Distance on a great circle

Percentage Accurate: 62.0% → 78.2%

Time: 41.9s

Alternatives: 20

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: 62.0% 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

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

FPCore

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

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi1 * 0.5))
	t_1 = sin(Float64(phi2 * -0.5))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = cos(Float64(phi2 * -0.5))
	t_4 = Float64(cos(phi2) * cos(phi1))
	t_5 = cos(Float64(phi1 * 0.5))
	return Float64(Float64(atan(sqrt(Float64(Float64(Float64(t_4 * t_2) * t_2) + (fma(t_1, t_5, Float64(t_0 * t_3)) ^ 2.0))), sqrt(Float64(1.0 - Float64(Float64(t_4 * fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5)) + (fma(t_0, t_3, Float64(t_5 * t_1)) ^ 2.0))))) * 2.0) * R)
end

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. div-inv N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 61.3

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

4. Applied rewrites 61.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. sin-sum N/A

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

   17. lift-sin.f64 N/A

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

   18. lower-fma.f64 N/A

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

6. Applied rewrites 77.3%

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

8. Applied rewrites 77.3%

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

10. Applied rewrites 77.4%

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

11. Final simplification 77.4%

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

Alternative 2: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_1 := \sin \left(\phi_2 \cdot -0.5\right)\\ t_2 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t\_3\right) \cdot t\_3\\ t_5 := \cos \left(\phi_2 \cdot -0.5\right)\\ t_6 := \left(\tan^{-1}_* \frac{\sqrt{t\_4 + {\left(\mathsf{fma}\left(t\_1, t\_2, t\_0 \cdot t\_5\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(t\_0, t\_5, t\_2 \cdot t\_1\right)\right)}^{2} + {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -8.5:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_4}}{\sqrt{1 - \mathsf{fma}\left(t\_1, \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \phi_1, t\_1\right), {\left(\cos \left(\lambda_2 \cdot 0.5\right) \cdot \sin \left(\lambda_1 \cdot 0.5\right) - \sin \left(\lambda_2 \cdot 0.5\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2} \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi1 0.5)))
        (t_1 (sin (* phi2 -0.5)))
        (t_2 (cos (* phi1 0.5)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (* (* (* (cos phi2) (cos phi1)) t_3) t_3))
        (t_5 (cos (* phi2 -0.5)))
        (t_6
         (*
          (*
           (atan2
            (sqrt (+ t_4 (pow (fma t_1 t_2 (* t_0 t_5)) 2.0)))
            (sqrt
             (-
              1.0
              (+
               (pow (fma t_0 t_5 (* t_2 t_1)) 2.0)
               (* (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0) (cos phi1))))))
           2.0)
          R)))
   (if (<= phi1 -8.5)
     t_6
     (if (<= phi1 1.25e-10)
       (*
        (*
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_4))
          (sqrt
           (-
            1.0
            (fma
             t_1
             (fma (cos (* 0.5 phi2)) phi1 t_1)
             (*
              (pow
               (-
                (* (cos (* lambda2 0.5)) (sin (* lambda1 0.5)))
                (* (sin (* lambda2 0.5)) (cos (* lambda1 0.5))))
               2.0)
              (cos phi2))))))
         2.0)
        R)
       t_6))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi1 * 0.5));
	double t_1 = sin((phi2 * -0.5));
	double t_2 = cos((phi1 * 0.5));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = ((cos(phi2) * cos(phi1)) * t_3) * t_3;
	double t_5 = cos((phi2 * -0.5));
	double t_6 = (atan2(sqrt((t_4 + pow(fma(t_1, t_2, (t_0 * t_5)), 2.0))), sqrt((1.0 - (pow(fma(t_0, t_5, (t_2 * t_1)), 2.0) + (pow(sin(((lambda1 - lambda2) * 0.5)), 2.0) * cos(phi1)))))) * 2.0) * R;
	double tmp;
	if (phi1 <= -8.5) {
		tmp = t_6;
	} else if (phi1 <= 1.25e-10) {
		tmp = (atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_4)), sqrt((1.0 - fma(t_1, fma(cos((0.5 * phi2)), phi1, t_1), (pow(((cos((lambda2 * 0.5)) * sin((lambda1 * 0.5))) - (sin((lambda2 * 0.5)) * cos((lambda1 * 0.5)))), 2.0) * cos(phi2)))))) * 2.0) * R;
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi1 * 0.5))
	t_1 = sin(Float64(phi2 * -0.5))
	t_2 = cos(Float64(phi1 * 0.5))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = Float64(Float64(Float64(cos(phi2) * cos(phi1)) * t_3) * t_3)
	t_5 = cos(Float64(phi2 * -0.5))
	t_6 = Float64(Float64(atan(sqrt(Float64(t_4 + (fma(t_1, t_2, Float64(t_0 * t_5)) ^ 2.0))), sqrt(Float64(1.0 - Float64((fma(t_0, t_5, Float64(t_2 * t_1)) ^ 2.0) + Float64((sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0) * cos(phi1)))))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -8.5)
		tmp = t_6;
	elseif (phi1 <= 1.25e-10)
		tmp = Float64(Float64(atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_4)), sqrt(Float64(1.0 - fma(t_1, fma(cos(Float64(0.5 * phi2)), phi1, t_1), Float64((Float64(Float64(cos(Float64(lambda2 * 0.5)) * sin(Float64(lambda1 * 0.5))) - Float64(sin(Float64(lambda2 * 0.5)) * cos(Float64(lambda1 * 0.5)))) ^ 2.0) * cos(phi2)))))) * 2.0) * R);
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 + N[Power[N[(t$95$1 * t$95$2 + N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[(t$95$0 * t$95$5 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -8.5], t$95$6, If[LessEqual[phi1, 1.25e-10], N[(N[(N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * phi1 + t$95$1), $MachinePrecision] + N[(N[Power[N[(N[(N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$6]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5 or 1.25000000000000008e-10 < phi1

