Equirectangular approximation to distance on a great circle

Average Error: 38.9 → 32.3

Time: 1.0min

Precision: binary64

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}\;\phi_2 \leq -623511300567565.1:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq -1.1432798135583093 \cdot 10^{-70}:\\ \;\;\;\;R \cdot \sqrt{{\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)}\\ \mathbf{elif}\;\phi_2 \leq -7.263758197549583 \cdot 10^{-158}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -1.4628256157457647 \cdot 10^{-182}:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq -1.3217438363980596 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \left(-\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -1.1623653209688043 \cdot 10^{-237}:\\ \;\;\;\;R \cdot \left(\left(0.5 \cdot \left(\frac{\phi_1 \cdot \phi_1}{\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} + \frac{\lambda_2 \cdot \left(\phi_2 \cdot \phi_2\right)}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(\lambda_1 \cdot \lambda_1\right)}\right) + \left(\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) + 0.5 \cdot \left(\frac{\lambda_2 \cdot \left(\phi_1 \cdot \phi_1\right)}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(\lambda_1 \cdot \lambda_1\right)} + \frac{\phi_2 \cdot \phi_2}{\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)}\right)\right)\right) - \left(\frac{\phi_2}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \cdot \left(\frac{\phi_1 \cdot \lambda_2}{\lambda_1 \cdot \lambda_1} + \frac{\phi_1}{\lambda_1}\right) + \lambda_2 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.303407807589607 \cdot 10^{-287}:\\ \;\;\;\;R \cdot \sqrt{{\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)}\\ \mathbf{elif}\;\phi_2 \leq 5.38115258344735 \cdot 10^{-250}:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 4.962564261877504 \cdot 10^{-88}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + {\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}^{2}}\\ \mathbf{elif}\;\phi_2 \leq 9.269945814378616 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.553943994428835 \cdot 10^{+59}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_2 \cdot \phi_2 + {\cos \left(\phi_2 \cdot 0.5\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1 + \lambda_2 \cdot \lambda_2\right)\right) - 2 \cdot \left({\cos \left(\phi_2 \cdot 0.5\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \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
 (if (<= phi2 -623511300567565.1)
   (* R (- phi1 phi2))
   (if (<= phi2 -1.1432798135583093e-70)
     (*
      R
      (sqrt
       (+
        (*
         (pow (- lambda1 lambda2) 2.0)
         (* (cos (/ (+ phi2 phi1) 2.0)) (cos (/ (+ phi2 phi1) 2.0))))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (if (<= phi2 -7.263758197549583e-158)
       (* R (* (- lambda1 lambda2) (cos (* (+ phi2 phi1) 0.5))))
       (if (<= phi2 -1.4628256157457647e-182)
         (* R (- phi1 phi2))
         (if (<= phi2 -1.3217438363980596e-213)
           (* R (- (* lambda1 (cos (* (+ phi2 phi1) 0.5)))))
           (if (<= phi2 -1.1623653209688043e-237)
             (*
              R
              (-
               (+
                (*
                 0.5
                 (+
                  (/ (* phi1 phi1) (* lambda1 (cos (* (+ phi2 phi1) 0.5))))
                  (/
                   (* lambda2 (* phi2 phi2))
                   (* (cos (* (+ phi2 phi1) 0.5)) (* lambda1 lambda1)))))
                (+
                 (* lambda1 (cos (* (+ phi2 phi1) 0.5)))
                 (*
                  0.5
                  (+
                   (/
                    (* lambda2 (* phi1 phi1))
                    (* (cos (* (+ phi2 phi1) 0.5)) (* lambda1 lambda1)))
                   (/
                    (* phi2 phi2)
                    (* lambda1 (cos (* (+ phi2 phi1) 0.5))))))))
               (+
                (*
                 (/ phi2 (cos (* (+ phi2 phi1) 0.5)))
                 (+ (/ (* phi1 lambda2) (* lambda1 lambda1)) (/ phi1 lambda1)))
                (* lambda2 (cos (* (+ phi2 phi1) 0.5))))))
             (if (<= phi2 9.303407807589607e-287)
               (*
                R
                (sqrt
                 (+
                  (*
                   (pow (- lambda1 lambda2) 2.0)
                   (* (cos (/ (+ phi2 phi1) 2.0)) (cos (/ (+ phi2 phi1) 2.0))))
                  (* (- phi1 phi2) (- phi1 phi2)))))
               (if (<= phi2 5.38115258344735e-250)
                 (* R (- phi1 phi2))
                 (if (<= phi2 4.962564261877504e-88)
                   (*
                    R
                    (sqrt
                     (+
                      (* (- phi1 phi2) (- phi1 phi2))
                      (pow
                       (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
                       2.0))))
                   (if (<= phi2 9.269945814378616e-22)
                     (* R (* (cos (* (+ phi2 phi1) 0.5)) (- lambda2 lambda1)))
                     (if (<= phi2 2.553943994428835e+59)
                       (*
                        R
                        (sqrt
                         (-
                          (+
                           (* phi2 phi2)
                           (*
                            (pow (cos (* phi2 0.5)) 2.0)
                            (+ (* lambda1 lambda1) (* lambda2 lambda2))))
                          (*
                           2.0
                           (*
                            (pow (cos (* phi2 0.5)) 2.0)
                            (* lambda1 lambda2))))))
                       (* R (- phi2 phi1))))))))))))))

