Distance on a great circle

Average Error: 24.7 → 24.5

Time: 1.3min

Precision: binary64

Cost: 99272

Math

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

↓

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \sin \left(-0.5 \cdot \phi_2\right)\\ t_5 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(0.5 \cdot \left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \phi_1\right) + t_4\right)}^{2} + \left(t_2 \cdot t_3\right) \cdot t_3}}{\sqrt{1 - \left({t_4}^{2} + \cos \phi_2 \cdot t_1\right)}}\right)\\ \mathbf{if}\;\phi_2 \leq -5.6 \cdot 10^{-26}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\phi_2 \leq 0.27:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot \left(t_3 \cdot t_3\right)}}{\sqrt{t_0 \cdot -1 + -1 \cdot \left(\cos \phi_1 \cdot t_1 - 1\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (+
      (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
      (*
       (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
       (sin (/ (- lambda1 lambda2) 2.0)))))
    (sqrt
     (-
      1.0
      (+
       (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
       (*
        (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
        (sin (/ (- lambda1 lambda2) 2.0))))))))))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (* (cos phi1) (cos phi2)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (sin (* -0.5 phi2)))
        (t_5
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (+
              (pow (+ (* 0.5 (* (cos (* -0.5 phi2)) phi1)) t_4) 2.0)
              (* (* t_2 t_3) t_3)))
            (sqrt (- 1.0 (+ (pow t_4 2.0) (* (cos phi2) t_1)))))))))
   (if (<= phi2 -5.6e-26)
     t_5
     (if (<= phi2 0.27)
       (*
        (atan2
         (sqrt (+ t_0 (* t_2 (* t_3 t_3))))
         (sqrt (+ (* t_0 -1.0) (* -1.0 (- (* (cos phi1) t_1) 1.0)))))
        (* R 2.0))
       t_5))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))))));
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = cos(phi1) * cos(phi2);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = sin((-0.5 * phi2));
	double t_5 = R * (2.0 * atan2(sqrt((pow(((0.5 * (cos((-0.5 * phi2)) * phi1)) + t_4), 2.0) + ((t_2 * t_3) * t_3))), sqrt((1.0 - (pow(t_4, 2.0) + (cos(phi2) * t_1))))));
	double tmp;
	if (phi2 <= -5.6e-26) {
		tmp = t_5;
	} else if (phi2 <= 0.27) {
		tmp = atan2(sqrt((t_0 + (t_2 * (t_3 * t_3)))), sqrt(((t_0 * -1.0) + (-1.0 * ((cos(phi1) * t_1) - 1.0))))) * (R * 2.0);
	} else {
		tmp = t_5;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0d0))) * sin(((lambda1 - lambda2) / 2.0d0))))), sqrt((1.0d0 - ((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0d0))) * sin(((lambda1 - lambda2) / 2.0d0))))))))
end function

↓

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_1 = sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0
    t_2 = cos(phi1) * cos(phi2)
    t_3 = sin(((lambda1 - lambda2) / 2.0d0))
    t_4 = sin(((-0.5d0) * phi2))
    t_5 = r * (2.0d0 * atan2(sqrt(((((0.5d0 * (cos(((-0.5d0) * phi2)) * phi1)) + t_4) ** 2.0d0) + ((t_2 * t_3) * t_3))), sqrt((1.0d0 - ((t_4 ** 2.0d0) + (cos(phi2) * t_1))))))
    if (phi2 <= (-5.6d-26)) then
        tmp = t_5
    else if (phi2 <= 0.27d0) then
        tmp = atan2(sqrt((t_0 + (t_2 * (t_3 * t_3)))), sqrt(((t_0 * (-1.0d0)) + ((-1.0d0) * ((cos(phi1) * t_1) - 1.0d0))))) * (r * 2.0d0)
    else
        tmp = t_5
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * Math.sin(((lambda1 - lambda2) / 2.0))) * Math.sin(((lambda1 - lambda2) / 2.0))))), Math.sqrt((1.0 - (Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * Math.sin(((lambda1 - lambda2) / 2.0))) * Math.sin(((lambda1 - lambda2) / 2.0))))))));
}

↓

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = Math.cos(phi1) * Math.cos(phi2);
	double t_3 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_4 = Math.sin((-0.5 * phi2));
	double t_5 = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(((0.5 * (Math.cos((-0.5 * phi2)) * phi1)) + t_4), 2.0) + ((t_2 * t_3) * t_3))), Math.sqrt((1.0 - (Math.pow(t_4, 2.0) + (Math.cos(phi2) * t_1))))));
	double tmp;
	if (phi2 <= -5.6e-26) {
		tmp = t_5;
	} else if (phi2 <= 0.27) {
		tmp = Math.atan2(Math.sqrt((t_0 + (t_2 * (t_3 * t_3)))), Math.sqrt(((t_0 * -1.0) + (-1.0 * ((Math.cos(phi1) * t_1) - 1.0))))) * (R * 2.0);
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * math.sin(((lambda1 - lambda2) / 2.0))) * math.sin(((lambda1 - lambda2) / 2.0))))), math.sqrt((1.0 - (math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * math.sin(((lambda1 - lambda2) / 2.0))) * math.sin(((lambda1 - lambda2) / 2.0))))))))

