Distance on a great circle

Average Error: 24.2 → 24.1

Time: 1.6min

Precision: binary64

Cost: 99272

Math

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

↓

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_3 \cdot t_3\right)\right)\\ t_5 := \sqrt{1 - \left(t_0 + \cos \phi_2 \cdot t_2\right)}\\ \mathbf{if}\;\phi_2 \leq -0.00015:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_4}}{t_5} \cdot \left(R \cdot 2\right)\\ \mathbf{elif}\;\phi_2 \leq 740:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\left(t_1 + \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-0.5 \cdot \phi_2\right)\right)}^{2} + t_4}}{\sqrt{1 - \left({t_1}^{2} + \cos \phi_1 \cdot t_2\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_0 + t_4}}{t_5} \cdot \left(R \cdot 2\right)\\ \end{array} \]

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 (* -0.5 phi2)) 2.0))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (* (cos phi1) (* (cos phi2) (* t_3 t_3))))
        (t_5 (sqrt (- 1.0 (+ t_0 (* (cos phi2) t_2))))))
   (if (<= phi2 -0.00015)
     (*
      (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_4)) t_5)
      (* R 2.0))
     (if (<= phi2 740.0)
       (*
        (atan2
         (sqrt (+ (pow (+ t_1 (* (cos (* 0.5 phi1)) (* -0.5 phi2))) 2.0) t_4))
         (sqrt (- 1.0 (+ (pow t_1 2.0) (* (cos phi1) t_2)))))
        (* R 2.0))
       (* (atan2 (sqrt (+ t_0 t_4)) t_5) (* R 2.0))))))

