Distance on a great circle

Average Error: 23.8 → 13.3

Time: 2.4min

Precision: binary64

Cost: 242944

Math

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

↓

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

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 (* (cos phi1) (cos phi2)))
        (t_1 (* (sin lambda1) (sin lambda2)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0))
        (t_4 (* (cos lambda1) (cos lambda2))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ t_3 (* t_2 (* t_0 t_2))))
      (sqrt
       (+
        1.0
        (-
         (*
          (/ t_0 2.0)
          (+
           (/
            (+ (pow t_4 3.0) (pow t_1 3.0))
            (+ (* t_4 t_4) (- (* t_1 t_1) (* t_4 t_1))))
           -1.0))
         t_3))))))))

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 = cos(phi1) * cos(phi2);
	double t_1 = sin(lambda1) * sin(lambda2);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double t_4 = cos(lambda1) * cos(lambda2);
	return R * (2.0 * atan2(sqrt((t_3 + (t_2 * (t_0 * t_2)))), sqrt((1.0 + (((t_0 / 2.0) * (((pow(t_4, 3.0) + pow(t_1, 3.0)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1)))) + -1.0)) - t_3)))));
}

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
    t_0 = cos(phi1) * cos(phi2)
    t_1 = sin(lambda1) * sin(lambda2)
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    t_3 = ((sin((phi1 * 0.5d0)) * cos((0.5d0 * phi2))) - (cos((phi1 * 0.5d0)) * sin((0.5d0 * phi2)))) ** 2.0d0
    t_4 = cos(lambda1) * cos(lambda2)
    code = r * (2.0d0 * atan2(sqrt((t_3 + (t_2 * (t_0 * t_2)))), sqrt((1.0d0 + (((t_0 / 2.0d0) * ((((t_4 ** 3.0d0) + (t_1 ** 3.0d0)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1)))) + (-1.0d0))) - t_3)))))
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.cos(phi1) * Math.cos(phi2);
	double t_1 = Math.sin(lambda1) * Math.sin(lambda2);
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_3 = Math.pow(((Math.sin((phi1 * 0.5)) * Math.cos((0.5 * phi2))) - (Math.cos((phi1 * 0.5)) * Math.sin((0.5 * phi2)))), 2.0);
	double t_4 = Math.cos(lambda1) * Math.cos(lambda2);
	return R * (2.0 * Math.atan2(Math.sqrt((t_3 + (t_2 * (t_0 * t_2)))), Math.sqrt((1.0 + (((t_0 / 2.0) * (((Math.pow(t_4, 3.0) + Math.pow(t_1, 3.0)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1)))) + -1.0)) - t_3)))));
}

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.cos(phi1) * math.cos(phi2)
	t_1 = math.sin(lambda1) * math.sin(lambda2)
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	t_3 = math.pow(((math.sin((phi1 * 0.5)) * math.cos((0.5 * phi2))) - (math.cos((phi1 * 0.5)) * math.sin((0.5 * phi2)))), 2.0)
	t_4 = math.cos(lambda1) * math.cos(lambda2)
	return R * (2.0 * math.atan2(math.sqrt((t_3 + (t_2 * (t_0 * t_2)))), math.sqrt((1.0 + (((t_0 / 2.0) * (((math.pow(t_4, 3.0) + math.pow(t_1, 3.0)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1)))) + -1.0)) - t_3)))))

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 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(lambda1) * sin(lambda2))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_4 = Float64(cos(lambda1) * cos(lambda2))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(t_2 * Float64(t_0 * t_2)))), sqrt(Float64(1.0 + Float64(Float64(Float64(t_0 / 2.0) * Float64(Float64(Float64((t_4 ^ 3.0) + (t_1 ^ 3.0)) / Float64(Float64(t_4 * t_4) + Float64(Float64(t_1 * t_1) - Float64(t_4 * t_1)))) + -1.0)) - t_3))))))
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 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	t_1 = sin(lambda1) * sin(lambda2);
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	t_3 = ((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))) ^ 2.0;
	t_4 = cos(lambda1) * cos(lambda2);
	tmp = R * (2.0 * atan2(sqrt((t_3 + (t_2 * (t_0 * t_2)))), sqrt((1.0 + (((t_0 / 2.0) * ((((t_4 ^ 3.0) + (t_1 ^ 3.0)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1)))) + -1.0)) - t_3)))));
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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(t$95$2 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 + N[(N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 23.8

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

2. Applied egg-rr 23.2

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

3. Applied egg-rr 13.7

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

4. Applied egg-rr 13.7

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

5. Simplified 13.7

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

   [Start] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 - \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - \cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 + \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}{2}\right)}}\right) \]
   associate-*l/ [<=] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\frac{\cos \phi_1 \cdot \cos \phi_2}{2} \cdot \left(\cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 - \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - \cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 + \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]
   *-commutative [<=] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\color{blue}{\cos \phi_2 \cdot \cos \phi_1}}{2} \cdot \left(\cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 - \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - \cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 + \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]
   +-inverses [=>] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\cos \phi_2 \cdot \cos \phi_1}{2} \cdot \left(\cos \color{blue}{0} - \cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 + \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]
   cos-0 [=>] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\cos \phi_2 \cdot \cos \phi_1}{2} \cdot \left(\color{blue}{1} - \cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5 + \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]
   distribute-lft-out [=>] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\cos \phi_2 \cdot \cos \phi_1}{2} \cdot \left(1 - \cos \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(0.5 + 0.5\right)\right)}\right)\right)}}\right) \]
   metadata-eval [=>] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\cos \phi_2 \cdot \cos \phi_1}{2} \cdot \left(1 - \cos \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}\right)\right)\right)}}\right) \]
   *-rgt-identity [=>] 13.7	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \frac{\cos \phi_2 \cdot \cos \phi_1}{2} \cdot \left(1 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}}\right) \]

