Equirectangular approximation to distance on a great circle

Average Error: 39.1 → 37.7

Time: 1.5m

Precision: 64

Math

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \le -2.4779403317554257 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\lambda_1} \cdot \frac{\phi_1}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} + \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \lambda_2\right) + \frac{\lambda_2}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \cdot \frac{\phi_2 \cdot \phi_1}{\lambda_1 \cdot \lambda_1}\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_2 \le -3.198787512097014 \cdot 10^{+178}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le -1.342595736819271 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\lambda_1} \cdot \frac{\phi_1}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} + \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \lambda_2\right) + \frac{\lambda_2}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \cdot \frac{\phi_2 \cdot \phi_1}{\lambda_1 \cdot \lambda_1}\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_2 \le -4.16714048268465 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\lambda_2 \le -1.7692204660992532 \cdot 10^{-87}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 1.1659851827648714 \cdot 10^{+126}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r6061688 = R;
        double r6061689 = lambda1;
        double r6061690 = lambda2;
        double r6061691 = r6061689 - r6061690;
        double r6061692 = phi1;
        double r6061693 = phi2;
        double r6061694 = r6061692 + r6061693;
        double r6061695 = 2.0;
        double r6061696 = r6061694 / r6061695;
        double r6061697 = cos(r6061696);
        double r6061698 = r6061691 * r6061697;
        double r6061699 = r6061698 * r6061698;
        double r6061700 = r6061692 - r6061693;
        double r6061701 = r6061700 * r6061700;
        double r6061702 = r6061699 + r6061701;
        double r6061703 = sqrt(r6061702);
        double r6061704 = r6061688 * r6061703;
        return r6061704;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r6061705 = lambda2;
        double r6061706 = -2.4779403317554257e+223;
        bool r6061707 = r6061705 <= r6061706;
        double r6061708 = phi2;
        double r6061709 = lambda1;
        double r6061710 = r6061708 / r6061709;
        double r6061711 = phi1;
        double r6061712 = r6061708 + r6061711;
        double r6061713 = 0.5;
        double r6061714 = r6061712 * r6061713;
        double r6061715 = cos(r6061714);
        double r6061716 = r6061711 / r6061715;
        double r6061717 = r6061710 * r6061716;
        double r6061718 = r6061715 * r6061705;
        double r6061719 = r6061717 + r6061718;
        double r6061720 = r6061705 / r6061715;
        double r6061721 = r6061708 * r6061711;
        double r6061722 = r6061709 * r6061709;
        double r6061723 = r6061721 / r6061722;
        double r6061724 = r6061720 * r6061723;
        double r6061725 = r6061719 + r6061724;
        double r6061726 = R;
        double r6061727 = -r6061726;
        double r6061728 = r6061725 * r6061727;
        double r6061729 = -3.198787512097014e+178;
        bool r6061730 = r6061705 <= r6061729;
        double r6061731 = r6061708 - r6061711;
        double r6061732 = r6061726 * r6061731;
        double r6061733 = -1.342595736819271e+154;
        bool r6061734 = r6061705 <= r6061733;
        double r6061735 = -4.16714048268465e-24;
        bool r6061736 = r6061705 <= r6061735;
        double r6061737 = r6061711 - r6061708;
        double r6061738 = r6061737 * r6061737;
        double r6061739 = 2.0;
        double r6061740 = r6061712 / r6061739;
        double r6061741 = cos(r6061740);
        double r6061742 = r6061709 - r6061705;
        double r6061743 = r6061741 * r6061742;
        double r6061744 = r6061742 * r6061743;
        double r6061745 = r6061741 * r6061744;
        double r6061746 = r6061738 + r6061745;
        double r6061747 = sqrt(r6061746);
        double r6061748 = r6061726 * r6061747;
        double r6061749 = -1.7692204660992532e-87;
        bool r6061750 = r6061705 <= r6061749;
        double r6061751 = 1.1659851827648714e+126;
        bool r6061752 = r6061705 <= r6061751;
        double r6061753 = r6061752 ? r6061748 : r6061732;
        double r6061754 = r6061750 ? r6061732 : r6061753;
        double r6061755 = r6061736 ? r6061748 : r6061754;
        double r6061756 = r6061734 ? r6061728 : r6061755;
        double r6061757 = r6061730 ? r6061732 : r6061756;
        double r6061758 = r6061707 ? r6061728 : r6061757;
        return r6061758;
}

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 lambda2 < -2.4779403317554257e+223 or -3.198787512097014e+178 < lambda2 < -1.342595736819271e+154

   1. Initial program 64.0

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   2. Using strategy rm

   3. Applied associate-*r* 64.0

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 64.0

      \[\leadsto R \cdot \sqrt{\left(\left(\color{blue}{\left(\left(\sqrt[3]{\lambda_1 - \lambda_2} \cdot \sqrt[3]{\lambda_1 - \lambda_2}\right) \cdot \sqrt[3]{\lambda_1 - \lambda_2}\right)} \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   6. Applied associate-*l* 64.0

      \[\leadsto R \cdot \sqrt{\left(\color{blue}{\left(\left(\sqrt[3]{\lambda_1 - \lambda_2} \cdot \sqrt[3]{\lambda_1 - \lambda_2}\right) \cdot \left(\sqrt[3]{\lambda_1 - \lambda_2} \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   7. Taylor expanded around inf 50.1

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

   8. Simplified 49.0

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

   if -2.4779403317554257e+223 < lambda2 < -3.198787512097014e+178 or -4.16714048268465e-24 < lambda2 < -1.7692204660992532e-87 or 1.1659851827648714e+126 < lambda2

   1. Initial program 53.0

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Taylor expanded around 0 51.2

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

   if -1.342595736819271e+154 < lambda2 < -4.16714048268465e-24 or -1.7692204660992532e-87 < lambda2 < 1.1659851827648714e+126

   1. Initial program 32.3

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   2. Using strategy rm

   3. Applied associate-*r* 32.3

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2.0}\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 37.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \le -2.4779403317554257 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\lambda_1} \cdot \frac{\phi_1}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} + \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \lambda_2\right) + \frac{\lambda_2}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \cdot \frac{\phi_2 \cdot \phi_1}{\lambda_1 \cdot \lambda_1}\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_2 \le -3.198787512097014 \cdot 10^{+178}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le -1.342595736819271 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\lambda_1} \cdot \frac{\phi_1}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} + \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \lambda_2\right) + \frac{\lambda_2}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \cdot \frac{\phi_2 \cdot \phi_1}{\lambda_1 \cdot \lambda_1}\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_2 \le -4.16714048268465 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\lambda_2 \le -1.7692204660992532 \cdot 10^{-87}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_2 \le 1.1659851827648714 \cdot 10^{+126}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2.0}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))))))