
        double r392936 = 2.7202990652582576e+32;
        bool r392937 = r392875 <= r392936;
        double r392938 = 1.891486701681941e+68;
        bool r392939 = r392875 <= r392938;
        double r392940 = 2.833616540079866e+153;
        bool r392941 = r392875 <= r392940;
        double r392942 = r392878 * r392897;
        double r392943 = r392941 ? r392918 : r392942;
        double r392944 = r392939 ? r392903 : r392943;
        double r392945 = r392937 ? r392918 : r392944;
        double r392946 = r392935 ? r392903 : r392945;
        double r392947 = r392922 ? r392933 : r392946;
        double r392948 = r392920 ? r392903 : r392947;
        double r392949 = r392905 ? r392918 : r392948;
        double r392950 = r392901 ? r392903 : r392949;
        double r392951 = r392877 ? r392899 : r392950;
        return r392951;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Split input into 5 regimes

2. if lambda2 < -2.621984700814708e+208

   1. Initial program 64.0

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Taylor expanded around inf 50.0

      \[\leadsto R \cdot \color{blue}{\left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)}\]

   if -2.621984700814708e+208 < lambda2 < -5.5068609279832755e+47 or -343.89931459918483 < lambda2 < -1.3990149607820454e-33 or 1.1505150553105537e-291 < lambda2 < 3.627341585498709e-162 or 2.7202990652582576e+32 < lambda2 < 1.891486701681941e+68

   1. Initial program 38.1

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Taylor expanded around 0 44.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]

   if -5.5068609279832755e+47 < lambda2 < -343.89931459918483 or 3.627341585498709e-162 < lambda2 < 2.7202990652582576e+32 or 1.891486701681941e+68 < lambda2 < 2.833616540079866e+153

   1. Initial program 31.6

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   2. Using strategy rm

   3. Applied associate-*l* 31.6

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   if -1.3990149607820454e-33 < lambda2 < 1.1505150553105537e-291

   1. Initial program 31.4

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   2. Using strategy rm

   3. Applied associate-*l* 31.4

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   4. Using strategy rm

   5. Applied flip-- 31.4

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\color{blue}{\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}} \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   6. Applied associate-*l/ 31.4

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \color{blue}{\frac{\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}{\lambda_1 + \lambda_2}}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   7. Applied associate-*r/ 31.4

      \[\leadsto R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}{\lambda_1 + \lambda_2}} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   if 2.833616540079866e+153 < lambda2

   1. Initial program 63.8

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Taylor expanded around -inf 52.6

      \[\leadsto R \cdot \color{blue}{\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 39.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \le -2.6219847008147082 \cdot 10^{208}:\\ \;\;\;\;R \cdot \left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \le -5.5068609279832755 \cdot 10^{47}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le -343.899314599184834:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le -1.39901496078204538 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 1.1505150553105537 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \frac{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}{\lambda_1 + \lambda_2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 3.62734158549870918 \cdot 10^{-162}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.7202990652582576 \cdot 10^{32}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \le 1.8914867016819411 \cdot 10^{68}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 2.83361654007986611 \cdot 10^{153}:\\ \;\;\;\;R \cdot \sqrt{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))