Distance on a great circle

Percentage Accurate: 61.8% → 78.2%

Time: 50.5s

Alternatives: 26

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

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

Python

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

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[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] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

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

Python

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

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[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] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_4 := \sin \left(\phi_2 \cdot 0.5\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_4, -t\_0, t\_1\right)\right)}^{2} + t\_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right)}}{\sqrt{1 - \left({\left(t\_1 - t\_4 \cdot t\_0\right)}^{2} + \left(\cos \phi_1 \cdot t\_3\right) \cdot \left(\cos \phi_2 \cdot t\_3\right)\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (sin (* 0.5 (- lambda1 lambda2))))
        (t_4 (sin (* phi2 0.5))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (fma t_4 (- t_0) t_1) 2.0)
        (* t_2 (* (* (cos phi1) (cos phi2)) t_2))))
      (sqrt
       (-
        1.0
        (+
         (pow (- t_1 (* t_4 t_0)) 2.0)
         (* (* (cos phi1) t_3) (* (cos phi2) t_3))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = sin((0.5 * (lambda1 - lambda2)));
	double t_4 = sin((phi2 * 0.5));
	return R * (2.0 * atan2(sqrt((pow(fma(t_4, -t_0, t_1), 2.0) + (t_2 * ((cos(phi1) * cos(phi2)) * t_2)))), sqrt((1.0 - (pow((t_1 - (t_4 * t_0)), 2.0) + ((cos(phi1) * t_3) * (cos(phi2) * t_3)))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_4 = sin(Float64(phi2 * 0.5))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_4, Float64(-t_0), t_1) ^ 2.0) + Float64(t_2 * Float64(Float64(cos(phi1) * cos(phi2)) * t_2)))), sqrt(Float64(1.0 - Float64((Float64(t_1 - Float64(t_4 * t_0)) ^ 2.0) + Float64(Float64(cos(phi1) * t_3) * Float64(cos(phi2) * t_3))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(t$95$4 * (-t$95$0) + t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$2 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[(t$95$1 - N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

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

   2. sin-diff N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-*.f64 N/A

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

   10. sin-diff N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

6. Applied rewrites 80.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

8. Applied rewrites 80.5%

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

9. Final simplification 80.5%

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

Alternative 2: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t\_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right)\\ t_4 := t\_3 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_5 := \cos \left(0.5 \cdot \phi_1\right)\\ t_6 := {\left(t\_1 - t\_0 \cdot t\_5\right)}^{2}\\ t_7 := {\left(\mathsf{fma}\left(t\_0, -t\_5, t\_1\right)\right)}^{2}\\ \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.2:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7 + t\_3}}{\sqrt{1 - \left(t\_6 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\right)}}{\sqrt{1 - \left(t\_3 + t\_6\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi2 0.5)))
        (t_1 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* t_2 (* (* (cos phi1) (cos phi2)) t_2)))
        (t_4 (+ t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (t_5 (cos (* 0.5 phi1)))
        (t_6 (pow (- t_1 (* t_0 t_5)) 2.0))
        (t_7 (pow (fma t_0 (- t_5) t_1) 2.0)))
   (if (<= (atan2 (sqrt t_4) (sqrt (- 1.0 t_4))) 0.2)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_7 t_3))
        (sqrt
         (-
          1.0
          (+
           t_6
           (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          t_7
          (*
           (cos phi1)
           (* (cos phi2) (fma -0.5 (cos (- lambda1 lambda2)) 0.5)))))
        (sqrt (- 1.0 (+ t_3 t_6)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 * 0.5));
	double t_1 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_2 * ((cos(phi1) * cos(phi2)) * t_2);
	double t_4 = t_3 + pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_5 = cos((0.5 * phi1));
	double t_6 = pow((t_1 - (t_0 * t_5)), 2.0);
	double t_7 = pow(fma(t_0, -t_5, t_1), 2.0);
	double tmp;
	if (atan2(sqrt(t_4), sqrt((1.0 - t_4))) <= 0.2) {
		tmp = R * (2.0 * atan2(sqrt((t_7 + t_3)), sqrt((1.0 - (t_6 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_7 + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda1 - lambda2)), 0.5))))), sqrt((1.0 - (t_3 + t_6)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 * 0.5))
	t_1 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(t_2 * Float64(Float64(cos(phi1) * cos(phi2)) * t_2))
	t_4 = Float64(t_3 + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))
	t_5 = cos(Float64(0.5 * phi1))
	t_6 = Float64(t_1 - Float64(t_0 * t_5)) ^ 2.0
	t_7 = fma(t_0, Float64(-t_5), t_1) ^ 2.0
	tmp = 0.0
	if (atan(sqrt(t_4), sqrt(Float64(1.0 - t_4))) <= 0.2)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_7 + t_3)), sqrt(Float64(1.0 - Float64(t_6 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_7 + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5))))), sqrt(Float64(1.0 - Float64(t_3 + t_6))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(t$95$1 - N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$7 = N[Power[N[(t$95$0 * (-t$95$5) + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.2], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$7 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$6 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$7 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$3 + t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))) < 0.20000000000000001

   1. Initial program 83.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 83.7

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

   4. Applied rewrites 83.7%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 86.2%

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

   7. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 86.2

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

   9. Applied rewrites 86.2%

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

   if 0.20000000000000001 < (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))

   1. Initial program 60.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 61.8

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

   4. Applied rewrites 61.8%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 79.7%

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

   8. Applied rewrites 79.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \leq 0.2:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\right)}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right)\\ t_2 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_3 := {\left(\mathsf{fma}\left(t\_0, -t\_1, t\_2\right)\right)}^{2}\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := t\_4 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_4\right)\\ t_6 := \sqrt{t\_3 + t\_5}\\ t_7 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_8 := \cos \phi_1 \cdot t\_7\\ t_9 := {\left(t\_2 - t\_0 \cdot t\_1\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.0042:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + t\_8}}{\sqrt{1 - \left(t\_5 + t\_9\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_6}{\sqrt{1 - \left(t\_9 + \cos \phi_2 \cdot t\_7\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_6}{\sqrt{1 - \left(t\_9 + t\_8\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi2 0.5)))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_3 (pow (fma t_0 (- t_1) t_2) 2.0))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (* t_4 (* (* (cos phi1) (cos phi2)) t_4)))
        (t_6 (sqrt (+ t_3 t_5)))
        (t_7 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_8 (* (cos phi1) t_7))
        (t_9 (pow (- t_2 (* t_0 t_1)) 2.0)))
   (if (<= phi1 -0.0042)
     (* R (* 2.0 (atan2 (sqrt (+ t_3 t_8)) (sqrt (- 1.0 (+ t_5 t_9))))))
     (if (<= phi1 45000000000000.0)
       (* R (* 2.0 (atan2 t_6 (sqrt (- 1.0 (+ t_9 (* (cos phi2) t_7)))))))
       (* R (* 2.0 (atan2 t_6 (sqrt (- 1.0 (+ t_9 t_8))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 * 0.5));
	double t_1 = cos((0.5 * phi1));
	double t_2 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_3 = pow(fma(t_0, -t_1, t_2), 2.0);
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = t_4 * ((cos(phi1) * cos(phi2)) * t_4);
	double t_6 = sqrt((t_3 + t_5));
	double t_7 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_8 = cos(phi1) * t_7;
	double t_9 = pow((t_2 - (t_0 * t_1)), 2.0);
	double tmp;
	if (phi1 <= -0.0042) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + t_8)), sqrt((1.0 - (t_5 + t_9)))));
	} else if (phi1 <= 45000000000000.0) {
		tmp = R * (2.0 * atan2(t_6, sqrt((1.0 - (t_9 + (cos(phi2) * t_7))))));
	} else {
		tmp = R * (2.0 * atan2(t_6, sqrt((1.0 - (t_9 + t_8)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 * 0.5))
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_3 = fma(t_0, Float64(-t_1), t_2) ^ 2.0
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = Float64(t_4 * Float64(Float64(cos(phi1) * cos(phi2)) * t_4))
	t_6 = sqrt(Float64(t_3 + t_5))
	t_7 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_8 = Float64(cos(phi1) * t_7)
	t_9 = Float64(t_2 - Float64(t_0 * t_1)) ^ 2.0
	tmp = 0.0
	if (phi1 <= -0.0042)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + t_8)), sqrt(Float64(1.0 - Float64(t_5 + t_9))))));
	elseif (phi1 <= 45000000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(t_6, sqrt(Float64(1.0 - Float64(t_9 + Float64(cos(phi2) * t_7)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_6, sqrt(Float64(1.0 - Float64(t_9 + t_8))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(t$95$0 * (-t$95$1) + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(t$95$3 + t$95$5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$8 = N[(N[Cos[phi1], $MachinePrecision] * t$95$7), $MachinePrecision]}, Block[{t$95$9 = N[Power[N[(t$95$2 - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[phi1, -0.0042], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$8), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$5 + t$95$9), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 45000000000000.0], N[(R * N[(2.0 * N[ArcTan[t$95$6 / N[Sqrt[N[(1.0 - N[(t$95$9 + N[(N[Cos[phi2], $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$6 / N[Sqrt[N[(1.0 - N[(t$95$9 + t$95$8), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.00419999999999999974

