Equirectangular approximation to distance on a great circle

Average Error: 39.2 → 28.9

Time: 9.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} = -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \le 1.40199345202389236 \cdot 10^{306}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array}\]

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2)))));
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double VAR;
	if ((((R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))))) <= -inf.0) || !((R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))))) <= 1.4019934520238924e+306))) {
		VAR = (R * (phi2 - phi1));
	} else {
		VAR = (R * sqrt(((((pow(lambda2, 2.0) * pow(((cos((phi2 * 0.5)) * cos((phi1 * 0.5))) - (sin((phi2 * 0.5)) * sin((phi1 * 0.5)))), 2.0)) + (pow(cos((0.5 * (phi2 + phi1))), 2.0) * pow(lambda1, 2.0))) - (2.0 * (lambda2 * (pow(cos((0.5 * (phi2 + phi1))), 2.0) * lambda1)))) + ((phi1 - phi2) * (phi1 - phi2)))));
	}
	return VAR;
}

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 1.4019934520238924e+306 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))))

   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 0 47.2

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

   if -inf.0 < (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))) < 1.4019934520238924e+306

   1. Initial program 1.8

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

   2. Taylor expanded around inf 1.8

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

   4. Applied distribute-lft-in 1.8

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

   5. Applied cos-sum 1.2

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

   6. Simplified 1.2

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

   7. Simplified 1.2

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

4. Final simplification 28.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} = -\infty \lor \neg \left(R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \le 1.40199345202389236 \cdot 10^{306}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt{\left(\left({\lambda_2}^{2} \cdot {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2} + {\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot {\lambda_1}^{2}\right) - 2 \cdot \left(\lambda_2 \cdot \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2} \cdot \lambda_1\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))))