↓

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_1 = math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)
	t_2 = math.cos(phi1) * math.cos(phi2)
	t_3 = math.sin(((lambda1 - lambda2) / 2.0))
	t_4 = math.sin((-0.5 * phi2))
	t_5 = R * (2.0 * math.atan2(math.sqrt((math.pow(((0.5 * (math.cos((-0.5 * phi2)) * phi1)) + t_4), 2.0) + ((t_2 * t_3) * t_3))), math.sqrt((1.0 - (math.pow(t_4, 2.0) + (math.cos(phi2) * t_1))))))
	tmp = 0
	if phi2 <= -5.6e-26:
		tmp = t_5
	elif phi2 <= 0.27:
		tmp = math.atan2(math.sqrt((t_0 + (t_2 * (t_3 * t_3)))), math.sqrt(((t_0 * -1.0) + (-1.0 * ((math.cos(phi1) * t_1) - 1.0))))) * (R * 2.0)
	else:
		tmp = t_5
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))))), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(Float64(lambda1 - lambda2) / 2.0)))))))))
end

↓

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64(cos(phi1) * cos(phi2))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = sin(Float64(-0.5 * phi2))
	t_5 = Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(Float64(0.5 * Float64(cos(Float64(-0.5 * phi2)) * phi1)) + t_4) ^ 2.0) + Float64(Float64(t_2 * t_3) * t_3))), sqrt(Float64(1.0 - Float64((t_4 ^ 2.0) + Float64(cos(phi2) * t_1)))))))
	tmp = 0.0
	if (phi2 <= -5.6e-26)
		tmp = t_5;
	elseif (phi2 <= 0.27)
		tmp = Float64(atan(sqrt(Float64(t_0 + Float64(t_2 * Float64(t_3 * t_3)))), sqrt(Float64(Float64(t_0 * -1.0) + Float64(-1.0 * Float64(Float64(cos(phi1) * t_1) - 1.0))))) * Float64(R * 2.0));
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))), sqrt((1.0 - ((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))))));
end

↓

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_1 = sin((0.5 * (lambda1 - lambda2))) ^ 2.0;
	t_2 = cos(phi1) * cos(phi2);
	t_3 = sin(((lambda1 - lambda2) / 2.0));
	t_4 = sin((-0.5 * phi2));
	t_5 = R * (2.0 * atan2(sqrt(((((0.5 * (cos((-0.5 * phi2)) * phi1)) + t_4) ^ 2.0) + ((t_2 * t_3) * t_3))), sqrt((1.0 - ((t_4 ^ 2.0) + (cos(phi2) * t_1))))));
	tmp = 0.0;
	if (phi2 <= -5.6e-26)
		tmp = t_5;
	elseif (phi2 <= 0.27)
		tmp = atan2(sqrt((t_0 + (t_2 * (t_3 * t_3)))), sqrt(((t_0 * -1.0) + (-1.0 * ((cos(phi1) * t_1) - 1.0))))) * (R * 2.0);
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[(0.5 * N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$2 * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -5.6e-26], t$95$5, If[LessEqual[phi2, 0.27], N[(N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$2 * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * -1.0), $MachinePrecision] + N[(-1.0 * N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -5.6000000000000002e-26 or 0.27000000000000002 < phi2

   1. Initial program 34.0

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0 33.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}}\right) \]

   3. Taylor expanded in phi1 around 0 33.5

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(0.5 \cdot \left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \phi_1\right) + \sin \left(-0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   if -5.6000000000000002e-26 < phi2 < 0.27000000000000002

   1. Initial program 14.3

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Simplified 14.3

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}} \cdot \left(R \cdot 2\right)} \]
      Proof

      [Start] 14.3	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 14.3	\[ R \cdot \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot 2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 14.3	\[ \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot \left(R \cdot 2\right)} \]

   3. Applied egg-rr 14.3

      \[\leadsto \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \cdot -1 + -1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right) + -1\right)}}} \cdot \left(R \cdot 2\right) \]

   4. Taylor expanded in phi2 around 0 14.3

      \[\leadsto \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \cdot -1 + -1 \cdot \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} - 1\right)}}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -5.6 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(0.5 \cdot \left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \phi_1\right) + \sin \left(-0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.27:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \cdot -1 + -1 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} - 1\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(0.5 \cdot \left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \phi_1\right) + \sin \left(-0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	24.7
Cost	99456