   1. Initial program 46.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 48.1

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

   4. Applied rewrites 48.1%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 78.9%

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

   7. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 58.3

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

   9. Applied rewrites 58.3%

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

   if -0.00419999999999999974 < phi1 < 4.5e13

   1. Initial program 79.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 79.9

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

   4. Applied rewrites 79.9%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 82.5%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 80.8

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

   9. Applied rewrites 80.8%

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

   if 4.5e13 < phi1

   1. Initial program 44.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 47.1

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

   4. Applied rewrites 47.1%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 77.9%

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

   7. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 57.5

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

   9. Applied rewrites 57.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0042:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_4 := {\left(t\_3 - t\_0 \cdot t\_1\right)}^{2}\\ t_5 := \sqrt{{\left(\mathsf{fma}\left(t\_0, -t\_1, t\_3\right)\right)}^{2} + t\_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right)}\\ t_6 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_7 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_5}{\sqrt{1 - \left(t\_4 + \cos \phi_1 \cdot t\_6\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -0.0145:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_5}{\sqrt{1 - \left(t\_4 + \cos \phi_2 \cdot t\_6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi2 0.5)))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_4 (pow (- t_3 (* t_0 t_1)) 2.0))
        (t_5
         (sqrt
          (+
           (pow (fma t_0 (- t_1) t_3) 2.0)
           (* t_2 (* (* (cos phi1) (cos phi2)) t_2)))))
        (t_6 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_7
         (* R (* 2.0 (atan2 t_5 (sqrt (- 1.0 (+ t_4 (* (cos phi1) t_6)))))))))
   (if (<= phi1 -0.0145)
     t_7
     (if (<= phi1 45000000000000.0)
       (* R (* 2.0 (atan2 t_5 (sqrt (- 1.0 (+ t_4 (* (cos phi2) t_6)))))))
       t_7))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 * 0.5));
	double t_1 = cos((0.5 * phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_4 = pow((t_3 - (t_0 * t_1)), 2.0);
	double t_5 = sqrt((pow(fma(t_0, -t_1, t_3), 2.0) + (t_2 * ((cos(phi1) * cos(phi2)) * t_2))));
	double t_6 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_7 = R * (2.0 * atan2(t_5, sqrt((1.0 - (t_4 + (cos(phi1) * t_6))))));
	double tmp;
	if (phi1 <= -0.0145) {
		tmp = t_7;
	} else if (phi1 <= 45000000000000.0) {
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - (t_4 + (cos(phi2) * t_6))))));
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 * 0.5))
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_4 = Float64(t_3 - Float64(t_0 * t_1)) ^ 2.0
	t_5 = sqrt(Float64((fma(t_0, Float64(-t_1), t_3) ^ 2.0) + Float64(t_2 * Float64(Float64(cos(phi1) * cos(phi2)) * t_2))))
	t_6 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_7 = Float64(R * Float64(2.0 * atan(t_5, sqrt(Float64(1.0 - Float64(t_4 + Float64(cos(phi1) * t_6)))))))
	tmp = 0.0
	if (phi1 <= -0.0145)
		tmp = t_7;
	elseif (phi1 <= 45000000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(t_5, sqrt(Float64(1.0 - Float64(t_4 + Float64(cos(phi2) * t_6)))))));
	else
		tmp = t_7;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(t$95$3 - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[Power[N[(t$95$0 * (-t$95$1) + t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$2 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$7 = N[(R * N[(2.0 * N[ArcTan[t$95$5 / N[Sqrt[N[(1.0 - N[(t$95$4 + N[(N[Cos[phi1], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.0145], t$95$7, If[LessEqual[phi1, 45000000000000.0], N[(R * N[(2.0 * N[ArcTan[t$95$5 / N[Sqrt[N[(1.0 - N[(t$95$4 + N[(N[Cos[phi2], $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$7]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0145000000000000007 or 4.5e13 < phi1

   1. Initial program 45.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 78.4%

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

   7. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 57.0

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

   9. Applied rewrites 57.0%

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

   if -0.0145000000000000007 < phi1 < 4.5e13

   1. Initial program 79.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 79.9

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

   4. Applied rewrites 79.9%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 82.5%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 80.8

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

   9. Applied rewrites 80.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0145:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \cos \left(0.5 \cdot \phi_1\right)\\ t_3 := -0.5 \cdot \left(-\lambda_1\right)\\ t_4 := \cos \left(\phi_2 \cdot 0.5\right) \cdot t\_1\\ t_5 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_6 := t\_5 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_5\right)\\ t_7 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_0, -t\_2, t\_4\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left(t\_6 + {\left(t\_4 - t\_0 \cdot t\_2\right)}^{2}\right)}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.08 \cdot 10^{-6}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;\phi_2 \leq 27.5:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot -0.5\right), \cos t\_3, \cos \left(\lambda_2 \cdot -0.5\right) \cdot \sin t\_3\right)\right)}^{2}, \cos \phi_1, t\_1 \cdot \mathsf{fma}\left(t\_2, -\phi_2, t\_1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi2 0.5)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (cos (* 0.5 phi1)))
        (t_3 (* -0.5 (- lambda1)))
        (t_4 (* (cos (* phi2 0.5)) t_1))
        (t_5 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_6 (* t_5 (* (* (cos phi1) (cos phi2)) t_5)))
        (t_7
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (+
              (pow (fma t_0 (- t_2) t_4) 2.0)
              (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
            (sqrt (- 1.0 (+ t_6 (pow (- t_4 (* t_0 t_2)) 2.0)))))))))
   (if (<= phi2 -1.08e-6)
     t_7
     (if (<= phi2 27.5)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_6 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (pow
              (fma
               (sin (* lambda2 -0.5))
               (cos t_3)
               (* (cos (* lambda2 -0.5)) (sin t_3)))
              2.0)
             (cos phi1)
             (* t_1 (fma t_2 (- phi2) t_1))))))))
       t_7))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 * 0.5));
	double t_1 = sin((0.5 * phi1));
	double t_2 = cos((0.5 * phi1));
	double t_3 = -0.5 * -lambda1;
	double t_4 = cos((phi2 * 0.5)) * t_1;
	double t_5 = sin(((lambda1 - lambda2) / 2.0));
	double t_6 = t_5 * ((cos(phi1) * cos(phi2)) * t_5);
	double t_7 = R * (2.0 * atan2(sqrt((pow(fma(t_0, -t_2, t_4), 2.0) + (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))), sqrt((1.0 - (t_6 + pow((t_4 - (t_0 * t_2)), 2.0))))));
	double tmp;
	if (phi2 <= -1.08e-6) {
		tmp = t_7;
	} else if (phi2 <= 27.5) {
		tmp = R * (2.0 * atan2(sqrt((t_6 + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - fma(pow(fma(sin((lambda2 * -0.5)), cos(t_3), (cos((lambda2 * -0.5)) * sin(t_3))), 2.0), cos(phi1), (t_1 * fma(t_2, -phi2, t_1)))))));
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 * 0.5))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = cos(Float64(0.5 * phi1))
	t_3 = Float64(-0.5 * Float64(-lambda1))
	t_4 = Float64(cos(Float64(phi2 * 0.5)) * t_1)
	t_5 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_6 = Float64(t_5 * Float64(Float64(cos(phi1) * cos(phi2)) * t_5))
	t_7 = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_0, Float64(-t_2), t_4) ^ 2.0) + Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))), sqrt(Float64(1.0 - Float64(t_6 + (Float64(t_4 - Float64(t_0 * t_2)) ^ 2.0)))))))
	tmp = 0.0
	if (phi2 <= -1.08e-6)
		tmp = t_7;
	elseif (phi2 <= 27.5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_6 + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - fma((fma(sin(Float64(lambda2 * -0.5)), cos(t_3), Float64(cos(Float64(lambda2 * -0.5)) * sin(t_3))) ^ 2.0), cos(phi1), Float64(t_1 * fma(t_2, Float64(-phi2), t_1))))))));
	else
		tmp = t_7;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * (-lambda1)), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(t$95$0 * (-t$95$2) + t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$6 + N[Power[N[(t$95$4 - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.08e-6], t$95$7, If[LessEqual[phi2, 27.5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$6 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[(N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision] + N[(N[Cos[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(t$95$1 * N[(t$95$2 * (-phi2) + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$7]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.08000000000000004e-6 or 27.5 < phi2

