Distance on a great circle

Average Error: 24.9 → 14.3

Time: 1.6min

Precision: binary64

Cost: 138564

\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_3 := {\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot t_2 + t_0\right)}^{2}\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := \sqrt{{\left(t_0 - \sin \left(\phi_2 \cdot 0.5\right) \cdot t_2\right)}^{2} + t_4 \cdot \left(t_4 \cdot t_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.000106:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right) + t_3\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - \left(t_3 + t_1 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

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 (* phi2 0.5)) (sin (* phi1 0.5))))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (cos (* phi1 0.5)))
        (t_3 (pow (+ (* (sin (* phi2 -0.5)) t_2) t_0) 2.0))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5
         (sqrt
          (+
           (pow (- t_0 (* (sin (* phi2 0.5)) t_2)) 2.0)
           (* t_4 (* t_4 t_1))))))
   (if (<= lambda1 -0.000106)
     (*
      R
      (*
       2.0
       (atan2
        t_5
        (sqrt
         (-
          1.0
          (+
           (* (cos phi2) (* (cos phi1) (pow (sin (* lambda1 0.5)) 2.0)))
           t_3))))))
     (*
      R
      (*
       2.0
       (atan2
        t_5
        (sqrt (- 1.0 (+ t_3 (* t_1 (pow (sin (* lambda2 -0.5)) 2.0)))))))))))

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((phi2 * 0.5)) * sin((phi1 * 0.5));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = cos((phi1 * 0.5));
	double t_3 = pow(((sin((phi2 * -0.5)) * t_2) + t_0), 2.0);
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = sqrt((pow((t_0 - (sin((phi2 * 0.5)) * t_2)), 2.0) + (t_4 * (t_4 * t_1))));
	double tmp;
	if (lambda1 <= -0.000106) {
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - ((cos(phi2) * (cos(phi1) * pow(sin((lambda1 * 0.5)), 2.0))) + t_3)))));
	} else {
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - (t_3 + (t_1 * pow(sin((lambda2 * -0.5)), 2.0)))))));
	}
	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) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = cos((phi2 * 0.5d0)) * sin((phi1 * 0.5d0))
    t_1 = cos(phi1) * cos(phi2)
    t_2 = cos((phi1 * 0.5d0))
    t_3 = ((sin((phi2 * (-0.5d0))) * t_2) + t_0) ** 2.0d0
    t_4 = sin(((lambda1 - lambda2) / 2.0d0))
    t_5 = sqrt((((t_0 - (sin((phi2 * 0.5d0)) * t_2)) ** 2.0d0) + (t_4 * (t_4 * t_1))))
    if (lambda1 <= (-0.000106d0)) then
        tmp = r * (2.0d0 * atan2(t_5, sqrt((1.0d0 - ((cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5d0)) ** 2.0d0))) + t_3)))))
    else
        tmp = r * (2.0d0 * atan2(t_5, sqrt((1.0d0 - (t_3 + (t_1 * (sin((lambda2 * (-0.5d0))) ** 2.0d0)))))))
    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.cos((phi2 * 0.5)) * Math.sin((phi1 * 0.5));
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double t_2 = Math.cos((phi1 * 0.5));
	double t_3 = Math.pow(((Math.sin((phi2 * -0.5)) * t_2) + t_0), 2.0);
	double t_4 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_5 = Math.sqrt((Math.pow((t_0 - (Math.sin((phi2 * 0.5)) * t_2)), 2.0) + (t_4 * (t_4 * t_1))));
	double tmp;
	if (lambda1 <= -0.000106) {
		tmp = R * (2.0 * Math.atan2(t_5, Math.sqrt((1.0 - ((Math.cos(phi2) * (Math.cos(phi1) * Math.pow(Math.sin((lambda1 * 0.5)), 2.0))) + t_3)))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_5, Math.sqrt((1.0 - (t_3 + (t_1 * Math.pow(Math.sin((lambda2 * -0.5)), 2.0)))))));
	}
	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.cos((phi2 * 0.5)) * math.sin((phi1 * 0.5))
	t_1 = math.cos(phi1) * math.cos(phi2)
	t_2 = math.cos((phi1 * 0.5))
	t_3 = math.pow(((math.sin((phi2 * -0.5)) * t_2) + t_0), 2.0)
	t_4 = math.sin(((lambda1 - lambda2) / 2.0))
	t_5 = math.sqrt((math.pow((t_0 - (math.sin((phi2 * 0.5)) * t_2)), 2.0) + (t_4 * (t_4 * t_1))))
	tmp = 0
	if lambda1 <= -0.000106:
		tmp = R * (2.0 * math.atan2(t_5, math.sqrt((1.0 - ((math.cos(phi2) * (math.cos(phi1) * math.pow(math.sin((lambda1 * 0.5)), 2.0))) + t_3)))))
	else:
		tmp = R * (2.0 * math.atan2(t_5, math.sqrt((1.0 - (t_3 + (t_1 * math.pow(math.sin((lambda2 * -0.5)), 2.0)))))))
	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 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(phi1 * 0.5)))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = cos(Float64(phi1 * 0.5))
	t_3 = Float64(Float64(sin(Float64(phi2 * -0.5)) * t_2) + t_0) ^ 2.0
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = sqrt(Float64((Float64(t_0 - Float64(sin(Float64(phi2 * 0.5)) * t_2)) ^ 2.0) + Float64(t_4 * Float64(t_4 * t_1))))
	tmp = 0.0
	if (lambda1 <= -0.000106)
		tmp = Float64(R * Float64(2.0 * atan(t_5, sqrt(Float64(1.0 - Float64(Float64(cos(phi2) * Float64(cos(phi1) * (sin(Float64(lambda1 * 0.5)) ^ 2.0))) + t_3))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_5, sqrt(Float64(1.0 - Float64(t_3 + Float64(t_1 * (sin(Float64(lambda2 * -0.5)) ^ 2.0))))))));
	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 = cos((phi2 * 0.5)) * sin((phi1 * 0.5));
	t_1 = cos(phi1) * cos(phi2);
	t_2 = cos((phi1 * 0.5));
	t_3 = ((sin((phi2 * -0.5)) * t_2) + t_0) ^ 2.0;
	t_4 = sin(((lambda1 - lambda2) / 2.0));
	t_5 = sqrt((((t_0 - (sin((phi2 * 0.5)) * t_2)) ^ 2.0) + (t_4 * (t_4 * t_1))));
	tmp = 0.0;
	if (lambda1 <= -0.000106)
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - ((cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5)) ^ 2.0))) + t_3)))));
	else
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - (t_3 + (t_1 * (sin((lambda2 * -0.5)) ^ 2.0)))))));
	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[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[Power[N[(t$95$0 - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$4 * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -0.000106], N[(R * N[(2.0 * N[ArcTan[t$95$5 / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$5 / N[Sqrt[N[(1.0 - N[(t$95$3 + N[(t$95$1 * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.06e-4

