Equirectangular approximation to distance on a great circle

Average Error: 39.3 → 29.9

Time: 9.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\phi_1 \le -3.159633297170386706240100837551728297364 \cdot 10^{143}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \le 1.725341016075638280892161287303491978895 \cdot 10^{102}:\\ \;\;\;\;\sqrt{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_1\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1566808 = R;
        double r1566809 = lambda1;
        double r1566810 = lambda2;
        double r1566811 = r1566809 - r1566810;
        double r1566812 = phi1;
        double r1566813 = phi2;
        double r1566814 = r1566812 + r1566813;
        double r1566815 = 2.0;
        double r1566816 = r1566814 / r1566815;
        double r1566817 = cos(r1566816);
        double r1566818 = r1566811 * r1566817;
        double r1566819 = r1566818 * r1566818;
        double r1566820 = r1566812 - r1566813;
        double r1566821 = r1566820 * r1566820;
        double r1566822 = r1566819 + r1566821;
        double r1566823 = sqrt(r1566822);
        double r1566824 = r1566808 * r1566823;
        return r1566824;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1566825 = phi1;
        double r1566826 = -3.1596332971703867e+143;
        bool r1566827 = r1566825 <= r1566826;
        double r1566828 = R;
        double r1566829 = -r1566828;
        double r1566830 = r1566825 * r1566829;
        double r1566831 = 1.7253410160756383e+102;
        bool r1566832 = r1566825 <= r1566831;
        double r1566833 = phi2;
        double r1566834 = r1566825 + r1566833;
        double r1566835 = 2.0;
        double r1566836 = r1566834 / r1566835;
        double r1566837 = cos(r1566836);
        double r1566838 = lambda1;
        double r1566839 = lambda2;
        double r1566840 = r1566838 - r1566839;
        double r1566841 = r1566840 * r1566837;
        double r1566842 = r1566837 * r1566841;
        double r1566843 = r1566842 * r1566840;
        double r1566844 = r1566825 - r1566833;
        double r1566845 = r1566844 * r1566844;
        double r1566846 = r1566843 + r1566845;
        double r1566847 = sqrt(r1566846);
        double r1566848 = r1566847 * r1566828;
        double r1566849 = r1566828 * r1566825;
        double r1566850 = r1566832 ? r1566848 : r1566849;
        double r1566851 = r1566827 ? r1566830 : r1566850;
        return r1566851;
}

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 3 regimes

2. if phi1 < -3.1596332971703867e+143

   1. Initial program 61.5

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

   2. Simplified 61.5

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

   4. Applied add-cube-cbrt 61.5

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right) \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)}} \cdot R\]

   5. Simplified 61.5

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)} \cdot R\]

   6. Simplified 61.5

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}}\right)} \cdot R\]
   7. Using strategy rm

   8. Applied add-exp-log 61.8

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)}\right)}} \cdot R\]
   9. Using strategy rm

   10. Applied pow1/2 61.8

       \[\leadsto e^{\log \color{blue}{\left({\left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)\right)}^{\frac{1}{2}}\right)}} \cdot R\]

   11. Applied log-pow 61.8

       \[\leadsto e^{\color{blue}{\frac{1}{2} \cdot \log \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)\right)}} \cdot R\]

   12. Applied exp-prod 61.8

       \[\leadsto \color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)\right)\right)}} \cdot R\]

   13. Taylor expanded around -inf 21.0

       \[\leadsto \color{blue}{\left(-1 \cdot \phi_1\right)} \cdot R\]

   if -3.1596332971703867e+143 < phi1 < 1.7253410160756383e+102

   1. Initial program 32.6

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

   2. Simplified 32.6

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

   if 1.7253410160756383e+102 < phi1

   1. Initial program 56.0

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

   2. Simplified 56.0

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

   4. Applied add-cube-cbrt 56.0

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)} \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right) \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)}} \cdot R\]

   5. Simplified 56.0

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)} \cdot R\]

   6. Simplified 56.0

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}}\right)} \cdot R\]
   7. Using strategy rm

   8. Applied add-exp-log 56.7

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)}\right)}} \cdot R\]
   9. Using strategy rm

   10. Applied pow1/2 56.7

       \[\leadsto e^{\log \color{blue}{\left({\left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)\right)}^{\frac{1}{2}}\right)}} \cdot R\]

   11. Applied log-pow 56.7

       \[\leadsto e^{\color{blue}{\frac{1}{2} \cdot \log \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)\right)}} \cdot R\]

   12. Applied exp-prod 56.8

       \[\leadsto \color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \sqrt[3]{\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right)\right)\right)}} \cdot R\]

   13. Taylor expanded around inf 23.3

       \[\leadsto \color{blue}{\phi_1} \cdot R\]
3. Recombined 3 regimes into one program.

4. Final simplification 29.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \le -3.159633297170386706240100837551728297364 \cdot 10^{143}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \le 1.725341016075638280892161287303491978895 \cdot 10^{102}:\\ \;\;\;\;\sqrt{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))