   1. Initial program 51.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 53.2

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

   4. Applied rewrites 53.2%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), \mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right)\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 62.5

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

   9. Applied rewrites 62.5%

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

   if -1.08000000000000004e-6 < phi2 < 27.5

   1. Initial program 75.7%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 75.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left({\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}, \cos \phi_1, \sin \left(0.5 \cdot \phi_1\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_1\right), -\phi_2, \sin \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. sin-sum N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 76.3

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

   7. Applied rewrites 76.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.08 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 27.5:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot -0.5\right), \cos \left(-0.5 \cdot \left(-\lambda_1\right)\right), \cos \left(\lambda_2 \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \left(-\lambda_1\right)\right)\right)\right)}^{2}, \cos \phi_1, \sin \left(0.5 \cdot \phi_1\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_1\right), -\phi_2, \sin \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \sin \left(\phi_2 \cdot 0.5\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_3, -t\_0, t\_1\right)\right)}^{2} + t\_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right)}}{\sqrt{1 - \left({\left(t\_1 - t\_3 \cdot t\_0\right)}^{2} + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\right)\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (sin (* phi2 0.5))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (fma t_3 (- t_0) t_1) 2.0)
        (* t_2 (* (* (cos phi1) (cos phi2)) t_2))))
      (sqrt
       (-
        1.0
        (+
         (pow (- t_1 (* t_3 t_0)) 2.0)
         (*
          (cos phi2)
          (* (cos phi1) (fma -0.5 (cos (- lambda1 lambda2)) 0.5)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = sin((phi2 * 0.5));
	return R * (2.0 * atan2(sqrt((pow(fma(t_3, -t_0, t_1), 2.0) + (t_2 * ((cos(phi1) * cos(phi2)) * t_2)))), sqrt((1.0 - (pow((t_1 - (t_3 * t_0)), 2.0) + (cos(phi2) * (cos(phi1) * fma(-0.5, cos((lambda1 - lambda2)), 0.5))))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = sin(Float64(phi2 * 0.5))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_3, Float64(-t_0), t_1) ^ 2.0) + Float64(t_2 * Float64(Float64(cos(phi1) * cos(phi2)) * t_2)))), sqrt(Float64(1.0 - Float64((Float64(t_1 - Float64(t_3 * t_0)) ^ 2.0) + Float64(cos(phi2) * Float64(cos(phi1) * fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5)))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(t$95$3 * (-t$95$0) + t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$2 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[(t$95$1 - N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

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

   2. sin-diff N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-*.f64 N/A

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

   10. sin-diff N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

6. Applied rewrites 80.5%

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

8. Applied rewrites 80.5%

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

9. Final simplification 80.5%

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

Alternative 7: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t\_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;t\_2 \leq 0.02:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - t\_3\right), {\sin t\_0}^{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(t\_3 - 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (- phi1 phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (* t_1 (* (* (cos phi1) (cos phi2)) t_1))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (t_3 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
   (if (<= t_2 0.02)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_2)
        (sqrt (- 1.0 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))))))
     (*
      (* R 2.0)
      (atan2
       (sqrt (fma (cos phi1) (* (cos phi2) (- 0.5 t_3)) (pow (sin t_0) 2.0)))
       (sqrt
        (+
         (+ 0.5 (* 0.5 (cos (* 2.0 t_0))))
         (* (cos phi1) (* (cos phi2) (- t_3 0.5))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * (phi1 - phi2);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (t_1 * ((cos(phi1) * cos(phi2)) * t_1)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = 0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))));
	double tmp;
	if (t_2 <= 0.02) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - pow(sin((-0.5 * (lambda2 - lambda1))), 2.0)))));
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - t_3)), pow(sin(t_0), 2.0))), sqrt(((0.5 + (0.5 * cos((2.0 * t_0)))) + (cos(phi1) * (cos(phi2) * (t_3 - 0.5))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * Float64(phi1 - phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))
	t_3 = Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (t_2 <= 0.02)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - t_3)), (sin(t_0) ^ 2.0))), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) + Float64(cos(phi1) * Float64(cos(phi2) * Float64(t_3 - 0.5)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.02], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - t$95$3), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 72.7%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-lft-in N/A

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

      11. neg-mul-1 N/A

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

      12. neg-mul-1 N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-pow.f64 N/A

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

   8. Applied rewrites 72.1%

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

   if 0.0200000000000000004 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 61.9%

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

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

   6. Applied rewrites 61.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 0.02:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_1 := 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t\_2 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;t\_3 \leq 0.02:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right), 0.5 - t\_1\right)}}{\sqrt{\left(0.5 + t\_1\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin t\_0}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (- lambda1 lambda2)))
        (t_1 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (* t_2 (* (* (cos phi1) (cos phi2)) t_2))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))))
   (if (<= t_3 0.02)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_3)
        (sqrt (- 1.0 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))))))
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (* (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 t_0)))))
         (- 0.5 t_1)))
       (sqrt
        (- (+ 0.5 t_1) (* (cos phi1) (* (cos phi2) (pow (sin t_0) 2.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * (lambda1 - lambda2);
	double t_1 = 0.5 * cos((2.0 * (0.5 * (phi1 - phi2))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = (t_2 * ((cos(phi1) * cos(phi2)) * t_2)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double tmp;
	if (t_3 <= 0.02) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - pow(sin((-0.5 * (lambda2 - lambda1))), 2.0)))));
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((2.0 * t_0))))), (0.5 - t_1))), sqrt(((0.5 + t_1) - (cos(phi1) * (cos(phi2) * pow(sin(t_0), 2.0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_1 = Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(Float64(t_2 * Float64(Float64(cos(phi1) * cos(phi2)) * t_2)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))
	tmp = 0.0
	if (t_3 <= 0.02)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0))))), Float64(0.5 - t_1))), sqrt(Float64(Float64(0.5 + t_1) - Float64(cos(phi1) * Float64(cos(phi2) * (sin(t_0) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.02], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + t$95$1), $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 72.7%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-lft-in N/A

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

      11. neg-mul-1 N/A

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

      12. neg-mul-1 N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-pow.f64 N/A

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

   8. Applied rewrites 72.1%

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

   if 0.0200000000000000004 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 61.9%