   1. Initial program 46.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. div-inv N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

      22. div-inv N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. sin-sum N/A

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

      17. lift-sin.f64 N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 78.8%

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. sub-neg N/A

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

       4. neg-mul-1 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. lower--.f64 N/A

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

       12. lower-cos.f64 61.0

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

   11. Applied rewrites 61.0%

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

   if -8.5 < phi1 < 1.25000000000000008e-10

   1. Initial program 74.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(\phi_1 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \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)\right)}^{2} + {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\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(\phi_2 \cdot -0.5\right), \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \phi_1, \sin \left(\phi_2 \cdot -0.5\right)\right), {\left(\cos \left(\lambda_2 \cdot 0.5\right) \cdot \sin \left(\lambda_1 \cdot 0.5\right) - \sin \left(\lambda_2 \cdot 0.5\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2} \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) + {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(\phi_1 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \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)\right)}^{2} + {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_1 := \cos \left(\phi_2 \cdot -0.5\right)\\ t_2 := \sin \left(\phi_2 \cdot -0.5\right)\\ t_3 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t\_4\right) \cdot t\_4\\ t_6 := \left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1 + {\left(\mathsf{fma}\left(t\_2, t\_3, t\_0 \cdot t\_1\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(t\_0, t\_1, t\_3 \cdot t\_2\right)\right)}^{2} + t\_5\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -8.5:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_5}}{\sqrt{1 - \mathsf{fma}\left(t\_2, \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \phi_1, t\_2\right), {\left(\cos \left(\lambda_2 \cdot 0.5\right) \cdot \sin \left(\lambda_1 \cdot 0.5\right) - \sin \left(\lambda_2 \cdot 0.5\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2} \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi1 0.5)))
        (t_1 (cos (* phi2 -0.5)))
        (t_2 (sin (* phi2 -0.5)))
        (t_3 (cos (* phi1 0.5)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (* (* (* (cos phi2) (cos phi1)) t_4) t_4))
        (t_6
         (*
          (*
           (atan2
            (sqrt
             (+
              (* (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0) (cos phi1))
              (pow (fma t_2 t_3 (* t_0 t_1)) 2.0)))
            (sqrt (- 1.0 (+ (pow (fma t_0 t_1 (* t_3 t_2)) 2.0) t_5))))
           2.0)
          R)))
   (if (<= phi1 -8.5)
     t_6
     (if (<= phi1 1.25e-10)
       (*
        (*
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_5))
          (sqrt
           (-
            1.0
            (fma
             t_2
             (fma (cos (* 0.5 phi2)) phi1 t_2)
             (*
              (pow
               (-
                (* (cos (* lambda2 0.5)) (sin (* lambda1 0.5)))
                (* (sin (* lambda2 0.5)) (cos (* lambda1 0.5))))
               2.0)
              (cos phi2))))))
         2.0)
        R)
       t_6))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi1 * 0.5));
	double t_1 = cos((phi2 * -0.5));
	double t_2 = sin((phi2 * -0.5));
	double t_3 = cos((phi1 * 0.5));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = ((cos(phi2) * cos(phi1)) * t_4) * t_4;
	double t_6 = (atan2(sqrt(((pow(sin(((lambda1 - lambda2) * 0.5)), 2.0) * cos(phi1)) + pow(fma(t_2, t_3, (t_0 * t_1)), 2.0))), sqrt((1.0 - (pow(fma(t_0, t_1, (t_3 * t_2)), 2.0) + t_5)))) * 2.0) * R;
	double tmp;
	if (phi1 <= -8.5) {
		tmp = t_6;
	} else if (phi1 <= 1.25e-10) {
		tmp = (atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_5)), sqrt((1.0 - fma(t_2, fma(cos((0.5 * phi2)), phi1, t_2), (pow(((cos((lambda2 * 0.5)) * sin((lambda1 * 0.5))) - (sin((lambda2 * 0.5)) * cos((lambda1 * 0.5)))), 2.0) * cos(phi2)))))) * 2.0) * R;
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi1 * 0.5))
	t_1 = cos(Float64(phi2 * -0.5))
	t_2 = sin(Float64(phi2 * -0.5))
	t_3 = cos(Float64(phi1 * 0.5))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = Float64(Float64(Float64(cos(phi2) * cos(phi1)) * t_4) * t_4)
	t_6 = Float64(Float64(atan(sqrt(Float64(Float64((sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0) * cos(phi1)) + (fma(t_2, t_3, Float64(t_0 * t_1)) ^ 2.0))), sqrt(Float64(1.0 - Float64((fma(t_0, t_1, Float64(t_3 * t_2)) ^ 2.0) + t_5)))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -8.5)
		tmp = t_6;
	elseif (phi1 <= 1.25e-10)
		tmp = Float64(Float64(atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_5)), sqrt(Float64(1.0 - fma(t_2, fma(cos(Float64(0.5 * phi2)), phi1, t_2), Float64((Float64(Float64(cos(Float64(lambda2 * 0.5)) * sin(Float64(lambda1 * 0.5))) - Float64(sin(Float64(lambda2 * 0.5)) * cos(Float64(lambda1 * 0.5)))) ^ 2.0) * cos(phi2)))))) * 2.0) * R);
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$2 * t$95$3 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[(t$95$0 * t$95$1 + N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -8.5], t$95$6, If[LessEqual[phi1, 1.25e-10], N[(N[(N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$2 * N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * phi1 + t$95$2), $MachinePrecision] + N[(N[Power[N[(N[(N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$6]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5 or 1.25000000000000008e-10 < phi1