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((-0.5 * phi2)), 2.0);
	double t_1 = sin((0.5 * phi1));
	double t_2 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = cos(phi1) * (cos(phi2) * (t_3 * t_3));
	double t_5 = sqrt((1.0 - (t_0 + (cos(phi2) * t_2))));
	double tmp;
	if (phi2 <= -0.00015) {
		tmp = atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_4)), t_5) * (R * 2.0);
	} else if (phi2 <= 740.0) {
		tmp = atan2(sqrt((pow((t_1 + (cos((0.5 * phi1)) * (-0.5 * phi2))), 2.0) + t_4)), sqrt((1.0 - (pow(t_1, 2.0) + (cos(phi1) * t_2))))) * (R * 2.0);
	} else {
		tmp = atan2(sqrt((t_0 + t_4)), t_5) * (R * 2.0);
	}
	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(((-0.5d0) * phi2)) ** 2.0d0
    t_1 = sin((0.5d0 * phi1))
    t_2 = sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0
    t_3 = sin(((lambda1 - lambda2) / 2.0d0))
    t_4 = cos(phi1) * (cos(phi2) * (t_3 * t_3))
    t_5 = sqrt((1.0d0 - (t_0 + (cos(phi2) * t_2))))
    if (phi2 <= (-0.00015d0)) then
        tmp = atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + t_4)), t_5) * (r * 2.0d0)
    else if (phi2 <= 740.0d0) then
        tmp = atan2(sqrt((((t_1 + (cos((0.5d0 * phi1)) * ((-0.5d0) * phi2))) ** 2.0d0) + t_4)), sqrt((1.0d0 - ((t_1 ** 2.0d0) + (cos(phi1) * t_2))))) * (r * 2.0d0)
    else
        tmp = atan2(sqrt((t_0 + t_4)), t_5) * (r * 2.0d0)
    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((-0.5 * phi2)), 2.0);
	double t_1 = Math.sin((0.5 * phi1));
	double t_2 = Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_3 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_4 = Math.cos(phi1) * (Math.cos(phi2) * (t_3 * t_3));
	double t_5 = Math.sqrt((1.0 - (t_0 + (Math.cos(phi2) * t_2))));
	double tmp;
	if (phi2 <= -0.00015) {
		tmp = Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_4)), t_5) * (R * 2.0);
	} else if (phi2 <= 740.0) {
		tmp = Math.atan2(Math.sqrt((Math.pow((t_1 + (Math.cos((0.5 * phi1)) * (-0.5 * phi2))), 2.0) + t_4)), Math.sqrt((1.0 - (Math.pow(t_1, 2.0) + (Math.cos(phi1) * t_2))))) * (R * 2.0);
	} else {
		tmp = Math.atan2(Math.sqrt((t_0 + t_4)), t_5) * (R * 2.0);
	}
	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((-0.5 * phi2)), 2.0)
	t_1 = math.sin((0.5 * phi1))
	t_2 = math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)
	t_3 = math.sin(((lambda1 - lambda2) / 2.0))
	t_4 = math.cos(phi1) * (math.cos(phi2) * (t_3 * t_3))
	t_5 = math.sqrt((1.0 - (t_0 + (math.cos(phi2) * t_2))))
	tmp = 0
	if phi2 <= -0.00015:
		tmp = math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_4)), t_5) * (R * 2.0)
	elif phi2 <= 740.0:
		tmp = math.atan2(math.sqrt((math.pow((t_1 + (math.cos((0.5 * phi1)) * (-0.5 * phi2))), 2.0) + t_4)), math.sqrt((1.0 - (math.pow(t_1, 2.0) + (math.cos(phi1) * t_2))))) * (R * 2.0)
	else:
		tmp = math.atan2(math.sqrt((t_0 + t_4)), t_5) * (R * 2.0)
	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(-0.5 * phi2)) ^ 2.0
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = Float64(cos(phi1) * Float64(cos(phi2) * Float64(t_3 * t_3)))
	t_5 = sqrt(Float64(1.0 - Float64(t_0 + Float64(cos(phi2) * t_2))))
	tmp = 0.0
	if (phi2 <= -0.00015)
		tmp = Float64(atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_4)), t_5) * Float64(R * 2.0));
	elseif (phi2 <= 740.0)
		tmp = Float64(atan(sqrt(Float64((Float64(t_1 + Float64(cos(Float64(0.5 * phi1)) * Float64(-0.5 * phi2))) ^ 2.0) + t_4)), sqrt(Float64(1.0 - Float64((t_1 ^ 2.0) + Float64(cos(phi1) * t_2))))) * Float64(R * 2.0));
	else
		tmp = Float64(atan(sqrt(Float64(t_0 + t_4)), t_5) * Float64(R * 2.0));
	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((-0.5 * phi2)) ^ 2.0;
	t_1 = sin((0.5 * phi1));
	t_2 = sin((0.5 * (lambda1 - lambda2))) ^ 2.0;
	t_3 = sin(((lambda1 - lambda2) / 2.0));
	t_4 = cos(phi1) * (cos(phi2) * (t_3 * t_3));
	t_5 = sqrt((1.0 - (t_0 + (cos(phi2) * t_2))));
	tmp = 0.0;
	if (phi2 <= -0.00015)
		tmp = atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + t_4)), t_5) * (R * 2.0);
	elseif (phi2 <= 740.0)
		tmp = atan2(sqrt((((t_1 + (cos((0.5 * phi1)) * (-0.5 * phi2))) ^ 2.0) + t_4)), sqrt((1.0 - ((t_1 ^ 2.0) + (cos(phi1) * t_2))))) * (R * 2.0);
	else
		tmp = atan2(sqrt((t_0 + t_4)), t_5) * (R * 2.0);
	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[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 - N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.00015], N[(N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$5], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 740.0], N[(N[ArcTan[N[Sqrt[N[(N[Power[N[(t$95$1 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(-0.5 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(t$95$0 + t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$5], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.49999999999999987e-4

   1. Initial program 34.6

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

   2. Simplified 34.5

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

      [Start] 34.6

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

      rational.json-simplify-43 [<=] 34.6

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

   3. Taylor expanded in phi1 around 0 34.0

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

   if -1.49999999999999987e-4 < phi2 < 740

   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

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)\right)}} \cdot \left(R \cdot 2\right)} \]
      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.json-simplify-43 [<=] 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. Taylor expanded in phi2 around 0 14.6

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

   4. Taylor expanded in phi2 around 0 14.6

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

   5. Simplified 14.6

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

      [Start] 14.6	\[ \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) + -0.5 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
      rational.json-simplify-43 [=>] 14.6	\[ \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) + \color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\phi_2 \cdot -0.5\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
      rational.json-simplify-2 [<=] 14.6	\[ \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) + \cos \left(0.5 \cdot \phi_1\right) \cdot \color{blue}{\left(-0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   if 740 < phi2