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

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

   6. Applied rewrites 61.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 0.02:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := t\_0 \cdot \left(t\_2 \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \cos \left(\phi_1 - \phi_2\right)\\ \mathbf{if}\;t\_3 \leq 0.02:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_1, \mathsf{fma}\left(-0.5, t\_4, 0.5\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \left(-\cos \phi_2\right), t\_1, \mathsf{fma}\left(0.5, t\_4, 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (fma -0.5 (cos (- lambda1 lambda2)) 0.5))
        (t_2 (* (cos phi1) (cos phi2)))
        (t_3 (+ (* t_0 (* t_2 t_0)) (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (t_4 (cos (- phi1 phi2))))
   (if (<= t_3 0.02)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_3)
        (sqrt (- 1.0 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))))))
     (*
      (* R 2.0)
      (atan2
       (sqrt (fma t_2 t_1 (fma -0.5 t_4 0.5)))
       (sqrt (fma (* (cos phi1) (- (cos phi2))) t_1 (fma 0.5 t_4 0.5))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = fma(-0.5, cos((lambda1 - lambda2)), 0.5);
	double t_2 = cos(phi1) * cos(phi2);
	double t_3 = (t_0 * (t_2 * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = cos((phi1 - phi2));
	double tmp;
	if (t_3 <= 0.02) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - pow(sin((-0.5 * (lambda2 - lambda1))), 2.0)))));
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(t_2, t_1, fma(-0.5, t_4, 0.5))), sqrt(fma((cos(phi1) * -cos(phi2)), t_1, fma(0.5, t_4, 0.5))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5)
	t_2 = Float64(cos(phi1) * cos(phi2))
	t_3 = Float64(Float64(t_0 * Float64(t_2 * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))
	t_4 = cos(Float64(phi1 - phi2))
	tmp = 0.0
	if (t_3 <= 0.02)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(t_2, t_1, fma(-0.5, t_4, 0.5))), sqrt(fma(Float64(cos(phi1) * Float64(-cos(phi2))), t_1, fma(0.5, t_4, 0.5)))));
	end
	return tmp
end

Wolfram

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[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.02], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$1 + N[(-0.5 * t$95$4 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[phi1], $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] * t$95$1 + N[(0.5 * t$95$4 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 72.7%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-lft-in N/A

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

      11. neg-mul-1 N/A

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

      12. neg-mul-1 N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-pow.f64 N/A

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

   8. Applied rewrites 72.1%

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

   if 0.0200000000000000004 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 61.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub N/A

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

      2. sin-diff N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sin.f64 N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 63.2

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

   4. Applied rewrites 63.2%

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

      1. div-sub N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. sin-diff N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

   6. Applied rewrites 80.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

   8. Applied rewrites 80.4%

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

   10. Applied rewrites 61.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 0.02:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \mathsf{fma}\left(-0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \left(-\cos \phi_2\right), \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \mathsf{fma}\left(0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + t\_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right)}}{\sqrt{\frac{\left(1 + t\_0\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \left(t\_0 + \cos \left(\phi_2 + \phi_1\right)\right)}{2}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- phi1 phi2))) (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow
         (fma
          (sin (* phi2 0.5))
          (- (cos (* 0.5 phi1)))
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
         2.0)
        (* t_1 (* (* (cos phi1) (cos phi2)) t_1))))
      (sqrt
       (/
        (-
         (+ 1.0 t_0)
         (*
          (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
          (+ t_0 (cos (+ phi2 phi1)))))
        2.0)))))))

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi1 - phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi2 * 0.5)), Float64(-cos(Float64(0.5 * phi1))), Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) ^ 2.0) + Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)))), sqrt(Float64(Float64(Float64(1.0 + t_0) - Float64(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5) * Float64(t_0 + cos(Float64(phi2 + phi1))))) / 2.0)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(1.0 + t$95$0), $MachinePrecision] - N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(t$95$0 + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

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

   2. sin-diff N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-*.f64 N/A

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

   10. sin-diff N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

6. Applied rewrites 80.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

8. Applied rewrites 80.5%

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

10. Applied rewrites 64.4%

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

11. Final simplification 64.4%

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

Alternative 11: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + t\_1 \cdot \left(t\_0 \cdot t\_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), t\_0, \mathsf{fma}\left(-0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow
         (fma
          (sin (* phi2 0.5))
          (- (cos (* 0.5 phi1)))
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
         2.0)
        (* t_1 (* t_0 t_1))))
      (sqrt
       (-
        1.0
        (fma
         (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
         t_0
         (fma -0.5 (cos (- phi1 phi2)) 0.5)))))))))

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi2 * 0.5)), Float64(-cos(Float64(0.5 * phi1))), Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) ^ 2.0) + Float64(t_1 * Float64(t_0 * t_1)))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5), t_0, fma(-0.5, cos(Float64(phi1 - phi2)), 0.5)))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * t$95$0 + N[(-0.5 * N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

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

   2. sin-diff N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-*.f64 N/A

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

   10. sin-diff N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

6. Applied rewrites 80.5%

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

8. Applied rewrites 63.8%

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

9. Final simplification 63.8%

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

Alternative 12: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + t\_1 \cdot \left(t\_0 \cdot t\_1\right)}}{\sqrt{\mathsf{fma}\left(0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right) - t\_0 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow
         (fma
          (sin (* phi2 0.5))
          (- (cos (* 0.5 phi1)))
          (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
         2.0)
        (* t_1 (* t_0 t_1))))
      (sqrt
       (-
        (fma 0.5 (cos (- phi1 phi2)) 0.5)
        (* t_0 (fma -0.5 (cos (- lambda1 lambda2)) 0.5)))))))))

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi2 * 0.5)), Float64(-cos(Float64(0.5 * phi1))), Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) ^ 2.0) + Float64(t_1 * Float64(t_0 * t_1)))), sqrt(Float64(fma(0.5, cos(Float64(phi1 - phi2)), 0.5) - Float64(t_0 * fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5)))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 * N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] - N[(t$95$0 * N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

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

   2. sin-diff N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. div-sub N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. lift-*.f64 N/A

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

   10. sin-diff N/A

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

   11. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lift-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. lift-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lift-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. sub-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. +-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 80.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

8. Applied rewrites 63.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(0.5, \cos \left(\left(\phi_1 - \phi_2\right) \cdot 1\right), 0.5\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}}}\right) \]

9. Final simplification 63.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\mathsf{fma}\left(0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)}}\right) \]
10. Add Preprocessing

Alternative 13: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{if}\;t\_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\lambda_1 \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right)\right)\right), t\_0\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), t\_0\right)}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (sqrt
          (+
           0.5
           (*
            (cos phi1)
            (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))))))))
   (if (<=
        (+
         (* t_1 (* (* (cos phi1) (cos phi2)) t_1))
         (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        2e-12)
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (*
          (cos phi2)
          (*
           lambda1
           (* lambda1 (fma lambda1 (* lambda1 -0.020833333333333332) 0.25))))
         t_0))
       t_2))
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (*
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
         t_0))
       t_2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2))))))));
	double tmp;
	if (((t_1 * ((cos(phi1) * cos(phi2)) * t_1)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0)) <= 2e-12) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (lambda1 * (lambda1 * fma(lambda1, (lambda1 * -0.020833333333333332), 0.25)))), t_0)), t_2);
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))), t_0)), t_2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2))))))))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)) <= 2e-12)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(lambda1 * Float64(lambda1 * fma(lambda1, Float64(lambda1 * -0.020833333333333332), 0.25)))), t_0)), t_2));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))), t_0)), t_2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-12], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(lambda1 * N[(lambda1 * N[(lambda1 * N[(lambda1 * -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 1.99999999999999996e-12

   1. Initial program 76.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 5.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 5.1

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 5.1%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 6.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 6.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left({\lambda_1}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right)\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       2. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       3. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right)} \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       6. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{\left(\frac{-1}{48} \cdot {\lambda_1}^{2} + \frac{1}{4}\right)} \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       7. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{{\lambda_1}^{2} \cdot \frac{-1}{48}} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       8. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{\left(\lambda_1 \cdot \lambda_1\right)} \cdot \frac{-1}{48} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       9. associate-*l* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{\lambda_1 \cdot \left(\lambda_1 \cdot \frac{-1}{48}\right)} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{48}, \frac{1}{4}\right)} \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       11. lower-*.f64 34.1