   1. Initial program 46.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. div-inv N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

      22. div-inv N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 48.2

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

   4. Applied rewrites 48.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. sin-sum N/A

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

      17. lift-sin.f64 N/A

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

      18. lower-fma.f64 N/A

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

   6. Applied rewrites 78.8%

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. sub-neg N/A

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

       4. neg-mul-1 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. neg-mul-1 N/A

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

       10. sub-neg N/A

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

       11. lower--.f64 N/A

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

       12. lower-cos.f64 60.5

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

   11. Applied rewrites 60.5%

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

   if -8.5 < phi1 < 1.25000000000000008e-10

   1. Initial program 74.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1 + {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(\phi_1 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \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)\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\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(\phi_2 \cdot -0.5\right), \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \phi_1, \sin \left(\phi_2 \cdot -0.5\right)\right), {\left(\cos \left(\lambda_2 \cdot 0.5\right) \cdot \sin \left(\lambda_1 \cdot 0.5\right) - \sin \left(\lambda_2 \cdot 0.5\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2} \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} \cdot \cos \phi_1 + {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(\phi_1 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \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)\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\phi_1 - \phi_2\right) \cdot 0.5\\ t_3 := \cos t\_2\\ \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_0 \cdot t\_1\right) \cdot t\_1 \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, t\_0, {\sin t\_2}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, t\_0, 0.5 - \cos \left(t\_2 \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_3, t\_3, \left(\left(-\cos \phi_1\right) \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (- phi1 phi2) 0.5))
        (t_3 (cos t_2)))
   (if (<=
        (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_0 t_1) t_1))
        0.00095)
     (*
      (atan2
       (sqrt (fma (- 0.5 (* (cos lambda1) 0.5)) t_0 (pow (sin t_2) 2.0)))
       (sqrt
        (+
         (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
         0.5)))
      (* 2.0 R))
     (*
      (atan2
       (sqrt
        (fma
         (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5))
         t_0
         (- 0.5 (* (cos (* t_2 2.0)) 0.5))))
       (sqrt
        (fma
         t_3
         t_3
         (*
          (* (- (cos phi1)) (cos phi2))
          (fma -0.5 (cos (- lambda1 lambda2)) 0.5)))))
      (* 2.0 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (phi1 - phi2) * 0.5;
	double t_3 = cos(t_2);
	double tmp;
	if ((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_0 * t_1) * t_1)) <= 0.00095) {
		tmp = atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), t_0, pow(sin(t_2), 2.0))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = atan2(sqrt(fma((0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, (0.5 - (cos((t_2 * 2.0)) * 0.5)))), sqrt(fma(t_3, t_3, ((-cos(phi1) * cos(phi2)) * fma(-0.5, cos((lambda1 - lambda2)), 0.5))))) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(phi1 - phi2) * 0.5)
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_0 * t_1) * t_1)) <= 0.00095)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), t_0, (sin(t_2) ^ 2.0))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, Float64(0.5 - Float64(cos(Float64(t_2 * 2.0)) * 0.5)))), sqrt(fma(t_3, t_3, Float64(Float64(Float64(-cos(phi1)) * cos(phi2)) * fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5))))) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.00095], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$3 * t$95$3 + N[(N[((-N[Cos[phi1], $MachinePrecision]) * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $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))))) < 9.49999999999999998e-4

   1. Initial program 65.7%

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

   4. Applied rewrites 12.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 15.8

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

   10. Applied rewrites 15.8%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift--.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. sqr-sin-a N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. lift-sin.f64 N/A

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

       18. unpow2 N/A

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

   12. Applied rewrites 45.5%

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

   if 9.49999999999999998e-4 < (+.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 59.9%

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

   4. Applied rewrites 59.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-cos-a N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \cos \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \left(\left(-\cos \phi_1\right) \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right)\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \left(\phi_1 - \phi_2\right) \cdot 0.5\\ t_4 := \cos \left(t\_3 \cdot 2\right) \cdot 0.5\\ \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_0 \cdot t\_2\right) \cdot t\_2 \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, t\_0, {\sin t\_3}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - t\_4\right)}}{\sqrt{\left(t\_4 + 0.5\right) - t\_1 \cdot t\_0}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* (- phi1 phi2) 0.5))
        (t_4 (* (cos (* t_3 2.0)) 0.5)))
   (if (<=
        (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_0 t_2) t_2))
        0.00095)
     (*
      (atan2
       (sqrt (fma (- 0.5 (* (cos lambda1) 0.5)) t_0 (pow (sin t_3) 2.0)))
       (sqrt
        (+
         (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
         0.5)))
      (* 2.0 R))
     (*
      (atan2
       (sqrt (fma t_1 t_0 (- 0.5 t_4)))
       (sqrt (- (+ t_4 0.5) (* t_1 t_0))))
      (* 2.0 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = (phi1 - phi2) * 0.5;
	double t_4 = cos((t_3 * 2.0)) * 0.5;
	double tmp;
	if ((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_0 * t_2) * t_2)) <= 0.00095) {
		tmp = atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), t_0, pow(sin(t_3), 2.0))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - t_4))), sqrt(((t_4 + 0.5) - (t_1 * t_0)))) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(Float64(phi1 - phi2) * 0.5)
	t_4 = Float64(cos(Float64(t_3 * 2.0)) * 0.5)
	tmp = 0.0
	if (Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_0 * t_2) * t_2)) <= 0.00095)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), t_0, (sin(t_3) ^ 2.0))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - t_4))), sqrt(Float64(Float64(t_4 + 0.5) - Float64(t_1 * t_0)))) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(t$95$3 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 0.00095], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Power[N[Sin[t$95$3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$4 + 0.5), $MachinePrecision] - N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $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))))) < 9.49999999999999998e-4

