Distance on a great circle

Average Accuracy: 62.6% → 78.6%

Time: 2.4min

Precision: binary64

Cost: 185664

Math

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

↓

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

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

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

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

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

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

Derivation

1. Initial program 62.6%

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

2. Applied egg-rr 63.5%

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

3. Applied egg-rr 78.6%

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

4. Applied egg-rr 78.6%

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

5. Final simplification 78.6%

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

Alternatives

Alternative 1
Accuracy	78.6%
Cost	185344

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

Alternative 2
Accuracy	78.6%
Cost	151360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot \left(t_0 \cdot t_2\right) + t_1}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left(t_0 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + t_1\right)\right)\right)}}\right) \end{array} \]

Alternative 3
Accuracy	71.1%
Cost	138696

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_3 := t_1 + t_2\\ t_4 := t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_5 := \sqrt{1 - t_3}\\ \mathbf{if}\;\lambda_2 \leq -0.26:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4}}{t_5}\right)\\ \mathbf{elif}\;\lambda_2 \leq 6.2 \cdot 10^{-81}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{t_5}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - t_4}}\right)\\ \end{array} \]

Alternative 4
Accuracy	73.1%
Cost	138696

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_3 := t_1 + t_2\\ t_4 := t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\lambda_2 \leq -0.26:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4}}{\sqrt{1 - t_3}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - \left(t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt[3]{{\left(t_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)\right)}^{1.5}}}{\sqrt{1 - t_4}}\right)\\ \end{array} \]

Alternative 5
Accuracy	78.4%
Cost	138696

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_3 := t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)\\ t_4 := t_0 \cdot \left(t_1 \cdot t_0\right) + t_2\\ t_5 := \sqrt{t_4}\\ \mathbf{if}\;\lambda_1 \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - t_3}}\right)\\ \mathbf{elif}\;\lambda_1 \leq 0.062:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - \left(t_2 + t_1 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - t_4}}\right)\\ \end{array} \]

Alternative 6
Accuracy	78.6%
Cost	138560

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

Alternative 7
Accuracy	63.5%
Cost	119040

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

Alternative 8
Accuracy	63.5%
Cost	119040

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

Alternative 9
Accuracy	63.5%
Cost	119040

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

Alternative 10
Accuracy	62.6%
Cost	112000

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 \cdot t_2 + t_0}}{\sqrt{1 - \left(t_0 + t_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}}\right) \end{array} \]

Alternative 11
Accuracy	49.8%
Cost	105545

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-16} \lor \neg \left(t_0 \leq 2 \cdot 10^{-128}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_1 \cdot \left(t_0 \cdot t_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right|}{\sqrt{\mathsf{fma}\left(t_1, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 12
Accuracy	62.6%
Cost	99200

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

Alternative 13
Accuracy	62.6%
Cost	98816

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

Alternative 14
Accuracy	62.6%
Cost	98760

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ t_2 := {t_1}^{2}\\ t_3 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - t_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.36 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot t_2}}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_1 \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \end{array} \]

Alternative 15
Accuracy	62.6%
Cost	98628

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := t_0 \cdot \left(t_2 \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -1.26 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t_1, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - t_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.36 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_2 \cdot \left(t_0 \cdot t_0\right)}}\right)\\ \end{array} \]

Alternative 16
Accuracy	62.6%
Cost	92361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.42 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 1.36 \cdot 10^{-21}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_0 \cdot \left(t_1 \cdot t_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 \cdot \left(t_0 \cdot t_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 17
Accuracy	62.6%
Cost	92360

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := t_0 \cdot \left(t_1 \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{1 - t_2}}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.36 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_1 \cdot \left(t_0 \cdot t_0\right)}}\right)\\ \end{array} \]

Alternative 18
Accuracy	49.5%
Cost	86728

\[\begin{array}{l} t_0 := -0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(t_2 \cdot t_2\right)}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)}}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 2 \cdot 10^{-225}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right|}{\sqrt{\mathsf{fma}\left(t_1, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{\left(0.5 + t_1 \cdot t_0\right) + 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot -0.5\right)\right)}}\right)\\ \end{array} \]

Alternative 19
Accuracy	49.5%
Cost	86537

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-15} \lor \neg \left(\lambda_1 - \lambda_2 \leq 2 \cdot 10^{-225}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(t_0 \cdot t_0\right)}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(-0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right|}{\sqrt{\mathsf{fma}\left(t_1, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 20
Accuracy	43.1%
Cost	86025

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 145000000000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2} + {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\left(0.5 + t_0 \cdot \left(-0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot -0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(t_1 \cdot t_1\right)}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 21
Accuracy	43.1%
Cost	85833

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.42 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 145000000000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2} + {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(-0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(t_1 \cdot t_1\right)}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 22
Accuracy	43.3%
Cost	85640

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sqrt{\mathsf{fma}\left(t_1, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}\\ \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 + -0.5 \cdot \cos \left(\phi_1 - \phi_2\right)}}{t_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{t_1}, t_0\right)}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|t_0\right|}{t_2}\right)\\ \end{array} \]

Alternative 23
Accuracy	40.0%
Cost	79625

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-15} \lor \neg \left(\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-88}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(t_0 \cdot t_0\right)}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right|}{\sqrt{\mathsf{fma}\left(t_1, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 24
Accuracy	43.3%
Cost	79368

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sqrt{\mathsf{fma}\left(t_1, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}\\ \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 + -0.5 \cdot \cos \left(\phi_1 - \phi_2\right)}}{t_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{t_1}, t_0\right)}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(-0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|t_0\right|}{t_2}\right)\\ \end{array} \]

Alternative 25
Accuracy	36.6%
Cost	72649

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.5 \cdot 10^{-15} \lor \neg \left(\phi_2 \leq 0.003\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left|t_1\right|}{\sqrt{\mathsf{fma}\left(t_2, \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right), 1\right) - {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t_0 \cdot \sqrt{t_2}, t_1\right)}{\sqrt{1 - \cos \phi_1 \cdot {t_0}^{2}}}\right)\\ \end{array} \]

Alternative 26
Accuracy	27.6%
Cost	72192

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

Alternative 27
Accuracy	13.8%
Cost	59456

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

Alternative 28
Accuracy	8.9%
Cost	52928

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

Alternative 29
Accuracy	11.6%
Cost	52928

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

Alternative 30
Accuracy	8.9%
Cost	52672

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

Alternative 31
Accuracy	8.9%
Cost	33280

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

Reproduce

herbie shell --seed 2023125 
(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))))))))))