6. Applied egg-rr 13.3

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

7. Final simplification 13.3

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 + \left(\frac{\cos \phi_1 \cdot \cos \phi_2}{2} \cdot \left(\frac{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} + -1\right) - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

Alternatives

Alternative 1
Error	13.3
Cost	151680

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

Alternative 2
Error	13.7
Cost	145024

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \left(t_0 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}{\sqrt{1 - \left(t_1 + \frac{t_0}{2} \cdot \left(1 - \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\right) \end{array} \]

Alternative 3
Error	13.9
Cost	132361

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \frac{t_1}{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \sqrt{t_0 + t_3 \cdot \left(t_1 \cdot t_3\right)}\\ \mathbf{if}\;\lambda_2 \leq -0.000192 \lor \neg \left(\lambda_2 \leq 3.6 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{1 + \left(t_2 \cdot \left(\cos \lambda_2 + -1\right) - t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{1 + \left(t_2 \cdot \left(\cos \lambda_1 + -1\right) - t_0\right)}}\right)\\ \end{array} \]

Alternative 4
Error	17.1
Cost	132360

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ t_4 := t_1 \cdot \left(t_3 \cdot t_1\right)\\ t_5 := \sqrt{t_0 + t_4}\\ t_6 := t_3 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{if}\;\lambda_2 \leq -0.0002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_6}}{\sqrt{\left(1 - t_2\right) - t_6}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 + \left(\frac{t_3}{2} \cdot \left(\cos \lambda_1 + -1\right) - t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - \left(t_4 + t_2\right)}}\right)\\ \end{array} \]

Alternative 5
Error	20.0
Cost	132297

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sin \left(0.5 \cdot \phi_2\right)\\ t_3 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot t_0 - \cos \left(\phi_1 \cdot 0.5\right) \cdot t_2\right)}^{2}\\ t_4 := t_3 + t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.8 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - t_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4}}{\sqrt{t_0 \cdot \left(t_0 + \phi_1 \cdot t_2\right) + \cos \phi_2 \cdot \left(0.5 \cdot \left(-1 + \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}}\right)\\ \end{array} \]

Alternative 6
Error	20.4
Cost	132297

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \sqrt{1 - \left(t_1 + t_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{+37} \lor \neg \left(\phi_1 \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_1 \cdot t_0}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_2 \cdot t_0}}{t_3}\right)\\ \end{array} \]

Alternative 7
Error	20.4
Cost	132296

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := t_0 + \cos \phi_1 \cdot t_1\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := t_0 + t_3 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_3\right)\\ t_5 := \sqrt{1 - t_4}\\ \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4}}{\sqrt{1 - t_2}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_2 \cdot t_1}}{t_5}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2}}{t_5}\right)\\ \end{array} \]

Alternative 8
Error	13.7
Cost	132224

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

Alternative 9
Error	23.2
Cost	125440

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left(t_1 + \sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{4}}\right)}}\right) \end{array} \]

Alternative 10
Error	23.2
Cost	125376

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left(t_1 + {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(\phi_1 \cdot 0.5\right)\right)\right)\right)}^{2}\right)}}\right) \end{array} \]

Alternative 11
Error	23.2
Cost	119040

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

Alternative 12
Error	23.2
Cost	119040

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

Alternative 13
Error	23.2
Cost	118720

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

Alternative 14
Error	23.5
Cost	111552

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 \cdot \left(t_0 \cdot t_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\left|-1 + \mathsf{fma}\left(t_0, {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\right|}}\right) \end{array} \]

Alternative 15
Error	23.8
Cost	92992

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

Alternative 16
Error	23.8
Cost	92864

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

Alternative 17
Error	23.8
Cost	92544

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

Alternative 18
Error	24.2
Cost	92361

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -4 \cdot 10^{+38} \lor \neg \left(\phi_1 \leq 0.00013\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_2}}{\sqrt{\left(1 - t_2\right) - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 19
Error	26.9
Cost	92233

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{+24} \lor \neg \left(\phi_1 \leq 6800000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_3 \cdot \sin \left(0.5 \cdot \lambda_1\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot t_3 + t_0}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_1\right)}}\right)\\ \end{array} \]

Alternative 20
Error	37.4
Cost	86153

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 10^{-22}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1}}{\sqrt{1 - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - t_1}}\right)\\ \end{array} \]

Alternative 21
Error	30.1
Cost	86089

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+79} \lor \neg \left(\phi_1 \leq 4.1 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 22
Error	39.0
Cost	72841

\[\begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -0.8 \lor \neg \left(\phi_2 \leq 4.5 \cdot 10^{+131}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - t_0}}\right)\\ \end{array} \]

Alternative 23
Error	47.8
Cost	65472

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

Alternative 24
Error	53.3
Cost	59264

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

Alternative 25
Error	53.3
Cost	59264

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

Alternative 26
Error	54.0
Cost	52740

\[\begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_1 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq 150000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_0}}\right)\\ \end{array} \]

Alternative 27
Error	54.8
Cost	46336

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

Alternative 28
Error	56.4
Cost	33088

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

Reproduce

herbie shell --seed 2023237 
(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))))))))))