   1. Initial program 65.7%

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

   4. Applied rewrites 12.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 15.8

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

   10. Applied rewrites 15.8%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift--.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. sqr-sin-a N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. lift-sin.f64 N/A

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

       18. unpow2 N/A

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

   12. Applied rewrites 45.5%

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

   if 9.49999999999999998e-4 < (+.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 59.9%

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

   4. Applied rewrites 59.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(\cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5 + 0.5\right) - \left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\phi_1 - \phi_2\right) \cdot 0.5\\ \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_0 \cdot t\_1\right) \cdot t\_1 \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, t\_0, {\sin t\_2}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, t\_0, 0.5 - \cos \left(t\_2 \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_1\right) \cdot \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (- phi1 phi2) 0.5)))
   (if (<=
        (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_0 t_1) t_1))
        0.00095)
     (*
      (atan2
       (sqrt (fma (- 0.5 (* (cos lambda1) 0.5)) t_0 (pow (sin t_2) 2.0)))
       (sqrt
        (+
         (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
         0.5)))
      (* 2.0 R))
     (*
      (atan2
       (sqrt
        (fma
         (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5))
         t_0
         (- 0.5 (* (cos (* t_2 2.0)) 0.5))))
       (sqrt
        (fma
         (fma -0.5 (cos (- lambda1 lambda2)) 0.5)
         (* (- (cos phi1)) (cos phi2))
         (fma (cos (- phi1 phi2)) 0.5 0.5))))
      (* 2.0 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (phi1 - phi2) * 0.5;
	double tmp;
	if ((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_0 * t_1) * t_1)) <= 0.00095) {
		tmp = atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), t_0, pow(sin(t_2), 2.0))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = atan2(sqrt(fma((0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, (0.5 - (cos((t_2 * 2.0)) * 0.5)))), sqrt(fma(fma(-0.5, cos((lambda1 - lambda2)), 0.5), (-cos(phi1) * cos(phi2)), fma(cos((phi1 - phi2)), 0.5, 0.5)))) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(phi1 - phi2) * 0.5)
	tmp = 0.0
	if (Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_0 * t_1) * t_1)) <= 0.00095)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), t_0, (sin(t_2) ^ 2.0))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, Float64(0.5 - Float64(cos(Float64(t_2 * 2.0)) * 0.5)))), sqrt(fma(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5), Float64(Float64(-cos(phi1)) * cos(phi2)), fma(cos(Float64(phi1 - phi2)), 0.5, 0.5)))) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.00095], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[((-N[Cos[phi1], $MachinePrecision]) * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $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))))) < 9.49999999999999998e-4

   1. Initial program 65.7%

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

   4. Applied rewrites 12.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 15.8

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

   10. Applied rewrites 15.8%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift--.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. sqr-sin-a N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. lift-sin.f64 N/A

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

       18. unpow2 N/A

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

   12. Applied rewrites 45.5%

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

   if 9.49999999999999998e-4 < (+.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 59.9%

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

   4. Applied rewrites 59.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 59.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right), \left(-\cos \phi_1\right) \cdot \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5\right)\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\phi_1 - \phi_2\right) \cdot 0.5\\ \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_0 \cdot t\_1\right) \cdot t\_1 \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, t\_0, {\sin t\_2}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, t\_0, 0.5 - \cos \left(t\_2 \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot t\_0\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (- phi1 phi2) 0.5)))
   (if (<=
        (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_0 t_1) t_1))
        0.00095)
     (*
      (atan2
       (sqrt (fma (- 0.5 (* (cos lambda1) 0.5)) t_0 (pow (sin t_2) 2.0)))
       (sqrt
        (+
         (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
         0.5)))
      (* 2.0 R))
     (*
      (atan2
       (sqrt
        (fma
         (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5))
         t_0
         (- 0.5 (* (cos (* t_2 2.0)) 0.5))))
       (sqrt
        (fma
         (cos (- phi1 phi2))
         0.5
         (- 0.5 (* (fma -0.5 (cos (- lambda1 lambda2)) 0.5) t_0)))))
      (* 2.0 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (phi1 - phi2) * 0.5;
	double tmp;
	if ((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_0 * t_1) * t_1)) <= 0.00095) {
		tmp = atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), t_0, pow(sin(t_2), 2.0))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = atan2(sqrt(fma((0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, (0.5 - (cos((t_2 * 2.0)) * 0.5)))), sqrt(fma(cos((phi1 - phi2)), 0.5, (0.5 - (fma(-0.5, cos((lambda1 - lambda2)), 0.5) * t_0))))) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(phi1 - phi2) * 0.5)
	tmp = 0.0
	if (Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_0 * t_1) * t_1)) <= 0.00095)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), t_0, (sin(t_2) ^ 2.0))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, Float64(0.5 - Float64(cos(Float64(t_2 * 2.0)) * 0.5)))), sqrt(fma(cos(Float64(phi1 - phi2)), 0.5, Float64(0.5 - Float64(fma(-0.5, cos(Float64(lambda1 - lambda2)), 0.5) * t_0))))) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.00095], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(0.5 - N[(N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $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))))) < 9.49999999999999998e-4

   1. Initial program 65.7%

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

   4. Applied rewrites 12.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 15.8

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

   10. Applied rewrites 15.8%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift--.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. sqr-sin-a N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. lift-sin.f64 N/A

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

       18. unpow2 N/A

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

   12. Applied rewrites 45.5%

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

   if 9.49999999999999998e-4 < (+.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 59.9%

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

   4. Applied rewrites 59.9%

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

   6. Applied rewrites 59.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 0.00095:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), 0.5, 0.5 - \mathsf{fma}\left(-0.5, \cos \left(\lambda_1 - \lambda_2\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. div-inv N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 61.3