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\mathsf{fma}\left(\lambda_1, \color{blue}{\lambda_1 \cdot -0.020833333333333332}, 0.25\right) \cdot \lambda_1\right) \cdot \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 34.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   if 1.99999999999999996e-12 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\lambda_1 \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{if}\;t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\lambda_1 \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \phi_1\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (+
           0.5
           (*
            (cos phi1)
            (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))))))))
   (if (<=
        (+
         (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
         (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        2e-12)
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (*
          (cos phi2)
          (*
           lambda1
           (* lambda1 (fma lambda1 (* lambda1 -0.020833333333333332) 0.25))))
         (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
       t_1))
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (*
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
         (- 0.5 (* 0.5 (cos phi1)))))
       t_1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2))))))));
	double tmp;
	if (((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0)) <= 2e-12) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (lambda1 * (lambda1 * fma(lambda1, (lambda1 * -0.020833333333333332), 0.25)))), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))), t_1);
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))), (0.5 - (0.5 * cos(phi1))))), t_1);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2))))))))
	tmp = 0.0
	if (Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)) <= 2e-12)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(lambda1 * Float64(lambda1 * fma(lambda1, Float64(lambda1 * -0.020833333333333332), 0.25)))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))), t_1));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))), Float64(0.5 - Float64(0.5 * cos(phi1))))), t_1));
	end
	return tmp
end

Wolfram

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[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-12], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(lambda1 * N[(lambda1 * N[(lambda1 * N[(lambda1 * -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 1.99999999999999996e-12

   1. Initial program 76.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 5.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 5.1

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 5.1%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 6.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 6.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left({\lambda_1}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right)\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       2. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       3. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right)} \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       6. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{\left(\frac{-1}{48} \cdot {\lambda_1}^{2} + \frac{1}{4}\right)} \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       7. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{{\lambda_1}^{2} \cdot \frac{-1}{48}} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       8. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{\left(\lambda_1 \cdot \lambda_1\right)} \cdot \frac{-1}{48} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       9. associate-*l* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{\lambda_1 \cdot \left(\lambda_1 \cdot \frac{-1}{48}\right)} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{48}, \frac{1}{4}\right)} \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       11. lower-*.f64 34.1

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\mathsf{fma}\left(\lambda_1, \color{blue}{\lambda_1 \cdot -0.020833333333333332}, 0.25\right) \cdot \lambda_1\right) \cdot \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 34.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   if 1.99999999999999996e-12 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower-cos.f64 42.1

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 42.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\lambda_1 \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{if}\;t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\lambda_1 \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \mathsf{fma}\left(-0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (+
           0.5
           (*
            (cos phi1)
            (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))))))))
   (if (<=
        (+
         (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
         (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        2e-12)
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (*
          (cos phi2)
          (*
           lambda1
           (* lambda1 (fma lambda1 (* lambda1 -0.020833333333333332) 0.25))))
         (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
       t_1))
     (*
      (* R 2.0)
      (atan2
       (sqrt
        (fma
         (cos phi1)
         (* (cos phi2) (- 0.5 (* 0.5 (cos lambda1))))
         (fma -0.5 (cos (- phi1 phi2)) 0.5)))
       t_1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2))))))));
	double tmp;
	if (((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0)) <= 2e-12) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (lambda1 * (lambda1 * fma(lambda1, (lambda1 * -0.020833333333333332), 0.25)))), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))), t_1);
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos(lambda1)))), fma(-0.5, cos((phi1 - phi2)), 0.5))), t_1);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2))))))))
	tmp = 0.0
	if (Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)) <= 2e-12)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(lambda1 * Float64(lambda1 * fma(lambda1, Float64(lambda1 * -0.020833333333333332), 0.25)))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))), t_1));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(lambda1)))), fma(-0.5, cos(Float64(phi1 - phi2)), 0.5))), t_1));
	end
	return tmp
end

Wolfram

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[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-12], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(lambda1 * N[(lambda1 * N[(lambda1 * N[(lambda1 * -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 1.99999999999999996e-12

   1. Initial program 76.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 5.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 5.1

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 5.1%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 6.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 6.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left({\lambda_1}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right)\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       2. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       3. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\left(\left(\frac{1}{4} + \frac{-1}{48} \cdot {\lambda_1}^{2}\right) \cdot \lambda_1\right)} \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       6. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{\left(\frac{-1}{48} \cdot {\lambda_1}^{2} + \frac{1}{4}\right)} \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       7. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{{\lambda_1}^{2} \cdot \frac{-1}{48}} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       8. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{\left(\lambda_1 \cdot \lambda_1\right)} \cdot \frac{-1}{48} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       9. associate-*l* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\left(\color{blue}{\lambda_1 \cdot \left(\lambda_1 \cdot \frac{-1}{48}\right)} + \frac{1}{4}\right) \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot \frac{-1}{48}, \frac{1}{4}\right)} \cdot \lambda_1\right) \cdot \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       11. lower-*.f64 34.1

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\mathsf{fma}\left(\lambda_1, \color{blue}{\lambda_1 \cdot -0.020833333333333332}, 0.25\right) \cdot \lambda_1\right) \cdot \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 34.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right) \cdot \lambda_1\right) \cdot \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   if 1.99999999999999996e-12 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 45.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 45.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 36.2

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 36.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in phi1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\phi_1 - \phi_2\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       2. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       3. neg-mul-1 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \left(\phi_1 + \color{blue}{-1 \cdot \phi_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       4. *-lft-identity N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \left(\color{blue}{1 \cdot \phi_1} + -1 \cdot \phi_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       5. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \left(\color{blue}{\left(-1 \cdot -1\right)} \cdot \phi_1 + -1 \cdot \phi_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       6. associate-*r* N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \left(\color{blue}{-1 \cdot \left(-1 \cdot \phi_1\right)} + -1 \cdot \phi_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       7. distribute-lft-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \color{blue}{\left(-1 \cdot \left(-1 \cdot \phi_1 + \phi_2\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       8. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \cos \left(-1 \cdot \color{blue}{\left(\phi_2 + -1 \cdot \phi_1\right)}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       9. *-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       10. sub-neg N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\right)\right)\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       11. +-commutative N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\phi_2 + -1 \cdot \phi_1\right)\right)\right)\right) + \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 36.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\lambda_1 \cdot \mathsf{fma}\left(\lambda_1, \lambda_1 \cdot -0.020833333333333332, 0.25\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \mathsf{fma}\left(-0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \frac{\mathsf{fma}\left(\cos \left(t\_0 + 0.5 \cdot \left(\phi_2 - \phi_1\right)\right) - \cos \left(2 \cdot t\_0\right), 2, 2 \cdot \left(\left(\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)}{4}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (- phi1 phi2))) (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (* t_1 (* (* (cos phi1) (cos phi2)) t_1))
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
      (sqrt
       (-
        1.0
        (/
         (fma
          (- (cos (+ t_0 (* 0.5 (- phi2 phi1)))) (cos (* 2.0 t_0)))
          2.0
          (*
           2.0
           (*
            (+ (cos (- phi1 phi2)) (cos (+ phi2 phi1)))
            (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))))
         4.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * (phi1 - phi2);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt(((t_1 * ((cos(phi1) * cos(phi2)) * t_1)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - (fma((cos((t_0 + (0.5 * (phi2 - phi1)))) - cos((2.0 * t_0))), 2.0, (2.0 * ((cos((phi1 - phi2)) + cos((phi2 + phi1))) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))))) / 4.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * Float64(phi1 - phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(fma(Float64(cos(Float64(t_0 + Float64(0.5 * Float64(phi2 - phi1)))) - cos(Float64(2.0 * t_0))), 2.0, Float64(2.0 * Float64(Float64(cos(Float64(phi1 - phi2)) + cos(Float64(phi2 + phi1))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))))) / 4.0))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[(N[Cos[N[(t$95$0 + N[(0.5 * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 63.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\left(\phi_1 - \phi_2\right) \cdot 0.5 - \left(\phi_1 - \phi_2\right) \cdot 0.5\right) - \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right), 2, 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)\right)}{4}}}}\right) \]

