Distance on a great circle

Average Error: 23.7 → 13.6

Time: 2.0min

Precision: binary64

Cost: 138880

Math

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

↓

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

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

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

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

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

Julia

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

↓

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

Derivation

1. Initial program 23.7

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

2. Applied egg-rr 23.2

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

3. Applied egg-rr 13.6

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

4. Final simplification 13.6

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

Alternatives

Alternative 1
Error	18.0
Cost	138697

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ t_3 := \sqrt{1 - \left(t_1 + t_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{+14} \lor \neg \left(\lambda_2 \leq 200000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{t_3}\right)\\ \end{array} \]

Alternative 2
Error	14.6
Cost	138697

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \sqrt{1 - \left(t_2 \cdot \left(t_0 \cdot t_2\right) + t_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -12 \lor \neg \left(\lambda_2 \leq 200000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{t_3}\right)\\ \end{array} \]

Alternative 3
Error	13.9
Cost	138697

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t_2 \cdot \left(t_0 \cdot t_2\right) + t_1\\ \mathbf{if}\;\lambda_2 \leq -12 \lor \neg \left(\lambda_2 \leq 200000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)\right)}}\right)\\ \end{array} \]

Alternative 4
Error	13.6
Cost	138560

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

Alternative 5
Error	23.1
Cost	119040

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

Alternative 6
Error	23.2
Cost	119040

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

Alternative 7
Error	23.6
Cost	93128

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t_2 \cdot \left(t_1 \cdot t_2\right)\\ t_4 := t_3 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -0.0285:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_1 \cdot \left(t_2 \cdot t_2\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + \left(0.5 - 0.5 \cdot \cos \left(\phi_1 - \phi_2\right)\right)}}{\sqrt{1 - t_4}}\right)\\ \end{array} \]

Alternative 8
Error	23.7
Cost	93120

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

Alternative 9
Error	23.5
Cost	92361

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -3.9 \cdot 10^{-16} \lor \neg \left(\phi_1 \leq 0.00026\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_2 \cdot t_0\right) + t_3}}{\sqrt{\left(1 - t_3\right) - \cos \phi_1 \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2 \cdot \left(t_0 \cdot t_0\right)}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_1\right)}}\right)\\ \end{array} \]

Alternative 10
Error	23.5
Cost	92361

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -3.9 \cdot 10^{-16} \lor \neg \left(\phi_1 \leq 5.7 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_2 \cdot t_0\right) + t_3}}{\sqrt{\left(1 - t_3\right) - \cos \phi_1 \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2 \cdot \left(t_0 \cdot t_0\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_1}}\right)\\ \end{array} \]

Alternative 11
Error	23.5
Cost	92361

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -0.03 \lor \neg \left(\phi_2 \leq 2.25 \cdot 10^{-9}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_1}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \end{array} \]

Alternative 12
Error	23.5
Cost	92361

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -0.0285 \lor \neg \left(\phi_2 \leq 2.25 \cdot 10^{-9}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_1 \cdot \left(t_2 \cdot t_2\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot \left(t_1 \cdot t_2\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_0}}\right)\\ \end{array} \]

Alternative 13
Error	26.5
Cost	92233

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_1 \leq -0.0004 \lor \neg \left(\phi_1 \leq 0.002\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot \left(t_3 \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot \left(t_3 \cdot t_3\right)}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_2\right)}}\right)\\ \end{array} \]

Alternative 14
Error	32.8
Cost	85824

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

Alternative 15
Error	41.9
Cost	78912

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

Alternative 16
Error	42.1
Cost	66249

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{if}\;\lambda_2 \leq -1.7 \cdot 10^{-33} \lor \neg \left(\lambda_2 \leq 2.9 \cdot 10^{-34}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \end{array} \]

Alternative 17
Error	41.9
Cost	66112

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

Alternative 18
Error	44.9
Cost	65984

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

Reproduce

herbie shell --seed 2023187 
(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))))))))))