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

4. Applied rewrites 61.3%

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

6. Applied rewrites 61.6%

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

7. Final simplification 61.6%

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

Alternative 9: 63.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. div-inv N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 61.3

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

4. Applied rewrites 61.3%

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

6. Applied rewrites 61.4%

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

7. Final simplification 61.4%

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

Alternative 10: 62.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 = cos(phi2) * cos(phi1)
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    code = (atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + ((t_0 * t_1) * t_1))), sqrt((((cos((((phi1 - phi2) * 0.5d0) * 2.0d0)) * 0.5d0) + 0.5d0) - ((0.5d0 - (cos((((lambda1 - lambda2) * 0.5d0) * 2.0d0)) * 0.5d0)) * t_0)))) * 2.0d0) * r
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * cos(phi1);
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	tmp = (atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + ((t_0 * t_1) * t_1))), sqrt((((cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5) + 0.5) - ((0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)) * t_0)))) * 2.0) * R;
end

Wolfram

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

Derivation

1. Initial program 60.3%

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

4. Applied rewrites 60.4%

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

5. Final simplification 60.4%

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

Alternative 11: 59.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

4. Applied rewrites 56.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. lift-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. lift-/.f64 N/A

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

   18. unpow2 N/A

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

   19. lift-pow.f64 58.6

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

6. Applied rewrites 58.6%

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

7. Final simplification 58.6%

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

Alternative 12: 59.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot 0.5\\ t_1 := 0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_4 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(t\_0 \cdot 2\right) \cdot 0.5, t\_2, t\_3\right)}}{\sqrt{t\_1 \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_2 \leq -1.16 \cdot 10^{-5}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin t\_0}^{2}, t\_2, t\_3\right)}}{\sqrt{t\_1 \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) 0.5))
        (t_1 (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_4
         (*
          (atan2
           (sqrt (fma (- 0.5 (* (cos (* t_0 2.0)) 0.5)) t_2 t_3))
           (sqrt (+ (* t_1 (cos phi2)) 0.5)))
          (* 2.0 R))))
   (if (<= phi2 -1.16e-5)
     t_4
     (if (<= phi2 8.5e+16)
       (*
        (atan2
         (sqrt (fma (pow (sin t_0) 2.0) t_2 t_3))
         (sqrt (+ (* t_1 (cos phi1)) 0.5)))
        (* 2.0 R))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * 0.5;
	double t_1 = 0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_4 = atan2(sqrt(fma((0.5 - (cos((t_0 * 2.0)) * 0.5)), t_2, t_3)), sqrt(((t_1 * cos(phi2)) + 0.5))) * (2.0 * R);
	double tmp;
	if (phi2 <= -1.16e-5) {
		tmp = t_4;
	} else if (phi2 <= 8.5e+16) {
		tmp = atan2(sqrt(fma(pow(sin(t_0), 2.0), t_2, t_3)), sqrt(((t_1 * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * 0.5)
	t_1 = Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_4 = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(t_0 * 2.0)) * 0.5)), t_2, t_3)), sqrt(Float64(Float64(t_1 * cos(phi2)) + 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi2 <= -1.16e-5)
		tmp = t_4;
	elseif (phi2 <= 8.5e+16)
		tmp = Float64(atan(sqrt(fma((sin(t_0) ^ 2.0), t_2, t_3)), sqrt(Float64(Float64(t_1 * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.16e-5], t$95$4, If[LessEqual[phi2, 8.5e+16], N[(N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.1600000000000001e-5 or 8.5e16 < phi2

   1. Initial program 47.4%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 47.7%

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

   if -1.1600000000000001e-5 < phi2 < 8.5e16

   1. Initial program 73.7%

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

   4. Applied rewrites 65.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 65.2

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

   7. Applied rewrites 65.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqr-sin-a N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lower-pow.f64 69.3

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

   9. Applied rewrites 69.3%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.16 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := 0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_3 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{t\_0 \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -8.5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 0.0014:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{t\_0 \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))))
        (t_1 (* (cos phi2) (cos phi1)))
        (t_2 (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5)))
        (t_3
         (*
          (atan2
           (sqrt (fma t_2 t_1 (- 0.5 (* (cos phi1) 0.5))))
           (sqrt (+ (* t_0 (cos phi1)) 0.5)))
          (* 2.0 R))))
   (if (<= phi1 -8.5)
     t_3
     (if (<= phi1 0.0014)
       (*
        (atan2
         (sqrt
          (fma t_2 t_1 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
         (sqrt (+ (* t_0 (cos phi2)) 0.5)))
        (* 2.0 R))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5));
	double t_1 = cos(phi2) * cos(phi1);
	double t_2 = 0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5);
	double t_3 = atan2(sqrt(fma(t_2, t_1, (0.5 - (cos(phi1) * 0.5)))), sqrt(((t_0 * cos(phi1)) + 0.5))) * (2.0 * R);
	double tmp;
	if (phi1 <= -8.5) {
		tmp = t_3;
	} else if (phi1 <= 0.0014) {
		tmp = atan2(sqrt(fma(t_2, t_1, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(((t_0 * cos(phi2)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)))
	t_1 = Float64(cos(phi2) * cos(phi1))
	t_2 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5))
	t_3 = Float64(atan(sqrt(fma(t_2, t_1, Float64(0.5 - Float64(cos(phi1) * 0.5)))), sqrt(Float64(Float64(t_0 * cos(phi1)) + 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -8.5)
		tmp = t_3;
	elseif (phi1 <= 0.0014)
		tmp = Float64(atan(sqrt(fma(t_2, t_1, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(Float64(Float64(t_0 * cos(phi2)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$1 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.5], t$95$3, If[LessEqual[phi1, 0.0014], N[(N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$1 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5 or 0.00139999999999999999 < phi1