5. Final simplification 63.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \frac{\mathsf{fma}\left(\cos \left(0.5 \cdot \left(\phi_1 - \phi_2\right) + 0.5 \cdot \left(\phi_2 - \phi_1\right)\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right), 2, 2 \cdot \left(\left(\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)}{4}}}\right) \]
6. Add Preprocessing

Alternative 17: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\frac{\left(1 + t\_0\right) - \left(t\_0 + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right)}{2}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- phi1 phi2))) (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (* t_1 (* (* (cos phi1) (cos phi2)) t_1))
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
      (sqrt
       (/
        (-
         (+ 1.0 t_0)
         (*
          (+ t_0 (cos (+ phi2 phi1)))
          (fma (cos (- lambda1 lambda2)) -0.5 0.5)))
        2.0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi1 - phi2));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt(((t_1 * ((cos(phi1) * cos(phi2)) * t_1)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((((1.0 + t_0) - ((t_0 + cos((phi2 + phi1))) * fma(cos((lambda1 - lambda2)), -0.5, 0.5))) / 2.0))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi1 - phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(Float64(Float64(1.0 + t_0) - Float64(Float64(t_0 + cos(Float64(phi2 + phi1))) * fma(cos(Float64(lambda1 - lambda2)), -0.5, 0.5))) / 2.0)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(1.0 + t$95$0), $MachinePrecision] - N[(N[(t$95$0 + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. div-inv N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-*.f64 64.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 64.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 63.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{\left(\cos \left(\left(\phi_1 - \phi_2\right) \cdot 1\right) + 1\right) - \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right), -0.5, 0.5\right)}{2}}}}\right) \]

7. Final simplification 63.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\frac{\left(1 + \cos \left(\phi_1 - \phi_2\right)\right) - \left(\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right)}{2}}}\right) \]
8. Add Preprocessing

Alternative 18: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right) \cdot \left(-\cos \phi_1\right), \cos \phi_2, \mathsf{fma}\left(0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
      (sqrt
       (fma
        (* (fma (cos (- lambda1 lambda2)) -0.5 0.5) (- (cos phi1)))
        (cos phi2)
        (fma 0.5 (cos (- phi1 phi2)) 0.5))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt(fma((fma(cos((lambda1 - lambda2)), -0.5, 0.5) * -cos(phi1)), cos(phi2), fma(0.5, cos((phi1 - phi2)), 0.5)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(fma(Float64(fma(cos(Float64(lambda1 - lambda2)), -0.5, 0.5) * Float64(-cos(phi1))), cos(phi2), fma(0.5, cos(Float64(phi1 - phi2)), 0.5))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * (-N[Cos[phi1], $MachinePrecision])), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(0.5 * N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. div-inv N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. div-inv N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-*.f64 64.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 64.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied rewrites 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\left(-\cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \left(\left(\lambda_1 - \lambda_2\right) \cdot 1\right), -0.5, 0.5\right), \cos \phi_2, \mathsf{fma}\left(0.5, \cos \left(\left(\phi_1 - \phi_2\right) \cdot 1\right), 0.5\right)\right)}}}\right) \]

7. Final simplification 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right) \cdot \left(-\cos \phi_1\right), \cos \phi_2, \mathsf{fma}\left(0.5, \cos \left(\phi_1 - \phi_2\right), 0.5\right)\right)}}\right) \]
8. Add Preprocessing

Alternative 19: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_1 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin t\_0}^{2}, {\sin t\_1}^{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 \cdot \cos \left(2 \cdot t\_0\right) - 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (- lambda1 lambda2))) (t_1 (* 0.5 (- phi1 phi2))))
   (*
    (atan2
     (sqrt
      (fma (cos phi1) (* (cos phi2) (pow (sin t_0) 2.0)) (pow (sin t_1) 2.0)))
     (sqrt
      (+
       (+ 0.5 (* 0.5 (cos (* 2.0 t_1))))
       (* (cos phi1) (* (cos phi2) (- (* 0.5 (cos (* 2.0 t_0))) 0.5))))))
    (* R 2.0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * (lambda1 - lambda2);
	double t_1 = 0.5 * (phi1 - phi2);
	return atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin(t_0), 2.0)), pow(sin(t_1), 2.0))), sqrt(((0.5 + (0.5 * cos((2.0 * t_1)))) + (cos(phi1) * (cos(phi2) * ((0.5 * cos((2.0 * t_0))) - 0.5)))))) * (R * 2.0);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_1 = Float64(0.5 * Float64(phi1 - phi2))
	return Float64(atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(t_0) ^ 2.0)), (sin(t_1) ^ 2.0))), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_1)))) + Float64(cos(phi1) * Float64(cos(phi2) * Float64(Float64(0.5 * cos(Float64(2.0 * t_0))) - 0.5)))))) * Float64(R * 2.0))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 57.2%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\phi_1 - \phi_2\right)} \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   4. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. sqr-sin-a N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   9. metadata-eval N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. div-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   12. lift-sin.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   14. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   15. div-inv N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   16. lift-/.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   17. lift-sin.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

6. Applied rewrites 60.0%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), \color{blue}{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   4. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. sqr-sin-a N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   8. lift-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   9. lift-sin.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. pow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}^{2}}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. lower-pow.f64 63.2

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

8. Applied rewrites 63.2%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

9. Final simplification 63.2%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
10. Add Preprocessing

Alternative 20: 58.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ t_2 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_3 := \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_2\right)\right), {\sin t\_1}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5\right)}}\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-54}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin t\_2}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot t\_0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* 0.5 (- phi1 phi2)))
        (t_2 (* 0.5 (- lambda1 lambda2)))
        (t_3
         (*
          (* R 2.0)
          (atan2
           (sqrt
            (fma
             (cos phi1)
             (* (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 t_2)))))
             (pow (sin t_1) 2.0)))
           (sqrt (fma (cos phi2) (- 0.5 (fma -0.5 t_0 0.5)) 0.5))))))
   (if (<= phi2 -3.7e-54)
     t_3
     (if (<= phi2 1.7e-14)
       (*
        (* R 2.0)
        (atan2
         (sqrt
          (fma
           (cos phi1)
           (* (cos phi2) (pow (sin t_2) 2.0))
           (- 0.5 (* 0.5 (cos (* 2.0 t_1))))))
         (sqrt (+ 0.5 (* (cos phi1) (- 0.5 (+ 0.5 (* -0.5 t_0))))))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = 0.5 * (phi1 - phi2);
	double t_2 = 0.5 * (lambda1 - lambda2);
	double t_3 = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((2.0 * t_2))))), pow(sin(t_1), 2.0))), sqrt(fma(cos(phi2), (0.5 - fma(-0.5, t_0, 0.5)), 0.5)));
	double tmp;
	if (phi2 <= -3.7e-54) {
		tmp = t_3;
	} else if (phi2 <= 1.7e-14) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin(t_2), 2.0)), (0.5 - (0.5 * cos((2.0 * t_1)))))), sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * t_0)))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(0.5 * Float64(phi1 - phi2))
	t_2 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_3 = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_2))))), (sin(t_1) ^ 2.0))), sqrt(fma(cos(phi2), Float64(0.5 - fma(-0.5, t_0, 0.5)), 0.5))))
	tmp = 0.0
	if (phi2 <= -3.7e-54)
		tmp = t_3;
	elseif (phi2 <= 1.7e-14)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(t_2) ^ 2.0)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))))), sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * t_0))))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.7e-54], t$95$3, If[LessEqual[phi2, 1.7e-14], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.7000000000000003e-54 or 1.70000000000000001e-14 < phi2