   1. Initial program 31.3

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

   2. Applied egg-rr 30.7

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

   3. Applied egg-rr 22.8

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

   4. Simplified 22.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
      Proof
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 -1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (Rewrite=> *-commutative_binary64 (*.f64 phi2 -1/2))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 phi2 (Rewrite<= metadata-eval (neg.f64 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 phi2 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (Rewrite=> sin-neg_binary64 (neg.f64 (sin.f64 (*.f64 phi2 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (cos.f64 (*.f64 1/2 phi1)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (neg.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi1 1/2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2))))): 8 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi1 1/2)))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2)))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in lambda2 around 0 22.9

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

   if -1.06e-4 < lambda1

   1. Initial program 19.8

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

   2. Applied egg-rr 19.3

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

   3. Applied egg-rr 7.3

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

   4. Simplified 7.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
      Proof
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 -1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (Rewrite=> *-commutative_binary64 (*.f64 phi2 -1/2))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 phi2 (Rewrite<= metadata-eval (neg.f64 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 phi2 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (Rewrite=> sin-neg_binary64 (neg.f64 (sin.f64 (*.f64 phi2 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (cos.f64 (*.f64 1/2 phi1)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (neg.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi1 1/2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2))))): 8 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi1 1/2)))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2)))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in lambda1 around 0 7.4

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

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.000106:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right) + {\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	14.2
Cost	151424

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

Alternative 2
Error	14.2
Cost	145152

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

Alternative 3
Error	14.2
Cost	138880

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

Alternative 4
Error	17.9
Cost	138564

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_2 := \cos \left(\phi_2 \cdot 0.5\right) \cdot t_1\\ t_3 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_4 := \sin \left(\phi_2 \cdot -0.5\right) \cdot t_3\\ t_5 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_2 \leq 9.5:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(t_2 - \sin \left(\phi_2 \cdot 0.5\right) \cdot t_3\right)}^{2} + t_5 \cdot \left(t_5 \cdot t_0\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right) + {\left(t_4 + t_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2} + {\left(t_4 + t_1 \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(t_5, \cos \phi_1 \cdot \left(t_5 \cdot \cos \phi_2\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 5
Error	24.3
Cost	112704

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

Alternative 6
Error	24.6
Cost	105408

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

Alternative 7
Error	24.9
Cost	92544

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

Alternative 8
Error	24.7
Cost	92356

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.00145:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{1 - \left(t_0 \cdot \left(t_0 \cdot t_1\right) + t_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1 \cdot \left(t_0 \cdot t_0\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 9
Error	27.4
Cost	92228

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.0056:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \left(t_1 \cdot t_2\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot \left(t_1 \cdot t_1\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 10
Error	24.7
Cost	92228

\[\begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_0}}\right)\\ \end{array} \]

Alternative 11
Error	33.9
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ t_2 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.015:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\left(\left(\phi_2 \cdot -0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) + \sin \left(\phi_1 \cdot 0.5\right) \cdot \left(1 + -0.125 \cdot \left(\phi_2 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \end{array} \]

Alternative 12
Error	28.2
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.0036:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t_0 \cdot t_1\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 \cdot \left(t_0 \cdot t_0\right) + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 13
Error	42.1
Cost	85248

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

Alternative 14
Error	42.1
Cost	85248

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

Alternative 15
Error	42.1
Cost	66112

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

Alternative 16
Error	57.9
Cost	60160

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot 0.0625 + -0.0026041666666666665 \cdot {\phi_1}^{3}\right)}{\sqrt{1 - \left(t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \end{array} \]

Alternative 17
Error	59.4
Cost	59840

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(\phi_1 \cdot 0.5\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.125\right)}{\sqrt{1 - \left(t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right) \end{array} \]

Alternative 18
Error	59.4
Cost	59840

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(\phi_1 \cdot 0.5\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1 \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.125\right)}{\sqrt{1 - \left(t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {t_1}^{2}\right)}}\right) \end{array} \]

Alternative 19
Error	60.2
Cost	53440

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot 0.0625\right)}{\sqrt{1 - \left(t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \end{array} \]

Reproduce

herbie shell --seed 2022330 
(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))))))))))