Equirectangular approximation to distance on a great circle

Average Error: 38.6 → 28.7

Time: 38.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} = -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \le 2.124445660313305561909369816980475178885 \cdot 10^{281}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array}\]

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r97295 = R;
        double r97296 = lambda1;
        double r97297 = lambda2;
        double r97298 = r97296 - r97297;
        double r97299 = phi1;
        double r97300 = phi2;
        double r97301 = r97299 + r97300;
        double r97302 = 2.0;
        double r97303 = r97301 / r97302;
        double r97304 = cos(r97303);
        double r97305 = r97298 * r97304;
        double r97306 = r97305 * r97305;
        double r97307 = r97299 - r97300;
        double r97308 = r97307 * r97307;
        double r97309 = r97306 + r97308;
        double r97310 = sqrt(r97309);
        double r97311 = r97295 * r97310;
        return r97311;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r97312 = R;
        double r97313 = lambda1;
        double r97314 = lambda2;
        double r97315 = r97313 - r97314;
        double r97316 = phi1;
        double r97317 = phi2;
        double r97318 = r97316 + r97317;
        double r97319 = 2.0;
        double r97320 = r97318 / r97319;
        double r97321 = cos(r97320);
        double r97322 = r97315 * r97321;
        double r97323 = r97322 * r97322;
        double r97324 = r97316 - r97317;
        double r97325 = r97324 * r97324;
        double r97326 = r97323 + r97325;
        double r97327 = sqrt(r97326);
        double r97328 = r97312 * r97327;
        double r97329 = -inf.0;
        bool r97330 = r97328 <= r97329;
        double r97331 = 2.1244456603133056e+281;
        bool r97332 = r97328 <= r97331;
        double r97333 = !r97332;
        bool r97334 = r97330 || r97333;
        double r97335 = r97317 - r97316;
        double r97336 = r97312 * r97335;
        double r97337 = 2.0;
        double r97338 = pow(r97314, r97337);
        double r97339 = 0.5;
        double r97340 = r97339 * r97317;
        double r97341 = cos(r97340);
        double r97342 = r97339 * r97316;
        double r97343 = cos(r97342);
        double r97344 = r97341 * r97343;
        double r97345 = sin(r97340);
        double r97346 = sin(r97342);
        double r97347 = r97345 * r97346;
        double r97348 = r97344 - r97347;
        double r97349 = pow(r97348, r97337);
        double r97350 = r97338 * r97349;
        double r97351 = r97317 * r97339;
        double r97352 = sin(r97351);
        double r97353 = r97316 * r97339;
        double r97354 = sin(r97353);
        double r97355 = r97352 * r97354;
        double r97356 = r97344 - r97355;
        double r97357 = pow(r97356, r97337);
        double r97358 = pow(r97313, r97337);
        double r97359 = r97357 * r97358;
        double r97360 = r97350 + r97359;
        double r97361 = r97317 + r97316;
        double r97362 = r97339 * r97361;
        double r97363 = cos(r97362);
        double r97364 = pow(r97363, r97337);
        double r97365 = r97364 * r97313;
        double r97366 = r97314 * r97365;
        double r97367 = r97337 * r97366;
        double r97368 = r97360 - r97367;
        double r97369 = r97368 + r97325;
        double r97370 = sqrt(r97369);
        double r97371 = r97312 * r97370;
        double r97372 = r97334 ? r97336 : r97371;
        return r97372;
}

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 (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))) < -inf.0 or 2.1244456603133056e+281 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))))

   1. Initial program 63.1

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

   2. Taylor expanded around 0 47.3

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

   if -inf.0 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))) < 2.1244456603133056e+281

   1. Initial program 1.9

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

   2. Taylor expanded around inf 1.9

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right)} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   3. Using strategy rm

   4. Applied distribute-lft-in 1.9

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

   5. Applied cos-sum 1.3

      \[\leadsto R \cdot \sqrt{\left(\left({\lambda_2}^{2} \cdot {\color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}^{2} + {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
   6. Using strategy rm

   7. Applied distribute-lft-in 1.3

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

   8. Applied cos-sum 0.7

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

   9. Simplified 0.7

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

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} = -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \le 2.124445660313305561909369816980475178885 \cdot 10^{281}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))))))