   1. Initial program 33.3

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

   2. Simplified 33.4

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

      [Start] 33.3

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

      rational.json-simplify-43 [<=] 33.4

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

   3. Taylor expanded in phi1 around 0 33.5

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

   4. Taylor expanded in phi1 around 0 32.9

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

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00015:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{elif}\;\phi_2 \leq 740:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) + \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	24.2
Cost	99200

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

Alternative 2
Error	24.2
Cost	99200

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

Alternative 3
Error	24.1
Cost	92552

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_2 \cdot t_2\right)\right)\\ t_4 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_3}\\ t_5 := \sqrt{1 - \left(t_0 + \cos \phi_2 \cdot t_1\right)}\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_5} \cdot \left(R \cdot 2\right)\\ \mathbf{elif}\;\phi_2 \leq 740:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{\sqrt{-\left(\cos \phi_1 \cdot t_1 + \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + -1\right)\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_0 + t_3}}{t_5} \cdot \left(R \cdot 2\right)\\ \end{array} \]

Alternative 4
Error	24.0
Cost	92488

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_1 \cdot t_2}}{\sqrt{1 - \left(t_0 + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(\cos \phi_1 \cdot t_1\right)\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_1 \cdot t_1\right)\right)}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot t_2\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	24.1
Cost	92488

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_2 \cdot t_2\right)\right)\\ t_4 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_3}\\ t_5 := \sqrt{1 - \left(t_0 + \cos \phi_2 \cdot t_1\right)}\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_5} \cdot \left(R \cdot 2\right)\\ \mathbf{elif}\;\phi_2 \leq 740:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot t_1\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_0 + t_3}}{t_5} \cdot \left(R \cdot 2\right)\\ \end{array} \]

Alternative 6
Error	36.0
Cost	92360

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}}}{\sqrt{1 - \left(t_1 + \left(\cos \phi_2 \cdot t_0\right) \cdot \left(\cos \phi_1 \cdot t_0\right)\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_0 \cdot t_0\right)\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	34.4
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(\cos \phi_1 \cdot t_1\right)\right)}}\right)\\ \mathbf{if}\;\lambda_1 \leq -0.00098:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 8000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_1 \cdot {\left(0.5 \cdot \left(\cos \left(-0.5 \cdot \lambda_2\right) \cdot \lambda_1\right) + \sin \left(-0.5 \cdot \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left(t_0 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	32.9
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_1 \cdot t_2}}{\sqrt{1 - \left(t_0 + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(\cos \phi_1 \cdot t_1\right)\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-40}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_1 \cdot t_1\right)\right)}}{\sqrt{1 - t_2}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	24.9
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_4 := \tan^{-1}_* \frac{\sqrt{t_3 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_2 \cdot t_2\right)\right)}}{\sqrt{1 - \left(t_3 + \cos \phi_2 \cdot t_1\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\phi_2 \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_1 \cdot t_1}}{\sqrt{1 - \left(t_0 + \left(\cos \phi_2 \cdot t_2\right) \cdot \left(\cos \phi_1 \cdot t_2\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 10
Error	37.3
Cost	92296

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_2 \cdot t_2\right)\right)}}{\sqrt{1 - t_0}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\lambda_1 \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_1 \cdot {\left(0.5 \cdot \left(\cos \left(-0.5 \cdot \lambda_2\right) \cdot \lambda_1\right) + \sin \left(-0.5 \cdot \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left(t_1 + \cos \phi_1 \cdot t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	40.8
Cost	86216

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_1 \cdot t_1\right)\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 10^{-17}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + {\lambda_1}^{2} \cdot \left(0.25 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left(t_0 + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(\cos \phi_1 \cdot t_1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	48.9
Cost	79236

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -2 \cdot 10^{-174}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1} \cdot \left(\sin \left(\lambda_2 \cdot 0.5\right) + \cos \left(\lambda_2 \cdot 0.5\right) \cdot \left(\lambda_1 \cdot -0.5\right)\right)}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot t_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_0}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot t_1\right)}}\right)\\ \end{array} \]

Alternative 13
Error	41.6
Cost	72576

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t_0 \cdot t_0\right)\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right) \end{array} \]

Alternative 14
Error	53.4
Cost	52736

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

Alternative 15
Error	53.4
Cost	52736

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

Alternative 16
Error	53.5
Cost	33088

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

Reproduce

herbie shell --seed 2023067 
(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))))))))))