   1. Initial program 46.2%

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 46.6

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-cos.f64 47.3

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

   10. Applied rewrites 47.3%

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

   if -8.5 < phi1 < 0.00139999999999999999

   1. Initial program 74.9%

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. cos-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 66.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0014:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_2 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_2 \leq -0.00172:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1
         (*
          (atan2
           (sqrt
            (fma
             (- 0.5 (* (cos lambda1) 0.5))
             t_0
             (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
           (sqrt
            (-
             0.5
             (*
              (- (- 0.5 (* (cos (- lambda1 lambda2)) 0.5)) 0.5)
              (cos phi2)))))
          (* 2.0 R))))
   (if (<= phi2 -0.00172)
     t_1
     (if (<= phi2 7.2e+19)
       (*
        (atan2
         (sqrt
          (fma
           (- 0.5 (* (cos (* (* (- lambda1 lambda2) 0.5) 2.0)) 0.5))
           t_0
           (- 0.5 (* (cos phi1) 0.5))))
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
           0.5)))
        (* 2.0 R))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt((0.5 - (((0.5 - (cos((lambda1 - lambda2)) * 0.5)) - 0.5) * cos(phi2))))) * (2.0 * R);
	double tmp;
	if (phi2 <= -0.00172) {
		tmp = t_1;
	} else if (phi2 <= 7.2e+19) {
		tmp = atan2(sqrt(fma((0.5 - (cos((((lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, (0.5 - (cos(phi1) * 0.5)))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(Float64(0.5 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)) - 0.5) * cos(phi2))))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi2 <= -0.00172)
		tmp = t_1;
	elseif (phi2 <= 7.2e+19)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(Float64(Float64(lambda1 - lambda2) * 0.5) * 2.0)) * 0.5)), t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[(N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00172], t$95$1, If[LessEqual[phi2, 7.2e+19], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.00171999999999999996 or 7.2e19 < phi2

   1. Initial program 47.6%

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 20.2

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

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 20.1

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

   10. Applied rewrites 20.1%

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

   11. Taylor expanded in phi1 around 0

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. cos-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-out-- N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower--.f64 N/A

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

       10. *-commutative N/A

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

       11. sub-neg N/A

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

       12. neg-mul-1 N/A

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

       13. cos-neg N/A

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

       14. neg-mul-1 N/A

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

       15. +-commutative N/A

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

       16. distribute-neg-in N/A

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

       17. remove-double-neg N/A

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

       18. sub-neg N/A

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

   13. Applied rewrites 39.0%

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

   if -0.00171999999999999996 < phi2 < 7.2e19

   1. Initial program 72.9%

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 64.3

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in phi2 around 0

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

      1. lower-cos.f64 64.4

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

   10. Applied rewrites 64.4%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00172:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_2 \cdot 0.5, t\_0, t\_1\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{+37}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, t\_0, t\_1\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5)))
        (t_2
         (*
          (atan2
           (sqrt (fma (- 0.5 (* (cos lambda2) 0.5)) t_0 t_1))
           (sqrt
            (+
             (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
             0.5)))
          (* 2.0 R))))
   (if (<= lambda2 -1.9e+37)
     t_2
     (if (<= lambda2 2.6e-25)
       (*
        (atan2
         (sqrt (fma (- 0.5 (* (cos lambda1) 0.5)) t_0 t_1))
         (sqrt
          (-
           0.5
           (* (- (- 0.5 (* (cos (- lambda1 lambda2)) 0.5)) 0.5) (cos phi2)))))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5);
	double t_2 = atan2(sqrt(fma((0.5 - (cos(lambda2) * 0.5)), t_0, t_1)), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	double tmp;
	if (lambda2 <= -1.9e+37) {
		tmp = t_2;
	} else if (lambda2 <= 2.6e-25) {
		tmp = atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), t_0, t_1)), sqrt((0.5 - (((0.5 - (cos((lambda1 - lambda2)) * 0.5)) - 0.5) * cos(phi2))))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5))
	t_2 = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda2) * 0.5)), t_0, t_1)), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (lambda2 <= -1.9e+37)
		tmp = t_2;
	elseif (lambda2 <= 2.6e-25)
		tmp = Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), t_0, t_1)), sqrt(Float64(0.5 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)) - 0.5) * cos(phi2))))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1.9e+37], t$95$2, If[LessEqual[lambda2, 2.6e-25], N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[(N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.89999999999999995e37 or 2.6e-25 < lambda2

   1. Initial program 50.6%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 41.0

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in lambda1 around 0

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

      1. cos-neg N/A

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

      2. lower-cos.f64 39.7

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

   10. Applied rewrites 39.7%

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

   if -1.89999999999999995e37 < lambda2 < 2.6e-25

   1. Initial program 72.1%

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 44.1

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

   7. Applied rewrites 44.1%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 44.1

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

   10. Applied rewrites 44.1%

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

   11. Taylor expanded in phi1 around 0

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. cos-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-out-- N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower--.f64 N/A

