Distance on a great circle

Average Error: 38.32% → 20.95%

Time: 1.7min

Precision: binary64

Cost: 151680

Math

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

↓

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

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

C

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

↓

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

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
    t_0 = cos(phi1) * cos(phi2)
    t_1 = ((sin((phi1 * 0.5d0)) * cos((0.5d0 * phi2))) - (cos((phi1 * 0.5d0)) * sin((0.5d0 * phi2)))) ** 2.0d0
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt((t_1 + (t_0 * (t_2 * t_2)))), sqrt(((1.0d0 - t_1) + (t_0 * ((0.5d0 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (-0.5d0)))))))
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.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_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * Math.atan2(Math.sqrt((t_1 + (t_0 * (t_2 * t_2)))), Math.sqrt(((1.0 - t_1) + (t_0 * ((0.5 * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))) + -0.5))))));
}

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

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(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_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + Float64(t_0 * Float64(t_2 * t_2)))), sqrt(Float64(Float64(1.0 - t_1) + Float64(t_0 * Float64(Float64(0.5 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + -0.5)))))))
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((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))) ^ 2.0;
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt((t_1 + (t_0 * (t_2 * t_2)))), sqrt(((1.0 - t_1) + (t_0 * ((0.5 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + -0.5))))));
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[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$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[(t$95$0 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] + N[(t$95$0 * N[(N[(0.5 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 38.32

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

2. Simplified 38.31

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

   [Start] 38.32

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

   associate-*l* [=>] 38.32

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

3. Applied egg-rr 37.39

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

4. Applied egg-rr 21.57

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

5. Applied egg-rr 21.54

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

6. Applied egg-rr 20.95

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

7. Final simplification 20.95

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

Alternatives

Alternative 1
Error	21.74%
Cost	132361

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := 1 - t_0\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ t_4 := \sqrt{t_0 + t_3 \cdot \left(t_1 \cdot t_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -7 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 0.00105\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{t_2 + t_3 \cdot \left(0.5 \cdot \cos \lambda_1 + -0.5\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{t_2 + t_3 \cdot \left(0.5 \cdot \cos \lambda_2 + -0.5\right)}}\right)\\ \end{array} \]

Alternative 2
Error	28.3%
Cost	132360

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := t_2 \cdot \left(t_1 \cdot t_1\right)\\ t_4 := \sqrt{t_0 + t_3}\\ \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - t_3}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot \left(t_1 \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{\left(1 - t_0\right) + t_2 \cdot \left(0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right) + -0.5\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(0.5 - \cos \left(\phi_1 - \phi_2\right) \cdot -0.5\right) - t_3}}\right)\\ \end{array} \]

Alternative 3
Error	27.2%
Cost	132360

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := t_2 \cdot \left(t_1 \cdot t_1\right)\\ t_4 := \sqrt{t_0 + t_3}\\ \mathbf{if}\;\lambda_2 \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - t_3}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(1 - t_0\right) + t_2 \cdot \left(0.5 \cdot \cos \lambda_1 + -0.5\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(0.5 - \cos \left(\phi_1 - \phi_2\right) \cdot -0.5\right) - t_3}}\right)\\ \end{array} \]

Alternative 4
Error	21.54%
Cost	132224

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

Alternative 5
Error	37.33%
Cost	112576

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

Alternative 6
Error	38.27%
Cost	92864

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

Alternative 7
Error	38.28%
Cost	92864

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

Alternative 8
Error	37.89%
Cost	92489

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot t_0}}{\sqrt{1 - t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_0\right)}}\right)\\ \end{array} \]

Alternative 9
Error	38.66%
Cost	92360

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_2 \cdot t_0\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_2\right)\\ t_4 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_3}\\ \mathbf{if}\;\phi_2 \leq -8.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(0.5 - t_1\right) + 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot -0.5\right)\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_1}}\right)\\ \end{array} \]

Alternative 10
Error	37.88%
Cost	92233

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_2\right)}}\right)\\ \end{array} \]

Alternative 11
Error	62.15%
Cost	86088

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\\ t_2 := t_0 \cdot t_1\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;t_0 \leq -0.27:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{t_4}\right)\\ \mathbf{elif}\;t_0 \leq 0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + t_2}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + t_1 \cdot \left|\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right|}}{t_4}\right)\\ \end{array} \]

Alternative 12
Error	58.13%
Cost	85961

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)}\\ \mathbf{if}\;\phi_2 \leq -1.62 \cdot 10^{-10} \lor \neg \left(\phi_2 \leq 4.1 \cdot 10^{-256}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \end{array} \]

Alternative 13
Error	56.91%
Cost	85897

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{if}\;\phi_2 \leq -3.05 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 1.6 \cdot 10^{-57}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \frac{1 - \cos \left(\phi_1 - \phi_2\right)}{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}}{\sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \end{array} \]

Alternative 14
Error	62.15%
Cost	85896

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}\\ t_3 := \sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;t_0 \leq -0.27:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{t_3}\right)\\ \mathbf{elif}\;t_0 \leq 0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{t_3}\right)\\ \end{array} \]

Alternative 15
Error	51.08%
Cost	85824

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

Alternative 16
Error	61.49%
Cost	79560

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := \cos \left(\phi_2 \cdot -0.5\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\left(\sin \left(\phi_2 \cdot -0.5\right) + 0.5 \cdot \left(\phi_1 \cdot t_3\right)\right)}^{2}}}{t_0}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2}}{\sqrt{{t_3}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\log \left(1 + \mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}^{2}}}{t_0}\right)\\ \end{array} \]

Alternative 17
Error	61.49%
Cost	79432

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := \cos \left(\phi_2 \cdot -0.5\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\left(\sin \left(\phi_2 \cdot -0.5\right) + 0.5 \cdot \left(\phi_1 \cdot t_3\right)\right)}^{2}}}{t_0}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2}}{\sqrt{{t_3}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\log \left(e^{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}\right)}^{2}}}{t_0}\right)\\ \end{array} \]

Alternative 18
Error	61.48%
Cost	73156

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2}\\ t_4 := \cos \left(\phi_2 \cdot -0.5\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\left(\sin \left(\phi_2 \cdot -0.5\right) + 0.5 \cdot \left(\phi_1 \cdot t_4\right)\right)}^{2}}}{t_0}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 5 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{{t_4}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{t_0}\right)\\ \end{array} \]

Alternative 19
Error	65.48%
Cost	66249

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)}\\ \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{-7} \lor \neg \left(\lambda_2 \leq 1.7 \cdot 10^{+20}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 - \cos \lambda_2 \cdot -0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \end{array} \]

Alternative 20
Error	65.19%
Cost	66112

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

Alternative 21
Error	70.43%
Cost	65984

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

Reproduce

herbie shell --seed 2023089 
(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))))))))))