Equirectangular approximation to distance on a great circle

Average Error: 39.5 → 23.1

Time: 23.2s

Precision: binary64

\[[lambda1, lambda2]=\mathsf{sort}([lambda1, lambda2])\]

\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(0.5 \cdot \phi_2\right)\\ t_3 := t_1 \cdot t_2\\ t_4 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3.250801521433806 \cdot 10^{+194}:\\ \;\;\;\;R \cdot \left(\left(t_4 \cdot t_0 - t_3\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_5 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.4797657161965226 \cdot 10^{+186}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.858726378974967 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \left(\left(t_1 \cdot \left(\lambda_1 \cdot t_2\right) + t_4 \cdot \left(\lambda_2 \cdot t_0\right)\right) - \left(\lambda_2 \cdot \log \left(e^{t_3}\right) + t_4 \cdot \left(\lambda_1 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.91600759841242 \cdot 10^{+52} \lor \neg \left(\lambda_1 - \lambda_2 \leq -1.3869924756902689 \cdot 10^{+45}\right):\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_6 := \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ t_7 := \sqrt[3]{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(t_6 \cdot t_6\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ R \cdot \left(\left|t_7\right| \cdot \sqrt{t_7}\right) \end{array}\\ \end{array}\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (sqrt
   (+
    (*
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
    (* (- phi1 phi2) (- phi1 phi2))))))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi2)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (sin (* 0.5 phi2)))
        (t_3 (* t_1 t_2))
        (t_4 (cos (* 0.5 phi1))))
   (if (<= (- lambda1 lambda2) -3.250801521433806e+194)
     (* R (* (- (* t_4 t_0) t_3) (- lambda2 lambda1)))
     (let* ((t_5 (* R (- phi2 phi1))))
       (if (<= (- lambda1 lambda2) -1.4797657161965226e+186)
         t_5
         (if (<= (- lambda1 lambda2) -1.858726378974967e+135)
           (*
            R
            (-
             (+ (* t_1 (* lambda1 t_2)) (* t_4 (* lambda2 t_0)))
             (+ (* lambda2 (log (exp t_3))) (* t_4 (* lambda1 t_0)))))
           (if (or (<= (- lambda1 lambda2) -2.91600759841242e+52)
                   (not (<= (- lambda1 lambda2) -1.3869924756902689e+45)))
             t_5
             (let* ((t_6 (cos (/ (+ phi1 phi2) 2.0)))
                    (t_7
                     (cbrt
                      (+
                       (* (pow (- lambda1 lambda2) 2.0) (* t_6 t_6))
                       (* (- phi1 phi2) (- phi1 phi2))))))
               (* R (* (fabs t_7) (sqrt t_7)))))))))))

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 t_0 = cos(0.5 * phi2);
	double t_1 = sin(0.5 * phi1);
	double t_2 = sin(0.5 * phi2);
	double t_3 = t_1 * t_2;
	double t_4 = cos(0.5 * phi1);
	double tmp;
	if ((lambda1 - lambda2) <= -3.250801521433806e+194) {
		tmp = R * (((t_4 * t_0) - t_3) * (lambda2 - lambda1));
	} else {
		double t_5 = R * (phi2 - phi1);
		double tmp_1;
		if ((lambda1 - lambda2) <= -1.4797657161965226e+186) {
			tmp_1 = t_5;
		} else if ((lambda1 - lambda2) <= -1.858726378974967e+135) {
			tmp_1 = R * (((t_1 * (lambda1 * t_2)) + (t_4 * (lambda2 * t_0))) - ((lambda2 * log(exp(t_3))) + (t_4 * (lambda1 * t_0))));
		} else if (((lambda1 - lambda2) <= -2.91600759841242e+52) || !((lambda1 - lambda2) <= -1.3869924756902689e+45)) {
			tmp_1 = t_5;
		} else {
			double t_6 = cos((phi1 + phi2) / 2.0);
			double t_7 = cbrt((pow((lambda1 - lambda2), 2.0) * (t_6 * t_6)) + ((phi1 - phi2) * (phi1 - phi2)));
			tmp_1 = R * (fabs(t_7) * sqrt(t_7));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 4 regimes

2. if (-.f64 lambda1 lambda2) < -3.25080152143380584e194

   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 -inf 33.0

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

   3. Simplified 33.0

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

   5. Applied distribute-rgt-in_binary64 33.0

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

   6. Applied cos-sum_binary64 28.5

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

   7. Simplified 28.5

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

   8. Simplified 28.5

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

   10. Applied cancel-sign-sub-inv_binary64 28.5

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

   11. Simplified 28.5

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

   if -3.25080152143380584e194 < (-.f64 lambda1 lambda2) < -1.4797657161965226e186 or -1.8587263789749671e135 < (-.f64 lambda1 lambda2) < -2.9160075984124197e52 or -1.3869924756902689e45 < (-.f64 lambda1 lambda2)

   1. Initial program 25.2

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

   2. Taylor expanded around -inf 18.1

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

   if -1.4797657161965226e186 < (-.f64 lambda1 lambda2) < -1.8587263789749671e135

   1. Initial program 51.2

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

   2. Taylor expanded around -inf 35.9

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

   3. Simplified 35.9

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

   5. Applied distribute-rgt-in_binary64 35.9

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

   6. Applied cos-sum_binary64 33.8

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

   7. Simplified 33.8

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

   8. Simplified 33.8

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

   9. Taylor expanded around 0 33.8

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

   10. Simplified 33.8

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

   12. Applied add-log-exp_binary64 33.8

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

   if -2.9160075984124197e52 < (-.f64 lambda1 lambda2) < -1.3869924756902689e45

   1. Initial program 22.8

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

   3. Applied add-cube-cbrt_binary64 23.2

      \[\leadsto R \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right) \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}} \]

   4. Applied sqrt-prod_binary64 23.2

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot \sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}} \cdot \sqrt{\sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right)} \]

   5. Simplified 23.2

      \[\leadsto R \cdot \left(\color{blue}{\left|\sqrt[3]{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right|} \cdot \sqrt{\sqrt[3]{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right) \]

   6. Simplified 23.2

      \[\leadsto R \cdot \left(\left|\sqrt[3]{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right| \cdot \color{blue}{\sqrt{\sqrt[3]{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3.250801521433806 \cdot 10^{+194}:\\ \;\;\;\;R \cdot \left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.4797657161965226 \cdot 10^{+186}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -1.858726378974967 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \left(\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 \cdot \sin \left(0.5 \cdot \phi_2\right)\right) + \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right) - \left(\lambda_2 \cdot \log \left(e^{\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)}\right) + \cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -2.91600759841242 \cdot 10^{+52} \lor \neg \left(\lambda_1 - \lambda_2 \leq -1.3869924756902689 \cdot 10^{+45}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left|\sqrt[3]{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right| \cdot \sqrt{\sqrt[3]{{\left(\lambda_1 - \lambda_2\right)}^{2} \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))