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

       10. *-commutative N/A

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

       11. sub-neg N/A

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

       12. neg-mul-1 N/A

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

       13. cos-neg N/A

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

       14. neg-mul-1 N/A

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

       15. +-commutative N/A

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

       16. distribute-neg-in N/A

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

       17. remove-double-neg N/A

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

       18. sub-neg N/A

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

   13. Applied rewrites 51.0%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{+37}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_2 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_2 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := 0.5 - \cos \lambda_1 \cdot 0.5\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -0.00042:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 0.0011:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_0, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (- 0.5 (* (cos lambda1) 0.5)))
        (t_2
         (*
          (atan2
           (sqrt (fma t_1 t_0 (- 0.5 (* (cos phi1) 0.5))))
           (sqrt
            (+
             (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
             0.5)))
          (* 2.0 R))))
   (if (<= phi1 -0.00042)
     t_2
     (if (<= phi1 0.0011)
       (*
        (atan2
         (sqrt
          (fma t_1 t_0 (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
         (sqrt
          (-
           0.5
           (* (- (- 0.5 (* (cos (- lambda1 lambda2)) 0.5)) 0.5) (cos phi2)))))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = 0.5 - (cos(lambda1) * 0.5);
	double t_2 = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos(phi1) * 0.5)))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
	double tmp;
	if (phi1 <= -0.00042) {
		tmp = t_2;
	} else if (phi1 <= 0.0011) {
		tmp = atan2(sqrt(fma(t_1, t_0, (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt((0.5 - (((0.5 - (cos((lambda1 - lambda2)) * 0.5)) - 0.5) * cos(phi2))))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(0.5 - Float64(cos(lambda1) * 0.5))
	t_2 = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -0.00042)
		tmp = t_2;
	elseif (phi1 <= 0.0011)
		tmp = Float64(atan(sqrt(fma(t_1, t_0, Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(Float64(0.5 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)) - 0.5) * cos(phi2))))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.00042], t$95$2, If[LessEqual[phi1, 0.0011], N[(N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[(N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.2000000000000002e-4 or 0.00110000000000000007 < phi1

   1. Initial program 46.0%

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 46.4

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

   7. Applied rewrites 46.4%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 34.2

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

   10. Applied rewrites 34.2%

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

   11. Taylor expanded in phi2 around 0

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

       1. lower-cos.f64 34.7

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

   13. Applied rewrites 34.7%

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

   if -4.2000000000000002e-4 < phi1 < 0.00110000000000000007

   1. Initial program 75.3%

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 38.2

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in lambda2 around 0

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

      1. lower-cos.f64 28.4

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

   10. Applied rewrites 28.4%

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

   11. Taylor expanded in phi1 around 0

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. cos-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-out-- N/A

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

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

       8. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

       9. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

       10. *-commutative N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       11. sub-neg N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       12. neg-mul-1 N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       13. cos-neg N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       14. neg-mul-1 N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       15. +-commutative N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       16. distribute-neg-in N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       17. remove-double-neg N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\lambda_2} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       18. sub-neg N/A

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 47.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_2 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00042:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0011:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{0.5 - \left(\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) - 0.5\right) \cdot \cos \phi_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}\\ t_2 := 0.5 - \cos \lambda_1 \cdot 0.5\\ t_3 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_0, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{t\_1} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_0, 0.5 - \cos \phi_2 \cdot 0.5\right)}}{t\_1} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1
         (sqrt
          (+
           (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1))
           0.5)))
        (t_2 (- 0.5 (* (cos lambda1) 0.5)))
        (t_3
         (*
          (atan2 (sqrt (fma t_2 t_0 (- 0.5 (* (cos phi1) 0.5)))) t_1)
          (* 2.0 R))))
   (if (<= phi1 -5.7e-5)
     t_3
     (if (<= phi1 2e-7)
       (*
        (atan2 (sqrt (fma t_2 t_0 (- 0.5 (* (cos phi2) 0.5)))) t_1)
        (* 2.0 R))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5));
	double t_2 = 0.5 - (cos(lambda1) * 0.5);
	double t_3 = atan2(sqrt(fma(t_2, t_0, (0.5 - (cos(phi1) * 0.5)))), t_1) * (2.0 * R);
	double tmp;
	if (phi1 <= -5.7e-5) {
		tmp = t_3;
	} else if (phi1 <= 2e-7) {
		tmp = atan2(sqrt(fma(t_2, t_0, (0.5 - (cos(phi2) * 0.5)))), t_1) * (2.0 * R);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))
	t_2 = Float64(0.5 - Float64(cos(lambda1) * 0.5))
	t_3 = Float64(atan(sqrt(fma(t_2, t_0, Float64(0.5 - Float64(cos(phi1) * 0.5)))), t_1) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -5.7e-5)
		tmp = t_3;
	elseif (phi1 <= 2e-7)
		tmp = Float64(atan(sqrt(fma(t_2, t_0, Float64(0.5 - Float64(cos(phi2) * 0.5)))), t_1) * Float64(2.0 * R));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$0 + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -5.7e-5], t$95$3, If[LessEqual[phi1, 2e-7], N[(N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$0 + N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.7000000000000003e-5 or 1.9999999999999999e-7 < phi1

   1. Initial program 46.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      18. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      20. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      21. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      22. lower--.f64 46.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. lower-cos.f64 34.2

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 34.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. lower-cos.f64 34.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 34.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   if -5.7000000000000003e-5 < phi1 < 1.9999999999999999e-7

   1. Initial program 75.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      4. distribute-lft-out-- N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

      8. lower--.f64 N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      11. sub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      12. +-commutative N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      13. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      14. associate-+l- N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      15. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      16. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      17. neg-sub0 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      18. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      20. mul-1-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      21. unsub-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

      22. lower--.f64 38.2

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. Applied rewrites 38.2%

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   9. Step-by-step derivation

      1. lower-cos.f64 28.4

         \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied rewrites 28.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
   12. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