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 tmp;
	if (phi2 <= -623511300567565.1) {
		tmp = R * (phi1 - phi2);
	} else if (phi2 <= -1.1432798135583093e-70) {
		tmp = R * sqrt((pow((lambda1 - lambda2), 2.0) * (cos((phi2 + phi1) / 2.0) * cos((phi2 + phi1) / 2.0))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if (phi2 <= -7.263758197549583e-158) {
		tmp = R * ((lambda1 - lambda2) * cos((phi2 + phi1) * 0.5));
	} else if (phi2 <= -1.4628256157457647e-182) {
		tmp = R * (phi1 - phi2);
	} else if (phi2 <= -1.3217438363980596e-213) {
		tmp = R * -(lambda1 * cos((phi2 + phi1) * 0.5));
	} else if (phi2 <= -1.1623653209688043e-237) {
		tmp = R * (((0.5 * (((phi1 * phi1) / (lambda1 * cos((phi2 + phi1) * 0.5))) + ((lambda2 * (phi2 * phi2)) / (cos((phi2 + phi1) * 0.5) * (lambda1 * lambda1))))) + ((lambda1 * cos((phi2 + phi1) * 0.5)) + (0.5 * (((lambda2 * (phi1 * phi1)) / (cos((phi2 + phi1) * 0.5) * (lambda1 * lambda1))) + ((phi2 * phi2) / (lambda1 * cos((phi2 + phi1) * 0.5))))))) - (((phi2 / cos((phi2 + phi1) * 0.5)) * (((phi1 * lambda2) / (lambda1 * lambda1)) + (phi1 / lambda1))) + (lambda2 * cos((phi2 + phi1) * 0.5))));
	} else if (phi2 <= 9.303407807589607e-287) {
		tmp = R * sqrt((pow((lambda1 - lambda2), 2.0) * (cos((phi2 + phi1) / 2.0) * cos((phi2 + phi1) / 2.0))) + ((phi1 - phi2) * (phi1 - phi2)));
	} else if (phi2 <= 5.38115258344735e-250) {
		tmp = R * (phi1 - phi2);
	} else if (phi2 <= 4.962564261877504e-88) {
		tmp = R * sqrt(((phi1 - phi2) * (phi1 - phi2)) + pow(((lambda1 - lambda2) * cos((phi2 + phi1) / 2.0)), 2.0));
	} else if (phi2 <= 9.269945814378616e-22) {
		tmp = R * (cos((phi2 + phi1) * 0.5) * (lambda2 - lambda1));
	} else if (phi2 <= 2.553943994428835e+59) {
		tmp = R * sqrt(((phi2 * phi2) + (pow(cos(phi2 * 0.5), 2.0) * ((lambda1 * lambda1) + (lambda2 * lambda2)))) - (2.0 * (pow(cos(phi2 * 0.5), 2.0) * (lambda1 * lambda2))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 9 regimes