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \tan^{-1}_* \frac{\sqrt{t_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1}}{\sqrt{t_2 \cdot -1 + -1 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right) + -1\right)}} \cdot \left(R \cdot 2\right) \end{array} \]

Alternative 2
Error	24.7
Cost	99200

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) \cdot t_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1}}{\sqrt{1 - t_1}}\right) \end{array} \]

Alternative 3
Error	24.7
Cost	99200

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ \tan^{-1}_* \frac{\sqrt{t_1}}{\sqrt{1 - t_1}} \cdot \left(R \cdot 2\right) \end{array} \]

Alternative 4
Error	24.7
Cost	92872

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := t_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_1 \cdot t_1\right)\\ t_4 := \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_0 \cdot \cos \phi_2}}{\sqrt{1 - t_3}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\phi_2 \leq -5.6 \cdot 10^{-26}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_2 \leq 0.00152:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{t_2 \cdot -1 + -1 \cdot \left(\cos \phi_1 \cdot t_0 - 1\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Error	24.4
Cost	92488

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_4 := \tan^{-1}_* \frac{\sqrt{t_3 + t_1 \cdot \left(t_2 \cdot t_2\right)}}{\sqrt{1 - \left(t_3 + \cos \phi_1 \cdot t_0\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\phi_1 \leq -0.0255:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t_1 \cdot t_2\right) \cdot t_2}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 6
Error	24.4
Cost	92488

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_2\right) \cdot t_2\\ t_4 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_3}\\ t_5 := \sqrt{1 - \left(t_0 + \cos \phi_2 \cdot t_1\right)}\\ \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{t_5}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.175:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot t_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_3}}{t_5}\right)\\ \end{array} \]

Alternative 7
Error	24.4
Cost	92488

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_0 \cdot \cos \phi_2}}{\sqrt{1 - \left(t_3 + t_1 \cdot \left(t_2 \cdot t_2\right)\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\phi_2 \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_2 \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + \left(t_1 \cdot t_2\right) \cdot t_2}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 8
Error	24.4
Cost	92488

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_1 \cdot t_1\right)\\ t_3 := \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_0 \cdot \cos \phi_2}}{\sqrt{1 - t_2}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\phi_2 \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_2}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot t_0\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	30.9
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\\ t_3 := t_2 \cdot t_1\\ t_4 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.048:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot \sin \left(0.5 \cdot \lambda_1\right)}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \cos \phi_1\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4 + t_3}}{\sqrt{1 - \left(t_4 + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_3}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 10
Error	27.9
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\\ t_3 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_4 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2 \cdot \sin \left(0.5 \cdot \lambda_1\right)}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot t_3\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot t_1}}{\sqrt{1 - \left(t_0 + \cos \phi_2 \cdot t_3\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 11
Error	25.2
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \left(t_2 \cdot t_3\right) \cdot t_3}}{\sqrt{1 - \left(t_0 + \cos \phi_2 \cdot t_1\right)}}\right)\\ t_5 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -5.4 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_2 \leq 0.175:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_5 + t_2 \cdot \left(t_3 \cdot t_3\right)}}{\sqrt{1 - \left(t_5 + \cos \phi_1 \cdot t_1\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 12
Error	36.0
Cost	86152

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + t_1 \cdot \cos \phi_2}}{\sqrt{1 - t_1}}\right)\\ \mathbf{if}\;t_0 \leq -0.0135:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 0.1:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) \cdot t_0}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	37.1
Cost	73096

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \sqrt{1 - t_2}\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + t_2 \cdot \cos \phi_2}}{t_3}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 50000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4 + t_1 \cdot \sin \left(0.5 \cdot \lambda_1\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4 + t_1 \cdot t_0}}{t_3}\right)\\ \end{array} \]

Alternative 14
Error	37.1
Cost	72968

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + t_0 \cdot \cos \phi_2}}{\sqrt{1 - t_0}}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2.5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 50000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	42.2
Cost	72712

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \sqrt{1 - t_1}\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + t_1 \cdot \cos \phi_2}}{t_2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.4 \cdot 10^{-23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{-88}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) \cdot t_0}}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	42.3
Cost	72712

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sqrt{1 - t_0}\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + t_0 \cdot \cos \phi_2}}{t_1}\right)\\ \mathbf{if}\;\lambda_2 \leq -2.65 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 1.45 \cdot 10^{-94}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)}}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	43.2
Cost	65600

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + t_0 \cdot \cos \phi_2}}{\sqrt{1 - t_0}}\right) \end{array} \]

Alternative 18
Error	55.0
Cost	46016

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1} \cdot t_0}{\sqrt{1 - {t_0}^{2}}}\right) \end{array} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))