Distance on a great circle

Average Error: 24.5 → 13.7

Time: 1.8min

Precision: binary64

Cost: 138564

\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_3 := t_0 \cdot \left(t_0 \cdot t_1\right) + t_2\\ \mathbf{if}\;\lambda_2 \leq 2.1336417472965286 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right) + t_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\lambda_2 \cdot -0.5\right)}^{2} \cdot t_1}}{\sqrt{1 - t_3}}\right)\\ \end{array} \]

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 (* (cos phi1) (cos phi2)))
        (t_2
         (pow
          (-
           (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
           (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
          2.0))
        (t_3 (+ (* t_0 (* t_0 t_1)) t_2)))
   (if (<= lambda2 2.1336417472965286e-5)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_3)
        (sqrt
         (-
          1.0
          (+
           (* (cos phi2) (* (cos phi1) (pow (sin (* 0.5 lambda1)) 2.0)))
           t_2))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (* (pow (sin (* lambda2 -0.5)) 2.0) t_1)))
        (sqrt (- 1.0 t_3))))))))

C

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

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0);
	double t_3 = (t_0 * (t_0 * t_1)) + t_2;
	double tmp;
	if (lambda2 <= 2.1336417472965286e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - ((cos(phi2) * (cos(phi1) * pow(sin((0.5 * lambda1)), 2.0))) + t_2)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (pow(sin((lambda2 * -0.5)), 2.0) * t_1))), sqrt((1.0 - t_3))));
	}
	return tmp;
}

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) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = cos(phi1) * cos(phi2)
    t_2 = ((cos((0.5d0 * phi2)) * sin((0.5d0 * phi1))) - (sin((0.5d0 * phi2)) * cos((0.5d0 * phi1)))) ** 2.0d0
    t_3 = (t_0 * (t_0 * t_1)) + t_2
    if (lambda2 <= 2.1336417472965286d-5) then
        tmp = r * (2.0d0 * atan2(sqrt(t_3), sqrt((1.0d0 - ((cos(phi2) * (cos(phi1) * (sin((0.5d0 * lambda1)) ** 2.0d0))) + t_2)))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_2 + ((sin((lambda2 * (-0.5d0))) ** 2.0d0) * t_1))), sqrt((1.0d0 - t_3))))
    end if
    code = tmp
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 = Math.cos(phi1) * Math.cos(phi2);
	double t_2 = Math.pow(((Math.cos((0.5 * phi2)) * Math.sin((0.5 * phi1))) - (Math.sin((0.5 * phi2)) * Math.cos((0.5 * phi1)))), 2.0);
	double t_3 = (t_0 * (t_0 * t_1)) + t_2;
	double tmp;
	if (lambda2 <= 2.1336417472965286e-5) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_3), Math.sqrt((1.0 - ((Math.cos(phi2) * (Math.cos(phi1) * Math.pow(Math.sin((0.5 * lambda1)), 2.0))) + t_2)))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_2 + (Math.pow(Math.sin((lambda2 * -0.5)), 2.0) * t_1))), Math.sqrt((1.0 - t_3))));
	}
	return tmp;
}

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 = math.cos(phi1) * math.cos(phi2)
	t_2 = math.pow(((math.cos((0.5 * phi2)) * math.sin((0.5 * phi1))) - (math.sin((0.5 * phi2)) * math.cos((0.5 * phi1)))), 2.0)
	t_3 = (t_0 * (t_0 * t_1)) + t_2
	tmp = 0
	if lambda2 <= 2.1336417472965286e-5:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_3), math.sqrt((1.0 - ((math.cos(phi2) * (math.cos(phi1) * math.pow(math.sin((0.5 * lambda1)), 2.0))) + t_2)))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_2 + (math.pow(math.sin((lambda2 * -0.5)), 2.0) * t_1))), math.sqrt((1.0 - t_3))))
	return tmp

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(cos(phi1) * cos(phi2))
	t_2 = Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0
	t_3 = Float64(Float64(t_0 * Float64(t_0 * t_1)) + t_2)
	tmp = 0.0
	if (lambda2 <= 2.1336417472965286e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - Float64(Float64(cos(phi2) * Float64(cos(phi1) * (sin(Float64(0.5 * lambda1)) ^ 2.0))) + t_2))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64((sin(Float64(lambda2 * -0.5)) ^ 2.0) * t_1))), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
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_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = cos(phi1) * cos(phi2);
	t_2 = ((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))) ^ 2.0;
	t_3 = (t_0 * (t_0 * t_1)) + t_2;
	tmp = 0.0;
	if (lambda2 <= 2.1336417472965286e-5)
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - ((cos(phi2) * (cos(phi1) * (sin((0.5 * lambda1)) ^ 2.0))) + t_2)))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_2 + ((sin((lambda2 * -0.5)) ^ 2.0) * t_1))), sqrt((1.0 - t_3))));
	end
	tmp_2 = tmp;
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[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[lambda2, 2.1336417472965286e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.13364174729652858e-5

   1. Initial program 19.4

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

   2. Applied egg-rr 18.8

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

   3. Applied egg-rr 6.7

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

   4. Taylor expanded in lambda2 around 0 6.7

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

   if 2.13364174729652858e-5 < lambda2

   1. Initial program 30.8

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

   2. Applied egg-rr 30.2

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

   3. Applied egg-rr 22.4

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

   4. Taylor expanded in lambda1 around 0 22.5

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

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.1336417472965286 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right) + {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + {\sin \left(\lambda_2 \cdot -0.5\right)}^{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	13.6
Cost	145216

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

Alternative 2
Error	13.6
Cost	138880

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

Alternative 3
Error	17.4
Cost	138564

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := t_0 \cdot \left(t_0 \cdot t_1\right)\\ t_3 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ \mathbf{if}\;\lambda_1 \leq -13947.983273143316:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right) + t_3\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_3}}{\sqrt{1 - \left(t_3 + {\sin \left(\lambda_2 \cdot -0.5\right)}^{2} \cdot t_1\right)}}\right)\\ \end{array} \]

Alternative 4
Error	23.8
Cost	112960

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

Alternative 5
Error	24.4
Cost	92864

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

Alternative 6
Error	31.8
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq 6.873882961343751 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 7
Error	30.6
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.3527832828207906:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 8
Error	28.5
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq 3.6479734147380136:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(0.5 \cdot \phi_2\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 9
Error	36.1
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ t_2 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq 2.228220978995804 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{t_2}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot t_2\right)}^{2}}}{\sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \end{array} \]

Alternative 10
Error	37.3
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.3527832828207906:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{0.5 + 0.5 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}}\right)\\ \end{array} \]

Alternative 11
Error	41.1
Cost	85952

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

Alternative 12
Error	41.6
Cost	85568

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

Alternative 13
Error	41.8
Cost	78912

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

Alternative 14
Error	42.0
Cost	66116

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\lambda_1 \leq -1.0327750563756326 \cdot 10^{-63}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right)\\ \end{array} \]

Alternative 15
Error	41.8
Cost	66112

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

Alternative 16
Error	45.0
Cost	65984

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

Alternative 17
Error	58.4
Cost	65856

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

Reproduce

herbie shell --seed 2022308 
(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))))))))))