Equirectangular approximation to distance on a great circle

Average Error: 39.1 → 33.8

Time: 9.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -4.62720478876289348 \cdot 10^{154}:\\ \;\;\;\;R \cdot e^{\frac{1}{2} \cdot \left(\log \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2}\right) - 2 \cdot \log \left(\frac{-1}{\lambda_1}\right)\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 4.0131744321398698 \cdot 10^{77}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 5.5353460225548844 \cdot 10^{147}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 2.24306291292195173 \cdot 10^{161}:\\ \;\;\;\;R \cdot \left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\frac{1}{2} \cdot \left(\log \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2}\right) - 2 \cdot \log \left(\frac{1}{\lambda_1}\right)\right)}\\ \end{array}\]

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 (((lambda1 - lambda2) <= -4.6272047887628935e+154)) {
		VAR = (R * exp((0.5 * (log(pow(cos((0.5 * (phi2 + phi1))), 2.0)) - (2.0 * log((-1.0 / lambda1)))))));
	} else {
		double VAR_1;
		if (((lambda1 - lambda2) <= 4.01317443213987e+77)) {
			VAR_1 = (R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2)))));
		} else {
			double VAR_2;
			if (((lambda1 - lambda2) <= 5.5353460225548844e+147)) {
				VAR_2 = (R * (phi2 - phi1));
			} else {
				double VAR_3;
				if (((lambda1 - lambda2) <= 2.2430629129219517e+161)) {
					VAR_3 = (R * -(((phi2 * (lambda2 * phi1)) / (cos((0.5 * (phi2 + phi1))) * pow(lambda1, 2.0))) + (((phi2 * phi1) / (cos((0.5 * (phi2 + phi1))) * lambda1)) + (lambda2 * cos((0.5 * (phi2 + phi1)))))));
				} else {
					VAR_3 = (R * exp((0.5 * (log(pow(cos((0.5 * (phi2 + phi1))), 2.0)) - (2.0 * log((1.0 / lambda1)))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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 5 regimes

2. if (- lambda1 lambda2) < -4.6272047887628935e+154

   1. Initial program 64.0

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

   3. Applied add-exp-log 64.0

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

   5. Applied pow1/2 64.0

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

   6. Applied log-pow 64.0

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

   7. Applied exp-prod 64.0

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

   8. Taylor expanded around -inf 44.9

      \[\leadsto R \cdot \color{blue}{e^{\frac{1}{2} \cdot \left(\log \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2}\right) - 2 \cdot \log \left(\frac{-1}{\lambda_1}\right)\right)}}\]

   if -4.6272047887628935e+154 < (- lambda1 lambda2) < 4.01317443213987e+77

   1. Initial program 23.1

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

   if 4.01317443213987e+77 < (- lambda1 lambda2) < 5.5353460225548844e+147

   1. Initial program 27.0

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

   2. Taylor expanded around 0 47.5

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

   if 5.5353460225548844e+147 < (- lambda1 lambda2) < 2.2430629129219517e+161

   1. Initial program 46.3

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

   2. Taylor expanded around inf 57.2

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

   if 2.2430629129219517e+161 < (- lambda1 lambda2)

   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. Using strategy rm

   3. Applied add-exp-log 64.0

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

   5. Applied pow1/2 64.0

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

   6. Applied log-pow 64.0

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

   7. Applied exp-prod 64.0

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

   8. Taylor expanded around inf 44.4

      \[\leadsto R \cdot \color{blue}{e^{\frac{1}{2} \cdot \left(\log \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2}\right) - 2 \cdot \log \left(\frac{1}{\lambda_1}\right)\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 33.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -4.62720478876289348 \cdot 10^{154}:\\ \;\;\;\;R \cdot e^{\frac{1}{2} \cdot \left(\log \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2}\right) - 2 \cdot \log \left(\frac{-1}{\lambda_1}\right)\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 4.0131744321398698 \cdot 10^{77}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 5.5353460225548844 \cdot 10^{147}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 2.24306291292195173 \cdot 10^{161}:\\ \;\;\;\;R \cdot \left(-\left(\frac{\phi_2 \cdot \left(\lambda_2 \cdot \phi_1\right)}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot {\lambda_1}^{2}} + \left(\frac{\phi_2 \cdot \phi_1}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1} + \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\frac{1}{2} \cdot \left(\log \left({\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)}^{2}\right) - 2 \cdot \log \left(\frac{1}{\lambda_1}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(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))))))