Equirectangular approximation to distance on a great circle

Average Error: 39.6 → 29.2

Time: 12.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 1.443474666294679403718523771330180153971 \cdot 10^{306}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r87824 = R;
        double r87825 = lambda1;
        double r87826 = lambda2;
        double r87827 = r87825 - r87826;
        double r87828 = phi1;
        double r87829 = phi2;
        double r87830 = r87828 + r87829;
        double r87831 = 2.0;
        double r87832 = r87830 / r87831;
        double r87833 = cos(r87832);
        double r87834 = r87827 * r87833;
        double r87835 = r87834 * r87834;
        double r87836 = r87828 - r87829;
        double r87837 = r87836 * r87836;
        double r87838 = r87835 + r87837;
        double r87839 = sqrt(r87838);
        double r87840 = r87824 * r87839;
        return r87840;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r87841 = lambda1;
        double r87842 = lambda2;
        double r87843 = r87841 - r87842;
        double r87844 = phi1;
        double r87845 = phi2;
        double r87846 = r87844 + r87845;
        double r87847 = 2.0;
        double r87848 = r87846 / r87847;
        double r87849 = cos(r87848);
        double r87850 = r87843 * r87849;
        double r87851 = r87850 * r87850;
        double r87852 = r87844 - r87845;
        double r87853 = r87852 * r87852;
        double r87854 = r87851 + r87853;
        double r87855 = 1.4434746662946794e+306;
        bool r87856 = r87854 <= r87855;
        double r87857 = R;
        double r87858 = sqrt(r87854);
        double r87859 = r87857 * r87858;
        double r87860 = r87845 - r87844;
        double r87861 = r87857 * r87860;
        double r87862 = r87856 ? r87859 : r87861;
        return r87862;
}

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 2 regimes

2. if (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))) < 1.4434746662946794e+306

   1. Initial program 1.9

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

   if 1.4434746662946794e+306 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))

   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 0 46.7

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 29.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 1.443474666294679403718523771330180153971 \cdot 10^{306}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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))))))