2. if phi2 < -623511300567565.125 or -7.26375819754958333e-158 < phi2 < -1.46282561574576469e-182 or 9.3034078075896073e-287 < phi2 < 5.3811525834473501e-250

   1. Initial program 45.3

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

   2. Taylor expanded around inf 30.8

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

   if -623511300567565.125 < phi2 < -1.1432798135583093e-70 or -1.16236532096880427e-237 < phi2 < 9.3034078075896073e-287

   1. Initial program 31.5

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

   3. Applied swap-sqr_binary64_1750 31.5

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

   4. Simplified 31.5

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

   if -1.1432798135583093e-70 < phi2 < -7.26375819754958333e-158

   1. Initial program 31.3

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

   2. Taylor expanded around inf 40.0

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

   3. Simplified 40.0

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

   if -1.46282561574576469e-182 < phi2 < -1.3217438363980596e-213

   1. Initial program 31.7

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

   2. Taylor expanded around -inf 50.2

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

   3. Simplified 50.2

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

   if -1.3217438363980596e-213 < phi2 < -1.16236532096880427e-237

   1. Initial program 32.2

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

   2. Taylor expanded around inf 52.7

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

   3. Simplified 53.1

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

   if 5.3811525834473501e-250 < phi2 < 4.9625642618775038e-88

   1. Initial program 31.8

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

   3. Applied pow2_binary64_1864 31.8

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

   if 4.9625642618775038e-88 < phi2 < 9.26994581437861598e-22

   1. Initial program 29.3

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

   2. Taylor expanded around -inf 42.4

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

   3. Simplified 42.4

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

   if 9.26994581437861598e-22 < phi2 < 2.55394399442883498e59

   1. Initial program 31.2

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

   2. Taylor expanded around 0 39.1

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

   3. Simplified 39.1

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

   if 2.55394399442883498e59 < phi2

   1. Initial program 51.3

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

   2. Taylor expanded around -inf 21.6

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
3. Recombined 9 regimes into one program.

4. Final simplification 32.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -623511300567565.1:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq -1.1432798135583093 \cdot 10^{-70}:\\ \;\;\;\;R \cdot \sqrt{{\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)}\\ \mathbf{elif}\;\phi_2 \leq -7.263758197549583 \cdot 10^{-158}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -1.4628256157457647 \cdot 10^{-182}:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq -1.3217438363980596 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \left(-\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -1.1623653209688043 \cdot 10^{-237}:\\ \;\;\;\;R \cdot \left(\left(0.5 \cdot \left(\frac{\phi_1 \cdot \phi_1}{\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} + \frac{\lambda_2 \cdot \left(\phi_2 \cdot \phi_2\right)}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(\lambda_1 \cdot \lambda_1\right)}\right) + \left(\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) + 0.5 \cdot \left(\frac{\lambda_2 \cdot \left(\phi_1 \cdot \phi_1\right)}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(\lambda_1 \cdot \lambda_1\right)} + \frac{\phi_2 \cdot \phi_2}{\lambda_1 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)}\right)\right)\right) - \left(\frac{\phi_2}{\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)} \cdot \left(\frac{\phi_1 \cdot \lambda_2}{\lambda_1 \cdot \lambda_1} + \frac{\phi_1}{\lambda_1}\right) + \lambda_2 \cdot \cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.303407807589607 \cdot 10^{-287}:\\ \;\;\;\;R \cdot \sqrt{{\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)}\\ \mathbf{elif}\;\phi_2 \leq 5.38115258344735 \cdot 10^{-250}:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 4.962564261877504 \cdot 10^{-88}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + {\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}^{2}}\\ \mathbf{elif}\;\phi_2 \leq 9.269945814378616 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.553943994428835 \cdot 10^{+59}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_2 \cdot \phi_2 + {\cos \left(\phi_2 \cdot 0.5\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_1 + \lambda_2 \cdot \lambda_2\right)\right) - 2 \cdot \left({\cos \left(\phi_2 \cdot 0.5\right)}^{2} \cdot \left(\lambda_1 \cdot \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021009 
(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))))))