   1. Initial program 52.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\phi_1 - \phi_2\right)} \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      10. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      15. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      17. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   6. Applied rewrites 51.9%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), \color{blue}{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \frac{1}{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   9. Applied rewrites 52.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}} \cdot \left(2 \cdot R\right) \]

   if -3.7000000000000003e-54 < phi2 < 1.70000000000000001e-14

   1. Initial program 78.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 67.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      10. pow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}^{2}}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. lower-pow.f64 74.5

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   9. Applied rewrites 74.5%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{-54}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}\\ \mathbf{elif}\;\phi_2 \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := 0.5 - \mathsf{fma}\left(-0.5, t\_1, 0.5\right)\\ t_3 := \sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}\\ \mathbf{if}\;\phi_1 \leq -0.0005:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_2, 0.5\right)}}\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_2, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot t\_1\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (- 0.5 (fma -0.5 t_1 0.5)))
        (t_3
         (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))
   (if (<= phi1 -0.0005)
     (* (* R 2.0) (atan2 t_3 (sqrt (fma (cos phi1) t_2 0.5))))
     (if (<= phi1 45000000000000.0)
       (* (* R 2.0) (atan2 t_3 (sqrt (fma (cos phi2) t_2 0.5))))
       (*
        (* R 2.0)
        (atan2
         (sqrt (fma (cos phi1) t_0 (- 0.5 (* 0.5 (cos phi1)))))
         (sqrt (+ 0.5 (* (cos phi1) (- 0.5 (+ 0.5 (* -0.5 t_1))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))));
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = 0.5 - fma(-0.5, t_1, 0.5);
	double t_3 = sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * (phi1 - phi2))), 2.0)));
	double tmp;
	if (phi1 <= -0.0005) {
		tmp = (R * 2.0) * atan2(t_3, sqrt(fma(cos(phi1), t_2, 0.5)));
	} else if (phi1 <= 45000000000000.0) {
		tmp = (R * 2.0) * atan2(t_3, sqrt(fma(cos(phi2), t_2, 0.5)));
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), t_0, (0.5 - (0.5 * cos(phi1))))), sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * t_1)))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(0.5 - fma(-0.5, t_1, 0.5))
	t_3 = sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)))
	tmp = 0.0
	if (phi1 <= -0.0005)
		tmp = Float64(Float64(R * 2.0) * atan(t_3, sqrt(fma(cos(phi1), t_2, 0.5))));
	elseif (phi1 <= 45000000000000.0)
		tmp = Float64(Float64(R * 2.0) * atan(t_3, sqrt(fma(cos(phi2), t_2, 0.5))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), t_0, Float64(0.5 - Float64(0.5 * cos(phi1))))), sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * t_1))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(-0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.0005], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$3 / N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$2 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 45000000000000.0], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$3 / N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$2 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.0000000000000001e-4

   1. Initial program 46.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\phi_1 - \phi_2\right)} \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      10. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      15. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      17. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   6. Applied rewrites 46.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), \color{blue}{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \frac{1}{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   9. Applied rewrites 46.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}} \cdot \left(2 \cdot R\right) \]

   if -5.0000000000000001e-4 < phi1 < 4.5e13

   1. Initial program 79.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\phi_1 - \phi_2\right)} \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      10. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      15. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      17. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   6. Applied rewrites 73.6%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), \color{blue}{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \frac{1}{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   9. Applied rewrites 73.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}} \cdot \left(2 \cdot R\right) \]

   if 4.5e13 < phi1

   1. Initial program 44.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 46.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower-cos.f64 46.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 46.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0005:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 59.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot t\_2\right)\right)}}\\ \mathbf{if}\;\phi_2 \leq -0.033:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 4.7 \cdot 10^{-8}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_1, {\sin t\_0}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - \mathsf{fma}\left(-0.5, t\_2, 0.5\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (- phi1 phi2)))
        (t_1
         (*
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3
         (*
          (* R 2.0)
          (atan2
           (sqrt (fma (cos phi1) t_1 (- 0.5 (* 0.5 (cos (* 2.0 t_0))))))
           (sqrt (+ 0.5 (* (cos phi2) (- 0.5 (+ 0.5 (* -0.5 t_2))))))))))
   (if (<= phi2 -0.033)
     t_3
     (if (<= phi2 4.7e-8)
       (*
        (* R 2.0)
        (atan2
         (sqrt (fma (cos phi1) t_1 (pow (sin t_0) 2.0)))
         (sqrt (fma (cos phi1) (- 0.5 (fma -0.5 t_2 0.5)) 0.5))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * (phi1 - phi2);
	double t_1 = cos(phi2) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))));
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = (R * 2.0) * atan2(sqrt(fma(cos(phi1), t_1, (0.5 - (0.5 * cos((2.0 * t_0)))))), sqrt((0.5 + (cos(phi2) * (0.5 - (0.5 + (-0.5 * t_2)))))));
	double tmp;
	if (phi2 <= -0.033) {
		tmp = t_3;
	} else if (phi2 <= 4.7e-8) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), t_1, pow(sin(t_0), 2.0))), sqrt(fma(cos(phi1), (0.5 - fma(-0.5, t_2, 0.5)), 0.5)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * Float64(phi1 - phi2))
	t_1 = Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), t_1, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0)))))), sqrt(Float64(0.5 + Float64(cos(phi2) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * t_2))))))))
	tmp = 0.0
	if (phi2 <= -0.033)
		tmp = t_3;
	elseif (phi2 <= 4.7e-8)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), t_1, (sin(t_0) ^ 2.0))), sqrt(fma(cos(phi1), Float64(0.5 - fma(-0.5, t_2, 0.5)), 0.5))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.033], t$95$3, If[LessEqual[phi2, 4.7e-8], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(-0.5 * t$95$2 + 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.033000000000000002 or 4.6999999999999997e-8 < phi2

   1. Initial program 50.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      9. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      15. lower--.f64 52.0

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 52.0%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   if -0.033000000000000002 < phi2 < 4.6999999999999997e-8

   1. Initial program 75.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\phi_1 - \phi_2\right)} \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. sqr-sin-a N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      10. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)} \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      15. div-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      17. lift-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   6. Applied rewrites 69.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), \color{blue}{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. +-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} + \frac{1}{2}}} \cdot \left(2 \cdot R\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), \frac{1}{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   9. Applied rewrites 69.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.033:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{elif}\;\phi_2 \leq 4.7 \cdot 10^{-8}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 56.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ t_1 := 0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sqrt{0.5 + \cos \phi_1 \cdot t\_1}\\ t_3 := \sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -0.00052:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_3}{t\_2}\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{0.5 + \cos \phi_2 \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, 0.5 - 0.5 \cdot \cos \phi_1\right)}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))))
        (t_1 (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))))
        (t_2 (sqrt (+ 0.5 (* (cos phi1) t_1))))
        (t_3
         (sqrt
          (fma
           (cos phi1)
           t_0
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))
   (if (<= phi1 -0.00052)
     (* (* R 2.0) (atan2 t_3 t_2))
     (if (<= phi1 45000000000000.0)
       (* (* R 2.0) (atan2 t_3 (sqrt (+ 0.5 (* (cos phi2) t_1)))))
       (*
        (* R 2.0)
        (atan2 (sqrt (fma (cos phi1) t_0 (- 0.5 (* 0.5 (cos phi1))))) t_2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))));
	double t_1 = 0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2))));
	double t_2 = sqrt((0.5 + (cos(phi1) * t_1)));
	double t_3 = sqrt(fma(cos(phi1), t_0, (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2))))))));
	double tmp;
	if (phi1 <= -0.00052) {
		tmp = (R * 2.0) * atan2(t_3, t_2);
	} else if (phi1 <= 45000000000000.0) {
		tmp = (R * 2.0) * atan2(t_3, sqrt((0.5 + (cos(phi2) * t_1))));
	} else {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), t_0, (0.5 - (0.5 * cos(phi1))))), t_2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))))
	t_1 = Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))))
	t_2 = sqrt(Float64(0.5 + Float64(cos(phi1) * t_1)))
	t_3 = sqrt(fma(cos(phi1), t_0, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))
	tmp = 0.0
	if (phi1 <= -0.00052)
		tmp = Float64(Float64(R * 2.0) * atan(t_3, t_2));
	elseif (phi1 <= 45000000000000.0)
		tmp = Float64(Float64(R * 2.0) * atan(t_3, sqrt(Float64(0.5 + Float64(cos(phi2) * t_1)))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), t_0, Float64(0.5 - Float64(0.5 * cos(phi1))))), t_2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.00052], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$3 / t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 45000000000000.0], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$3 / N[Sqrt[N[(0.5 + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.19999999999999954e-4