       2. lower-cos.f64 28.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\cos \phi_2}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. Applied rewrites 28.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\cos \phi_2}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_2 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, 0.5\right)}} \cdot \left(2 \cdot R\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (atan2
   (sqrt
    (fma
     (- 0.5 (* (cos lambda1) 0.5))
     (* (cos phi2) (cos phi1))
     (- 0.5 (* (cos (* (* (- phi1 phi2) 0.5) 2.0)) 0.5))))
   (sqrt (fma (* (cos (- lambda2 lambda1)) 0.5) (cos phi1) 0.5)))
  (* 2.0 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), (cos(phi2) * cos(phi1)), (0.5 - (cos((((phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma((cos((lambda2 - lambda1)) * 0.5), cos(phi1), 0.5))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(cos(Float64(Float64(Float64(phi1 - phi2) * 0.5) * 2.0)) * 0.5)))), sqrt(fma(Float64(cos(Float64(lambda2 - lambda1)) * 0.5), cos(phi1), 0.5))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

   9. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. sub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   12. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. neg-sub0 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   14. associate-+l- N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   15. unsub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   16. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   17. neg-sub0 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   18. cos-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   20. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   21. unsub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   22. lower--.f64 42.4

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

7. Applied rewrites 42.4%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
9. Step-by-step derivation

   1. lower-cos.f64 31.4

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

10. Applied rewrites 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
11. Step-by-step derivation

12. Applied rewrites 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, 0.5\right)}}} \cdot \left(R \cdot 2\right) \]

13. Final simplification 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right) \cdot 2\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, 0.5\right)}} \cdot \left(2 \cdot R\right) \]
14. Add Preprocessing

Alternative 19: 32.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (atan2
   (sqrt
    (fma
     (- 0.5 (* (cos lambda1) 0.5))
     (* (cos phi2) (cos phi1))
     (fma (cos (- phi1 phi2)) -0.5 0.5)))
   (sqrt
    (+ (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1)) 0.5)))
  (* 2.0 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), (cos(phi2) * cos(phi1)), fma(cos((phi1 - phi2)), -0.5, 0.5))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), Float64(cos(phi2) * cos(phi1)), fma(cos(Float64(phi1 - phi2)), -0.5, 0.5))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

   9. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. sub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   12. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. neg-sub0 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   14. associate-+l- N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   15. unsub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   16. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   17. neg-sub0 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   18. cos-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   20. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   21. unsub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   22. lower--.f64 42.4

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

7. Applied rewrites 42.4%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
9. Step-by-step derivation

   1. lower-cos.f64 31.4

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

10. Applied rewrites 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]
11. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    2. sub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right)}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    3. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right) + \frac{1}{2}}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    4. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)\right) + \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    5. *-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \left(\mathsf{neg}\left(\color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right) \cdot \frac{1}{2}}\right)\right) + \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    6. distribute-rgt-neg-in N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    7. metadata-eval N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right) \cdot \color{blue}{\frac{-1}{2}} + \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    8. lower-fma.f64 31.4

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right), -0.5, 0.5\right)}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    10. lift-*.f64 N/A

        \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    11. associate-*r* N/A

        \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 - \phi_2\right)\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    12. metadata-eval N/A

        \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\color{blue}{1} \cdot \left(\phi_1 - \phi_2\right)\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

    13. *-lft-identity 31.4

        \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_1 - \phi_2\right)}, -0.5, 0.5\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

12. Applied rewrites 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \color{blue}{\mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), -0.5, 0.5\right)}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

13. Final simplification 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right) \]
14. Add Preprocessing

Alternative 20: 29.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (atan2
   (sqrt
    (fma
     (- 0.5 (* (cos lambda1) 0.5))
     (* (cos phi2) (cos phi1))
     (- 0.5 (* (cos phi1) 0.5))))
   (sqrt
    (+ (* (- 0.5 (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))) (cos phi1)) 0.5)))
  (* 2.0 R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sqrt(fma((0.5 - (cos(lambda1) * 0.5)), (cos(phi2) * cos(phi1)), (0.5 - (cos(phi1) * 0.5)))), sqrt((((0.5 - (0.5 - (cos((lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(lambda1) * 0.5)), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(cos(phi1) * 0.5)))), sqrt(Float64(Float64(Float64(0.5 - Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5))) * cos(phi1)) + 0.5))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[lambda1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[phi1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing

4. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot \left(R \cdot 2\right)} \]

5. Taylor expanded in phi2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \phi_1\right) - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}} \cdot \left(R \cdot 2\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_1 - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \frac{1}{2}} - \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   4. distribute-lft-out-- N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\cos \phi_1} \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \cdot \left(R \cdot 2\right) \]

   7. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot \left(R \cdot 2\right) \]

   8. lower--.f64 N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}} \cdot \left(R \cdot 2\right) \]

   9. *-commutative N/A

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   11. sub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   12. +-commutative N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   13. neg-sub0 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\color{blue}{\left(0 - \lambda_2\right)} + \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   14. associate-+l- N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(0 - \left(\lambda_2 - \lambda_1\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   15. unsub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   16. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(0 - \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   17. neg-sub0 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   18. cos-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   20. mul-1-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   21. unsub-neg N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]

   22. lower--.f64 42.4

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

7. Applied rewrites 42.4%

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\color{blue}{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}}} \cdot \left(R \cdot 2\right) \]

8. Taylor expanded in lambda2 around 0

   \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
9. Step-by-step derivation

   1. lower-cos.f64 31.4

      \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

10. Applied rewrites 31.4%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \color{blue}{\cos \lambda_1}, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in phi2 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}\right)\right)}} \cdot \left(R \cdot 2\right) \]
12. Step-by-step derivation

    1. lower-cos.f64 27.7

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

13. Applied rewrites 27.7%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\cos \phi_1}\right)}}{\sqrt{0.5 + \cos \phi_1 \cdot \left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right)}} \cdot \left(R \cdot 2\right) \]

14. Final simplification 27.7%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \lambda_1 \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \phi_1 \cdot 0.5\right)}}{\sqrt{\left(0.5 - \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right)\right) \cdot \cos \phi_1 + 0.5}} \cdot \left(2 \cdot R\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))))))))