Distance on a great circle

Average Error: 24.6 → 14.0

Time: 1.2min

Precision: binary64

Cost: 198464

Math

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

↓

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (+
      (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
      (*
       (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
       (sin (/ (- lambda1 lambda2) 2.0)))))
    (sqrt
     (-
      1.0
      (+
       (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
       (*
        (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
        (sin (/ (- lambda1 lambda2) 2.0))))))))))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0))
        (t_1 (- 1.0 t_0))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* (cos phi1) (cos phi2)))
        (t_4 (* t_3 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ t_0 (* t_3 (* t_2 t_2))))
      (sqrt (* (- (pow t_1 2.0) (pow t_4 2.0)) (/ 1.0 (+ 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 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double t_1 = 1.0 - t_0;
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = cos(phi1) * cos(phi2);
	double t_4 = t_3 * (0.5 - (0.5 * cos((lambda1 - lambda2))));
	return R * (2.0 * atan2(sqrt((t_0 + (t_3 * (t_2 * t_2)))), sqrt(((pow(t_1, 2.0) - pow(t_4, 2.0)) * (1.0 / (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 = ((sin((phi1 * 0.5d0)) * cos((0.5d0 * phi2))) - (cos((phi1 * 0.5d0)) * sin((0.5d0 * phi2)))) ** 2.0d0
    t_1 = 1.0d0 - t_0
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    t_3 = cos(phi1) * cos(phi2)
    t_4 = t_3 * (0.5d0 - (0.5d0 * cos((lambda1 - lambda2))))
    code = r * (2.0d0 * atan2(sqrt((t_0 + (t_3 * (t_2 * t_2)))), sqrt((((t_1 ** 2.0d0) - (t_4 ** 2.0d0)) * (1.0d0 / (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.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_1 = 1.0 - t_0;
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_3 = Math.cos(phi1) * Math.cos(phi2);
	double t_4 = t_3 * (0.5 - (0.5 * Math.cos((lambda1 - lambda2))));
	return R * (2.0 * Math.atan2(Math.sqrt((t_0 + (t_3 * (t_2 * t_2)))), Math.sqrt(((Math.pow(t_1, 2.0) - Math.pow(t_4, 2.0)) * (1.0 / (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.pow(((math.sin((phi1 * 0.5)) * math.cos((0.5 * phi2))) - (math.cos((phi1 * 0.5)) * math.sin((0.5 * phi2)))), 2.0)
	t_1 = 1.0 - t_0
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	t_3 = math.cos(phi1) * math.cos(phi2)
	t_4 = t_3 * (0.5 - (0.5 * math.cos((lambda1 - lambda2))))
	return R * (2.0 * math.atan2(math.sqrt((t_0 + (t_3 * (t_2 * t_2)))), math.sqrt(((math.pow(t_1, 2.0) - math.pow(t_4, 2.0)) * (1.0 / (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(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_1 = Float64(1.0 - t_0)
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(cos(phi1) * cos(phi2))
	t_4 = Float64(t_3 * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(t_3 * Float64(t_2 * t_2)))), sqrt(Float64(Float64((t_1 ^ 2.0) - (t_4 ^ 2.0)) * Float64(1.0 / Float64(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 = ((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))) ^ 2.0;
	t_1 = 1.0 - t_0;
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	t_3 = cos(phi1) * cos(phi2);
	t_4 = t_3 * (0.5 - (0.5 * cos((lambda1 - lambda2))));
	tmp = R * (2.0 * atan2(sqrt((t_0 + (t_3 * (t_2 * t_2)))), sqrt((((t_1 ^ 2.0) - (t_4 ^ 2.0)) * (1.0 / (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[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$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$3 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$1 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 24.6

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

2. Simplified 24.6

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

   [Start] 24.6

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

   associate-*l* [=>] 24.6

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

3. Applied egg-rr 24.0

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

4. Applied egg-rr 14.0

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

5. Applied egg-rr 14.0

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

6. Final simplification 14.0

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

Alternatives

Alternative 1
Error	14.0
Cost	132224

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

Alternative 2
Error	23.9
Cost	112832

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

Alternative 3
Error	24.5
Cost	92544

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

Alternative 4
Error	24.3
Cost	92360

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}\\ t_3 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ t_4 := \sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_3}\\ \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_1}}{t_4}\right)\\ \mathbf{elif}\;\phi_2 \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\left(0.5 + 0.5 \cdot \cos \phi_1\right) - \cos \phi_1 \cdot t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{t_4}\right)\\ \end{array} \]

Alternative 5
Error	24.3
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}\\ t_3 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ t_4 := \cos \phi_2 \cdot t_3\\ \mathbf{if}\;\phi_2 \leq -0.000155:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_1}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\left(0.5 + 0.5 \cdot \cos \phi_1\right) - \cos \phi_1 \cdot t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - t_4\right)}}\right)\\ \end{array} \]

Alternative 6
Error	24.3
Cost	86089

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)}\\ t_2 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -1.5 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{\left(0.5 + 0.5 \cdot \cos \phi_1\right) - \cos \phi_1 \cdot t_2}}\right)\\ \end{array} \]

Alternative 7
Error	33.2
Cost	85824

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

Alternative 8
Error	41.8
Cost	78912

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

Alternative 9
Error	41.9
Cost	66249

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)}\\ \mathbf{if}\;\lambda_1 \leq -1 \cdot 10^{-7} \lor \neg \left(\lambda_1 \leq 4.4 \cdot 10^{-31}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right)\\ \end{array} \]

Alternative 10
Error	41.8
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 11
Error	44.8
Cost	65984

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

Reproduce

herbie shell --seed 2023018 
(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))))))))))