   1. Initial program 46.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 46.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 46.7%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   if -5.19999999999999954e-4 < phi1 < 4.5e13

   1. Initial program 79.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\frac{1}{2} \cdot \color{blue}{\cos \phi_2} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_2 \cdot \frac{1}{2}} - \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      5. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      9. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      15. lower--.f64 68.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   if 4.5e13 < phi1

   1. Initial program 44.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 46.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower-cos.f64 46.8

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 46.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00052:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{elif}\;\phi_1 \leq 45000000000000:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 43.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \lambda_1\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_2 := \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - t\_0\right), t\_1\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot t\_0}}\\ \mathbf{if}\;\lambda_1 \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_2\right), t\_1\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos lambda1)))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))
        (t_2
         (*
          (* R 2.0)
          (atan2
           (sqrt (fma (cos phi1) (* (cos phi2) (- 0.5 t_0)) t_1))
           (sqrt (+ 0.5 (* (cos phi1) t_0)))))))
   (if (<= lambda1 -9.8e-8)
     t_2
     (if (<= lambda1 1.35e-6)
       (*
        (* R 2.0)
        (atan2
         (sqrt
          (fma (cos phi1) (* (cos phi2) (- 0.5 (* 0.5 (cos lambda2)))) t_1))
         (sqrt
          (+
           0.5
           (*
            (cos phi1)
            (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * cos(lambda1);
	double t_1 = 0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))));
	double t_2 = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - t_0)), t_1)), sqrt((0.5 + (cos(phi1) * t_0))));
	double tmp;
	if (lambda1 <= -9.8e-8) {
		tmp = t_2;
	} else if (lambda1 <= 1.35e-6) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos(lambda2)))), t_1)), sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2)))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * cos(lambda1))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))
	t_2 = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - t_0)), t_1)), sqrt(Float64(0.5 + Float64(cos(phi1) * t_0)))))
	tmp = 0.0
	if (lambda1 <= -9.8e-8)
		tmp = t_2;
	elseif (lambda1 <= 1.35e-6)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(lambda2)))), t_1)), sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2))))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -9.8e-8], t$95$2, If[LessEqual[lambda1, 1.35e-6], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -9.8000000000000004e-8 or 1.34999999999999999e-6 < lambda1

   1. Initial program 49.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 41.6

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 41.6%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 41.6

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 41.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \lambda_1\right)}}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \lambda_1\right)}}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 41.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 \cdot \color{blue}{\cos \lambda_1}\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 41.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \color{blue}{\left(0.5 \cdot \cos \lambda_1\right)}}} \cdot \left(2 \cdot R\right) \]

   if -9.8000000000000004e-8 < lambda1 < 1.34999999999999999e-6

   1. Initial program 76.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 43.3

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda1 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_2}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-cos.f64 43.3

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 43.3%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 \cdot \cos \lambda_1\right)}}\\ \mathbf{elif}\;\lambda_1 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_2\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 \cdot \cos \lambda_1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 32.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ t_1 := \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \mathsf{fma}\left(-0.5, \cos \phi_1, 0.5\right)\right)}}{t\_0}\\ \mathbf{if}\;\lambda_1 \leq -1100:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_1 \leq 1.22 \cdot 10^{-8}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (sqrt
          (+
           0.5
           (* (cos phi1) (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2)))))))))
        (t_1
         (*
          (* R 2.0)
          (atan2
           (sqrt
            (fma
             (cos phi1)
             (* (cos phi2) (- 0.5 (* 0.5 (cos lambda1))))
             (fma -0.5 (cos phi1) 0.5)))
           t_0))))
   (if (<= lambda1 -1100.0)
     t_1
     (if (<= lambda1 1.22e-8)
       (*
        (* R 2.0)
        (atan2
         (sqrt
          (fma
           (cos phi1)
           (* (cos phi2) (* 0.25 (* lambda1 lambda1)))
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
         t_0))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2))))))));
	double t_1 = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos(lambda1)))), fma(-0.5, cos(phi1), 0.5))), t_0);
	double tmp;
	if (lambda1 <= -1100.0) {
		tmp = t_1;
	} else if (lambda1 <= 1.22e-8) {
		tmp = (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.25 * (lambda1 * lambda1))), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2))))))))
	t_1 = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(lambda1)))), fma(-0.5, cos(phi1), 0.5))), t_0))
	tmp = 0.0
	if (lambda1 <= -1100.0)
		tmp = t_1;
	elseif (lambda1 <= 1.22e-8)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.25 * Float64(lambda1 * lambda1))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))), t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Cos[phi1], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1100.0], t$95$1, If[LessEqual[lambda1, 1.22e-8], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.25 * N[(lambda1 * lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1100 or 1.22e-8 < lambda1

   1. Initial program 50.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 42.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 42.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 42.4

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 42.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. cancel-sign-sub-inv N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       2. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \phi_1\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       3. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\frac{-1}{2} \cdot \cos \phi_1 + \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right), \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \phi_1, \frac{1}{2}\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 40.3

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \mathsf{fma}\left(-0.5, \color{blue}{\cos \phi_1}, 0.5\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 40.3%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \color{blue}{\mathsf{fma}\left(-0.5, \cos \phi_1, 0.5\right)}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   if -1100 < lambda1 < 1.22e-8

   1. Initial program 75.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

      8. sub-neg N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

      14. lower--.f64 42.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. Applied rewrites 42.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

      3. lower-cos.f64 25.2

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   10. Applied rewrites 25.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. Taylor expanded in lambda1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{4} \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{4} \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       2. unpow2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-*.f64 29.6

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.25 \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. Applied rewrites 29.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1100:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \mathsf{fma}\left(-0.5, \cos \phi_1, 0.5\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{elif}\;\lambda_1 \leq 1.22 \cdot 10^{-8}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), \mathsf{fma}\left(-0.5, \cos \phi_1, 0.5\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 21.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (* R 2.0)
  (atan2
   (sqrt
    (fma
     (cos phi1)
     (* (cos phi2) (* 0.25 (* lambda1 lambda1)))
     (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
   (sqrt
    (+
     0.5
     (* (cos phi1) (- 0.5 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (R * 2.0) * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.25 * (lambda1 * lambda1))), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))), sqrt((0.5 + (cos(phi1) * (0.5 - (0.5 + (-0.5 * cos((lambda1 - lambda2)))))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(R * 2.0) * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.25 * Float64(lambda1 * lambda1))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))), sqrt(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2))))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.25 * N[(lambda1 * lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 57.2%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\right)}} \cdot \left(2 \cdot R\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(2 \cdot R\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

   8. sub-neg N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}} \cdot \left(2 \cdot R\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   11. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

   14. lower--.f64 42.4

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)\right)}} \cdot \left(2 \cdot R\right) \]

7. Applied rewrites 42.4%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(2 \cdot R\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

   3. lower-cos.f64 33.8

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

10. Applied rewrites 33.8%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \lambda_1\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

11. Taylor expanded in lambda1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{4} \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{4} \cdot {\lambda_1}^{2}\right)}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

    2. unpow2 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

    3. lower-*.f64 21.4

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.25 \cdot \color{blue}{\left(\lambda_1 \cdot \lambda_1\right)}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

13. Applied rewrites 21.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right)}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(2 \cdot R\right) \]

14. Final simplification 21.4%

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.25 \cdot \left(\lambda_1 \cdot \lambda_1\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))))))))